AN ABSTRACT OF THE THESIS OF
Rodrigo H. Nunez for the degree of Master of Science in Oceanography presented on August 3, 1990.
Title: Prediction of Tidal Propagation and Circulation in Chilean InlandSeas Using a Frequency Domain Model
Redacted for privacy Abstract approved: / Steve Neshyb(
The calculated flow field and tide wave propagation in the Chilean Inland Seas is modeled with a non-linear tidal embayment method (TEA-NL, Westerink et al, 1988) and assimilation of tide records of the Chilean HydrographicInstitute. Model isvalidated for two testcases with assimilated error-freeboundary conditions.
An Inverse Tidal Method (I. T. M.) developed as a time-domain method by Bennett and McIntosh in 1982, is used to calculate the forcing functions, but in the frequency-domain instead.
The model is first run using all available tide data. A second run with a subset of five stations shows how the choice and location of stations affect the prediction of tide wave propagation and flow fields.The boundary conditions calculated by the I. T. M. for the cases of seven and five stations reproduced the known data at the stations with like error magnitudes.However, for the seven stations case, the tide wave propagation along the seaward side of Chiloe Island (Pacific Ocean) propagated from South to North (the known direction of propagation is from North to South).After disregarding two stations, the model gave tide waves along the seaward side of ChloeIsland propagating from North to South. The model is clearly sensitive to perturbations introducedby hydraulics factors other than pure tide wave propagation.
Charts of model simulation of flow fields are shown at 3, 6, 9and 12 hours of the M2 tidal cycle. Charts of phases and amplitudes are alsoshown.
Conclusions from the study are that the I. T. M. is a good dataassimilation technique to calculate the forcing functions of a numericalmodel, even when known data is scarce.Further, the I. T. M. is a valuable aid in chosing locations for new and useful data to improve the simulation.
Next steps required to improve the understanding of theChilean Inland Seas are a good field survey (based on theresults of the I. T. M.), the development of a transport model against which measured flow data canbe compared, the inclusion of river discharge in the boundary conditions, the preparationof trajectory charts todemonstrate particlediffusion and advection,the expansiontothree- dimensional models, etc. Prediction of Tidal Propagation andCirculation in Chilean Inland Seas Using a FrequencyDomain Model by Rodrigo H. Nunez
A THESIS submitted to Oregon State University
in partial fulfillment of the requirements for the degree of Master of Science
Completed August 3, 1990 Commencement June 1991 APPROVED: Redacted for privacy in cha4 of major If Redacted for privacy
Head of Co1l'ee of
Redacted for privacy
Dean of Graduate Scf
Date thesis is presented August 3, 1990
Typed by Donna Kent forRodrigo H. Nunez Acknowledgements
There are several people without whom this work wouldhave been impos- sible.First, I would like to thank Dr. S. Nesbyba for his supportand counsel as my major professor, which wasessential to my training in physical oceanography.
I would like to thank Dr. A. Baptista of Oregon GraduateInstitute of Science and Technology for all his suggestions, guidance and patienceduring the last two years.
Without the aid of Dr. .J. Davis and Dr. R. Miller the InverseMethod could not have been an integral part of this thesis. Theirwillingness to consult with me was great, and in turn it isgreatly appreciated.Dr. A. Bennett also made valuable suggestions in this section at the beginning of thestudy, and his enthusiasm towards my work was very encouraging.
Special thanks to all my friends who helped make mytime in Oregon unforgettable and who gave me their support during this study.Thank you, Ricardo and Anita, Yerko and Cuti, Marcio and Maria, and Jorge.
Thanks to Randy and Vassilis who were very special friends to me.They were always ready to help me forget mywork and to force me to have a good time.
The typing was done by Donna Kent, who I thank forher patience and efforts made to understand my handwriting and especially myEnglish.
Funding was provided by the Chilean Navy through the NavalHydrographic Institute. P
TABLE OF CONTENTS
CHAPTER I: INTRODUCTION CHAPTER II: DESCRIPTION OF METHODS 6 2.1 Review of the Circulation Model 6 2.2 Inverse Tidal Method 12 2.2.1 Application of the Inverse Tidal Method to the Solution of a Practical Problem 12 2.2.2 Mathematical Formulation of the Inverse Tidal Method 14
CHAPTER III: VALIDATION OF METHODS 22 3.1Rectangular Embayment 24 3.2 North Sea 30 CHAPTER IV: CHILEAN INLAND SEAWATER SYSTEM 62 4.1Analysis of Tidal Data 62 4.2Definition of Boundary Conditions 71 4.3Analysis of Model Results 75 CHAPTERV: CONCLUSIONS 102 5.1Conclusions 102 5.2Outline of the Future Field Survey 103 5.3Outline of Next Steps in Modeling 105 BIBLIOGRAPHY 107 APPENDIX A 109 Governing Equations 109 APPENDIX B 118 Inverse Tidal Method (I. T. M.) 118 LIST OF FIGURES
Figure Page
1. Chilean Inland Seawater System of Gulf of Ancud. Geographical Location la
2. Iterative non-linear scheme 10
3. Schematic diagram for the Inverse Tidal Method 15
4. Rectangular embayment, finite element grid. 25
5. Known and calculated flow fields using linear and trigonometricbasis functions for the rectangular embayment at instant=11 hours. 29
6. North Sea, finite element grid 31
7. North Sea, location of reference stations 32
8. Elevations. Known data vs. calculated data (Expel45) at station St. Malo for a period of 13 hours 35
9. Elevations. Known data vs. calculated data (Expel4S) at station Cherbourg for a period of 13 hours 35
10.Elevations. Known data vs. calculated data (Expel45) at station Christchurch for a period of 13 hours 36
11.Elevations. Known data vs. calculated data (Expel45) at station Dieppe for a period of 13 hours 36
12.Elevations. Known data vs. calculated data (Expel4S) at station Boulogne for a period of 13 hours 37
13.Elevations. Known data vs. calculated data (Expel45) at station Dover for a period of 13 hours 37
14.Elevations. Known data vs. calculated data (Expel45) at station Calais for a period of 13 hours 38 15.Elevations. Known data vs. calculated data (Expel45) atstation Walton for a period of 13 hours 38
16.Elevations. Known data vs. calculated data (Expel45) atstation Zeebrugge for a period of 13 hours 39
17.Elevations. Known data vs. calculated data (Expel45) at station Lowestoft for a period of 13 hours 39
18.Elevations. Known data vs. calculated data (Expel45) at station Hoek Van Holland for a period of 13 hours 40
19.Elevations. Known data vs. calculated data (Expe28O) at station St. Mao for a period of 13 hours 40
20.Elevations. Known data vs. calculated data (Expe28O) atstation Cherbourg for a period of 13 hours 41
21.Elevations. Known data vs. calculated data (Expe28O) at station Christchurch for a period of 13 hours 41
22.Elevations. Known data vs. calculated data (Expe28O) at station Dieppe for a period of 13 hours 42
23.Elevations. Known data vs. calculated data (Expe28O) at station Boulogne for a period of 13 hours 42
24.Elevations. Known data vs. calculated data (Expe28O) at station Dover for a period of 13 hours 43
25.Elevations. Known data vs. calculated data (Expe28O) at station Calais for a period of 13 hours 43
26.Elevations. Known data vs. calculated data (Expe28O) at station Walton for a period of 13 hours 44
27.Elevations. Known data vs. calculated data (Expe28O) atstation Zeebrugge for a period of 13 hours 44 28.Elevations. Known data vs. calculated data (Expe28O) atstation Lowestoft for a period of 13 hours 45
29.Elevations. Known data vs. calculated data (Expe28O) atstation Hoek Van Holland for a period of 13 hours 45
30.Elevations. Known data vs. calculated data (Expe294) atstation St. Malo for a period of 13 hours 46 31. Elevations. Known data vs. calculated data (Expe294) atstation Cherbourg for a period of 13 hours 46
32.Elevations. Known data vs. calculated data (Expe294) atstation Christchurch for a period of 13 hours 47
33.Elevations. Known data vs. calculated data (Expe294) at station Dieppe for a period of 13 hours 47
34.Elevations. Known data vs. calculated data (Expe294) atstation Boulogne for a period of 13 hours 48
35.Elevations. Known data vs. calculated data (Expe294) atstation Dover for a period of 13 hours 48
36.Elevations. Known data vs. calculated data (Expe294) atstation Calais for a period of 13 hours 49
37.Elevations. Known data vs. calculated data (Expe294) atstation Walton for a period of 13 hours 49
38.Elevations. Known data vs. calculated data (Expe294) atstation Zeebrugge for a period of 13 hours 50
39.Elevations. Known data vs. calculated data (Expe294) atstation Lowestoft for a period of 13 hours 50
40.Elevations. Known data vs. calculated data (Expe294) atstation Hoek Van Holland for a period of 13 hours 51 41. Known flow field (Expel66) and calculated flow fields (Expel4S-Expe28O-Expe294) for the North Sea at instant = 3 hours. 53 42. Known flow field (Expel66) and calculated flow fields (Expel45-Expe28O-Expe294) for the North Sea at instant = 6 hours. 54 43. Known flow field (Expel66) and calculated flow fields (Expel45-Expe28O-Expe294) for the North Sea at instant = 9 hours. 55 44. Known flow field (Expel66) and calculated flowfields (Expel4S-Expe28O-Expe294) for the North Sea at instant = 12 hours. 56
45.Isolines of amplitudes for Expel66 (known data) and Expel45-Expe28O-Expe294 (calculated data) for North Sea. 57
46.Isolines of phases for Expel66 (known data) and Expel45-Expe28O-Expe294 (calculated data) for North Sea. 58
47.Residual flow fields for the North Sea. Difference between known flow field (Expel66) and three Experiments
(Expel45-Expe28O-Expe294) 60 48. Chilean Inland Seawater System of Gulf of Ancud. Location of reference stations 63
49.Definition sketch for the boundaries in the Chilean Inland Seawater System of Gulf of Ancud 72
50.Schematic drawing of plane used to calculate the values ofamplitude and phase along the western boundary for the basis functions 74
51.Elevations. Known data vs. calculated data at station Tautil for a period of 13 hours using seven stations 76
52.Elevations. Known data vs. calculated data at station PuertoMontt for a period of 13 hours using seven stations 76
53.Elevations. Known data vs. calculated data at station Chacao for a period of 13 hours using seven stations 77 54.Elevations. Known data vs. calculated data at station Chumilden for a period of 13 hours using seven stations 77
55.Elevations. Known data vs. calculated data at station Carelmapu for a period of 13 hours using seven stations 78 56. Elevations. Known data vs. calculated data at station Faro Corona for a period of 13 hours using seven stations 78
57.Elevations. Known data vs. calculated data at station Castro for a period of 13 hours using seven stations 79
58.Elevations. Known data vs. calculated data at station Tautil for a period of 13 hours using five stations 79
59.Elevations. Known data vs. calculated data at station Puerto Montt for a period of 13 hours using five stations 80
60.Elevations. Known data vs. calculated data at station Chacao for a period of 13 hours using five stations 80
61.Elevations. Known data vs. calculated data at station Chumilden for a period of 13 hours using five stations 81
62.Elevations. Known data vs. calculated data at station Carelmapu for a period of 13 hours using five stations 81
63.Elevations. Known data vs. calculated data at station Faro Corona for a period of 13 hours using five stations 82 64. Elevations. Known data vs. calculated data at station Castro for a period of 13 hours using five stations 82 65. RMS-Error at the seven reference stations for a period of 13 hours using seven stations 83 66. RMS-Error at the seven reference stations for a period of 13 hours using five stations 84 67.Calculated flow field for the Chilean Inland Seawater Systemof Gulf of Ancud for frequencies Msf, 01, K1, M2, S2, M4, and MS4 at instant = 6 hours using seven stations 88
68.Calculated flow field for the Chilean Inland Seawater System of Gulf of Ancud for frequencies Msf, 01, K1, M2, S2, M4, and MS4 at instant = 3 hours using five stations 89
69.Calculated flow field for the Chilean Inland Seawater System of Gulf of Ancud for frequencies Msf, 01, K1, M2, S2, M4, and MS4 at instant = 6 hours using five stations 90
70.Calculated flow field for the Chilean Inland Seawater System of Gulf of Ancud for frequencies Msf, 01, K1, M2, S2, M4, and MS4 at instant = 9 hours using five stations 91
71.Calculated flow field for the Chilean Inland Seawater System of Gulf of Ancud for frequencies Msf, 01, K1, M2, S2, M4, and MS4 at instant 12 hours using five stations 92
72.Isolines of calculated amplitudes for frequency M2 in the Chilean Inland Seawater System of Gulf of Ancud using seven stations 94
73.Isolines of calculated phases for frequency M2 in the Chilean Inland Seawater System of Gulf of Ancud using seven stations 95
74.Isolines of calculated amplitudes for frequency M2 in the Chilean Inland Seawater System of Gulf of Ancud using fivestations 96
75.Isolines of calculated phases for frequency M2 in the Chilean Inland Seawater System of Gulf of Ancud using fivestations 97 (Gulf of Ancud and Gulf of Corcovado)
76.Isolines of calculated phases for frequency M2 in the Chilean Inland Seawater System of Gulf of Ancud using fivestations 98 (seaward shore of Chiloe Island) 77.Isolines of calculated phases for frequency 01 in the Chilean Inland Seawater System of Gulf of Ancud using fivestations 99
78.Isolines of calculated phases for frequency K1 in the Chilean Inland Seawater System of Gulf of Ancud using five stations 100
79.Pathlines of particles obtained from the flow field calculated using five stations, superposed with an hypothetical river discharge at Reloncavi Estuary 106
80.Definition sketch for typical elevation prescribed(r,7) and flux prescribe (FQ) boundaries. (From Westerink, 1984) 116 81. Flow chart for the Inverse Tidal Method. 127 LIST OF TABLES
1. Amplitudes and phases at grid point 30 for linear basis functions 26
2. Values of weights that multiply the linear basis functions to correct the boundary conditions 26
3. Amplitudes and phases at grid point 30 for a combination of linear and trigonometric basis functions 28
4. Values of weights that multiply the linear and trigonometric basis functions to correct the boundary conditions. 28
5. RMS-Error at the eleven reference stations in the North Sea for frequency M2 for a period of 13 hours. 52
6. Known amplitudes and phases at the reference stations in the Chilean Inland Seawater System of Gulf of Ancud 64
7. Amplitude of leading astronomic constituents relative to M2 for Faro Corona and Puerto Montt 66
8. Maximum and minimum known amplitudes and tidal amplification factors in the Chilean Inland Seawater System of Gulf of Ancud 68
9. RMS-Error at the seven reference stations in the Chilean Inland Seawater System of Gulf of Ancud for frequencies Msf, 01, K1, M2, S2, M4, MS4 for a period of 13 hours using seven stations and five stations in the I.T.M. 85 PREDICTION OF TIDAL PROPAGATION AND CIRCULATION IN CHILEAN INLAND SEAS USING A FREQUENCY DOMAIN MODEL
CHAPTER I
INTRODUCTION
The Chilean Inland Seawater System of Gulf of Ancudis located on the Pacific Coast of South America (Figure 1).It includes the Gulf of Ancud, Corcovado Gulf, Reloncavi Sound, Chacao Canal, the Chiloe Island and other, smaller islands.
For many years this region was inhabited only by native tribes who made their living by fishing.In the second half of the 19th century the Chilean government started a new settlement in the northwestern part of Reloncavi Sound; subsequently, this settlement became the base for the colonization of the southern territories.
Because of the lateness of itscolonization and the difficulties of accessibility, oceanographic data available for the area are scarce and are concentrated near the northern bound of Gulf of Ancud.As a consequence, patterns of circulation and wave propagation are poorly known. There is, however, an immediate need for understanding such patterns.Indeed, a large marine-aquaculture industry has in recent years developed in the fjords and channels of Reloncavi Sound and Gulf of Ancud. This industry, and especially the salmon aquacultures, need to define strategies to prevent damages caused by tidal propagation of contaminants or microrganisms, such as are responsible for red tide events.
The understanding of tidal propagation and of general circulation is central to la
74- _73_ J Reloncavl- '' (2i Sound Chacao Canal
(1q
- - ofAcuj
7 -- o a k
Gulf of Corcovado Y_'-,\j',1
44 , C
/ f
5.
Figure 1. Chilean Inland Seawater System of Gulf of Ancud. GeographicaJ Lbcation (from Clement et al.,1988) 2 any such developmentalstrategies,becausetidesarea dominant forcing mechanism for the circulation in the area.Because the structure of the Gulfs of Ancud and Corcovado magnify the tidal wave oscillations at Puerto Montt, for example, the M2 (semi-diurnal lunar tide component), has a tidal range of 5.5 meters.
Therefore, the first goal of this thesis is to improve the current knowledge of tidal propagation in the Chilean Inland Seawater System of Gulf of Ancud using simulation methods that assimilate known tidal data.
Our approach relies on the use of an existing numerical model for tidal propagation and circulation in shallow water bodies (Westerink et aL, 1988), as a tool to integrate and interpret the scarce data base available for the region.
Because of the scarcity and poor geographic location of available data, we were forced to concentrate a significant effort in the definition of a sensible scenario for tidal forcings at the boundaries (ocean boundary and communication with Gulf of Corcovado).With that purpose, we adopted and extended a technique to solve the "Inverse Problem" in tidal dynamics, proposed earlier by Bennett and McIntosh in 1982. In this technique, we used the available tidal data to obtain a set of boundary conditions that can be used as the forcing functions of our numerical model.
Three keydifferencesexistbetween Bennett and McIntosh'soriginal technique and the one used here:
(a)Our solution to obtain boundary conditions was made in the frequency domain. A set of weights for the basis functions was calculated at each one of the tidal frequencies used in the simulation, for amplitudes and phases.These were 3 combined to get the boundary condition at each frequency. Bennet and McIntosh solved their problem in the time domain.
(b) We ran the numerical model using the non-linear shallow Navier-Stokes equations.After each iteration, a new corrected set of boundary conditions was calculated to improve the fitting at the reference stations. Bennett and McIntosh used the linear shallow Navier-Stokes equations.
(c)The shallow water equations were solved forcing the system to strictly conserve mass and momentum at the boundaries.Small residuals in elevation were allowed at the reference stations that were within the range of the estimated errors of the available tidal data (see section 2.2.2).
The solution of the shallow water equations by Bennett and McIntosh allowed small residuals in elevations and velocities at the boundaries that were within the range of the estimated errors of the respective data.
The differences between these two solutions are based on the available oceanographic data. In our study where only tidal data (elevations) were available, solving the equations to conserve mass and momentum looked like a more natural solution.In the study done by Bennett and McIntosh where tidal data and velocities were available, solving the equations allowing small residuals in elevations and velocities at the boundaries and at the reference stations looked like a more natural solution for them. In order to compare both approaches would be necessary to run our numerical model under one set of forcing functions and two different sets of boundary conditions (in one case conserving mass and momentum at the boundaries and in the other case allowing small residuals in elevations and velocities at the boundaries).This comparison would not be possible until a new field survey of the area is finished and more data is available. 4
Chapter 2 explains the reasons why a two-dimensional frequency domain was chosen as the circulation model, and briefly reviews the model. This chapter also presents the mathematical formulation used to calculate the boundary forcing functions and describes the "Inverse Tidal Method" (I. T. M.) used to incorporate the observational data into the boundary conditions through a least-squares fit.
Chapter 3 presents the validation of the I. T. M. The validation is performed applying the fequency domain model TEA-NL (Westerink, 1984), with a corrected set of boundary functions obtained using the I. T. M., to two different cases:
(a) A rectangularchannelwithflatbottom, chosen becauseofits simplicity.
(b) The North Sea, chosen because of the quality and reliability of the available data set.The North Sea is a well-controlled environment that allows testing of the method for several combination of boundary conditions.
Chapter 4 describes the modeling of tidal propagation in the Chilean Inland Seawater System of Gulf of Ancud. We first summarize and analyze the available tidal records to provide as complete an overall picture as possible of the circulation in the area under study.This circulation is dominated by tides (Clement et al., 1988), and therefore our discussion centers around the description of the spatial and temporal variability of the tidal constituents.
We then apply the I. T. M. to get a reasonable scenario for the boundary conditions of the region under study and we critically discuss the method and its results.Next we use the model results to hypothesize general characteristics of the tidal propagation in the region.
In Chapter 5 we use the model results to go one step further and discuss how 5 they can be used to reach a better understanding of thephysical, biological and chemical oceanography of the area.The steps required for a future transport model are outlined. CHAPTER II
DESCRIPTION OF METHODS
2.1 Review of the Circulation Model
The model chosen to get the circulation in the Chilean Inland Seawater of Gulf of Ancud is called TEA-NL (Non-Linear Tidal Embayment Analysis).It was developed at the Massachusetts .Institude of Technology (Westerink et al., 1988).
The model solves the two dimensional non-linear shallow-water equations using a frequency-time domain finite element technique.The derivation of the governing equations used in this model is presented in Appendix A.
The model is forced by tides and it gives as the output the amplitude and phase at each node of a finite element grid.Then, it is extrapolated to yield the sea elevations and velocities at any given point in space and time.
The main considerations taken into account to choose a two-dimensional model were:
a) The lack of information about the vertical structure in the area.The only available data on annual change in vertical structure comes from the Lund expedition (Brattström and Dali, 1951-52) for the Northern most bay of the system (Reloncavi Sound).These data show that between April and September there is a relatively well-mixed layer for the whole depth of the bay (Clement et al., 1988). During austrai Fall and Summer when the fresh water run-off is largest because of snow melt, the surface mixed layer is about 20 meters thick.
b) The assumption of vertical homogeneity as a first step allows us to calculate the flow field and get an estimate of the circulation pattern in the region 7 that can be used until all the necessary information is available to run a three- dimensional model.
The main advantage of the finite element method over the more traditional finite difference method is that it allows a greater versatility in grid discretization whichisespecially important for the small scaleof coastal embayments. (Westerink et al, 1984)This feature permits a better fitting of the irregular boundaries and allows refinement of the grid in such areas as high flow and bottom gradient regions and at narrow mouths connecting to the open ocean.
The use of harmonic analysis in conjunction with finite elements to solve the shallow water equations, is intrinsically a natural solution procedure because of the periodic nature of the tidal phenomenon.In the frequency-domain there are no time-stepping limitations since this procedure generates a set ofquasi-steady (or time independent) equations.Eliminating the time dependence from the governing equations reduces them from the equations of the difficult and time consuming hyperbolic type to that of the elliptic type which are more easily solved by finite element methods. The harmonic method also offers the potential of economically performing realistic long term simulations in tidal embayments and calculation the residual circulation. (Westerink et al., 1988)
One difficulty which arises in implementing this frequency domain model is that non linear terms generate additional responses at frequencies other than the base forcing frequency. TEA-NL handles the harmonic coupling produced by the non-linear terms (finite amplitude, convective acceleration and bottom friction terms) using an iterative procedure.The iterative scheme generates a finite spectral series representing the pseudo forcings due to all non-linear components of the shallow water equations.A linear solution is then used to evaluate the response to each pseudo forcing component(at predetermined frequencies). Pseudo forcings are then updated using the updated responses in elevation and velocities until convergence is achieved.
However, inherent to any solution scheme that iteratively updates the non- linearities as right hand side loadings is the limitation that the relative magnitude of the right hand side non-linear term must be small compared to the left hand side linear term. The implication of this is that the non-linear solution must be a perturbed linear solution. (Westerink et al., 1988)
For the case of tidal estuaries, the magnitude of the non-linear friction term far exceeds that of the other non-linear terms. In order to minimize friction as a right hand side term in the equations (where all the non-linear terms are located), a close approximation of the linear part of the friction termwill be included on the left hand side of these equations (a major part of the friction term is linear as shown in Appendix A). The non-linearities are treated in an iterative fashion and calculated at each cycle to appear as a constant on the right hand side of the equations during the solution of the next cycle. A linear term is added to both sides of the momentum equations to provide iterative stability and improve convergency rates.Equations A.13, A.14 and A.15 (see appendix A) are then reduced to their harmonic form (equations A.18, A.19 and A.20) using Fourier series.This leads toNfsets of time independent equations (equations A.21, A.22 and A.23) that can be easily solved using finite elements.Each system of equations is coupled to the other Nf1 system through the non-linear loading terms.This non-linear coupling between harmonics is handled using an iterative solution strategy by updating the right hand side load terms at each cycle.
To improve convergency rates and to insure iterative stability, the linearized friction factor is calculated at each node using the bathymetric data available kand then multiplied by a relaxation factor prior to starting the full non-linear iterations.Non-linear load terms are generated at each cycle of the iteration in the time domain using previously computed harmonic responses (obtained superpositioning equations 2.1 and 2.2 for all frequencies of interest). These time histories of the non-linear terms are then harmonically decomposed using a least- squares harmonic analysis method. A schematic of the iterativescheme is shown in figure 2.The iterative solution strategy starts out with the assumption that the non-linear loading terms are zero. Each of the Nf sets of equations are then solved for the boundary loadings imposed (equations A.24 and A.25).Time histories can then be generated for velocity and elevation with the following equations:
(Nf i(.Q.t' ti(t) = Re .e 2.1 I.j=1j J (N u(t) = ReEii.e , 2.2 j=1 )
This in turn allows time histories of the non-linear pseudo loading vectors P(t) and P(t) to be produced.These time domain pseudo forcings can now be approximated as harmonic series. Hence the total non-linear loadings for continuity arid momentum are distributed to all or some of the sets of frequency domain equations. Now each of these sets of equations is solved again and the entire procedure is repeated until convergency is reached.Each of the sets of equations are linear at each cycle of the iteration although they are coupled through the non-linear loading terms.Solving each set of linear equations is the most important step of the non-linear solution scheme. The harmonic analysis of the non-linear pseudo forcings is of vital importance for the efficiency, accuracy COU ND
a
FOR AU.
UNEAR SOLU
GENERATE TB HISTORY BY
OF SOL(JTIO u() A
GENERATE ThIE HISTORY LOA?P'G
pOI(t)USING TIME HISTORY RESPOrSJ
RAR1(ONIC ANALYSIS OF
TO GENERATE
FREQ. DOMAIN LOADING P.j j t. ,1
'1'
SELECT SET OF a FOR
WHICH UNEAR CORE IS IUJN
Figure 2. Iterative non-linear scheme 11 and generality of the full non-linear model. The efficiency of the model is strongly influenced by the number of time history points required for the harmonic analysis procedure, since the procedure must be applied at every node in the grid at each iteration.The method used by this model is the least-squares harmonic analysis.This method consists of a common least-squares error minimization procedure which uses a harmonic series as the fitting function.The harmonic series only contains frequencies which are known to exist in the time hisry record.(Westerink's model defines the time history record as a time series that containsallthe necessary information in the time domain to resolve the frequencies that we are using in the frequency domain.) The method is able to extract extremely close and irregularly spaced frequency information, yet only requires a number of time history points equal to twice the number of frequencies contained in the time history record.
The squared error between the sampling points and the harmonic fitting function is normally defined as:
E cosjt + b sin wt) 2.3 =j f(tj)12 where
f(t) = value of the time history record at time sampling point t N = number of sampling points M = number of frequencies a, b. = unknown harmonic coefficients
The error minimization is had by setting to zero the partial derivatives of E with respect to each of the coefficientsaand b. The model stops when it reaches 12 convergency between two consecutive cycles in elevation and velocities according to the criteria of convergency previously defined, or when it reaches a number of cycles that is equal to a limit already defined at the start of the run.
The final output from the model are the sea elevations and velocities at each of the nodes and elements of the finite grid, respectively. These results allow the user of the model to generate a time series of elevation and velocities for any given time using the equation A.16a.
2.2 Inverse Tidal Method
2.2.1 Application f Inverse Tidal Method th Solution of a Practical Problem
The Inverse Tidal Method (I. T. M.) can be defined as the least-squares fitting technique that uses a singular value decomposition to solve the problem of defining the values of the boundary conditions for a tidal driven circulation model (Bennett and McIntosh, 1982).
The I. T. M. is used to calculate the tidal elevations at the boundaries.It is a data assimilation technique that uses the known information inside the domain and the results obtained from a model forced with a set of arbitrary basis functions to generate a new set of boundaries that will recreate, within some selected criteria, the observed data at known stations.
In a practical case the set of equations that we solve is: A.cd where: 13
1) Ais the rectangular matrix with the values of each one of the basis functions at each one of the grid points of the finite element grid that corresponds to a known station. These values are obtained from a linear run of TEA-NL.
2) cis the matrix that contains the weightscjfor each one of the basis functions.It is the only unknown.
3) d is the matrix that contains the perturbated known data.
Matrix d is obtained from the following equation:
d = non-linear guess - known data where the non-linear guess is the value of any arbitrary function at the given stations obtained from a full non-linear run of TEA-NL.
Using matrix d defined in this way, instead of
d = known data, saved computer time and reduced the number of iterations required for a given convergency criterion. Of course, this statement is based on theassumption that our first guess is close to the right solution.In case that we do not have enough information to estimate the first guess, setting the non-linear guess equal to zero may be a wise choice because it does not introduce unnecessarycorrections to the boundaries.
The solution of this set of equations gave us the weights cthat multiply the 14 basis functions and made possible the computation of a new set of boundary conditions.
After each iteration we computed matrix d using the previous results of the model as the non-linear guess.If the solution is converging the value of the elements of matrix d decreased with each iteration.If matrix d decreased then the weights cjalso decreased. Matrix A always remained the same, because its elements depend only on the values obtained from the model for the arbitrary sets of boundary conditions (basis functions) and not from the new set of boundary conditions.Figure 3 is a schematic diagram that explains how the Inverse Tidal Problem is solved for a general case.
Appendix B has a complete description, step by step, of the application of the I. T. M. for a general case.It also contains the Fortran codes used for this computation, samples of input files for the case of the Chilean Inland Seawater System of Gulf of Ancud and a flow chart that describes the process.
2.2.2 Mathematical Formulationf The. Inverse Tidal Method
This section reviews the mathematical formulation of the least squares data fitting and singular value decomposition used to perform the I. T. M.
Suppose we have m given data points
(t,y) i=1,... ,m wheretis the independent variable andyis the dependent variable that satisfies the following unknown equatibn:
y=y(t1).
16
Suppose now that y can be approximated by a linear combination of n given basis functionsc6, then:
y(t) (t) = c11(t) + c22(t) +.. . +cnç5n(t)
The problem to be solved is to choose the n coefficientsc1, ..., Cn sothat the model fits the data in some physical or mathematical sense. The model is linear because the coefficients appear linearly, although the basis functions may be nonlinear functions of t.
A common linear model is a polynomial
(t) = c1 + c2t + c3t2 +... + wherek(t) = ckt.
Another example using trigonometric approximations is:
5(t) =c1 sin t + c2 cost + c3 sin 2t + c4 cos 2t + . + Cn sin nt + c+i cos nt wherek(t) = ck sin kt + ck+1 cos kt.
In our particular case weshallconsider the case where m, the number of data points, is smaller than or equal ton,the number of unknown coefficients. Then the problem of choosing the coefficients is well defined, and it is usually possible to have the model fit exactly the data.
The criteria for determining the coefficientscj'swill be the method of least- 17
squares. For any choice of cj's the residual at thetbdata point is
= c(t) y 3=1 or
=5(t) The least-squares criterion specifies that the cj's must be chosen to minimize the sum of the squares of the residuals, or in other words, thecj's have to be chosen to fit the data using a combination of basis functions with the smallest possible error.
(m 2 Minimizeover c
The least-squares criterion necessarily determine a unique set of coefficients, unless the number of data points m is larger than the number of unknown coefficients.If the basis functions are linearly dependent at the data points, which means that there are non-zero coefficients for which
j=1JJ7...(t.) = 0i = 1,. ., m
then any multiple of the 7j's can be added to the cj's without changing the sum of the squares of the residuals.
In order to compute a set of coefficients which give a minimum sum of squares we will have to minimize z, or equivalently, to minimize
1/2 2 'm'n yi i=1 j=1 18
So we require that
?=O fork=1,...,n. Ck
Taking the derivatives and interchanging orders of summation, we obtain
(t)k(t))c yik(ti). j1(i1 =j
This is the set of n simultaneous linear equations for the n unknown c1's. It can be written in matrix form as:
P.c=q
where1kj 1k(ti)(t) andkj =
1=1
Notice that thematrix Pdepends only on the basis functions and itis symmetric.
There are some serious drawbacks to the use of these equations to compute the coefficients Cj. It turns out that thematrix Poften has a very high condition number, so that no matter how the set of equations is solved, errors in the data and round-off errors introduced during the solution are excessively magnified in the computed coefficients. In the extreme situation where two of the basis functions (t)are linearly dependent, it can be shown that P issingular, 19 and the condition number is infinite. (See last paragraph, page 23.)The most reliable method for computing the coefficients for least-squares problems is based on a matrix factorization known as the singular value decomposition (S.V.D.). There are other methods which require less computer time and storage, but they are less effective in dealing with errors in the data, round-off errors, and linear dependence (Forsythe et al., 1977).
The singular value decomposition begins with a rectangular matrix A with m rows and n columns whose elements are:
=
If y denotes the m vector with elements and c denotes the n vector with components Cj then the approximate equations are:
c,(t)= i = 1,..., m and can be written as
A.c=y. 11 1 mXn nXl mXl
The S.V.D. takesAas the input and returns three matrices A, U and V as output. The matrixAis diagonal with non-negative diagonal entries, which are known as the singular values of A. The matrices U and V are used to decomposeAinto: A=uAVT 20
where Uis an orthogonal matrix and V is a transposed orthogonal matrix. (Note:if we use complex matrices, instead of real matrices, the transposed matrices become transposed conjugated matrices.)
"It is known that a matrix can be obtained as the product of three factors: the orthogonal matrixU,the diagonal matrixA,and the transposed orthogonal matrix (Cornelius Lanczos, 1961)
So using the S.V.D. we can transform the equation
Ac=y into an equivalent diagonal set of equations:
= 7
where =vT. c
and 7=uT..
It is important to recall that multiplication by orthogonal matrices does not change such important geometrical quantities as the length of a vector or the angle between two vectors.
Letc, j= 1,..., nbe the diagonal entries ofA.In principle, if none of thecrj'sare zero, the transformed equations could be solved by setting
y. j=1,...,n. J .3 21
However, this is not always desirable in practice if some oj's are small. The are non-zero if and only if the basis functions are linearly independent at the data points.So the key to the proper use of the S.V.D. is to choose a tolerancer which reflects the accuracy of the original data and the accuracy of the floating-point arithmetic being used. Any greater than ris acceptable and the corresponding is computed as Any less than T is negligible and the corresponding can be set to zero.
The ratiocrmax/omjn,where°maxis the largest singular value and0minis the smallest singular value, is defined as the condition number ofmatrix A.
Neglecting o'sless than r is the same as decreasing the condition number to cmax/r.Since the condition number is an error magnification factor, this results in a more reliable determination of thecj's. The cost of this increased reliability is usually an increase in the size of the residuals. (Forsythe et al., 1977) 22
CHAPTER III VALIDATION OF METHODS
Because of the uncertainties surrounding the data for the Chilean Inland Seawater System of Gulf of Ancud, it is essential to "prove" the predictive utility of this model by validating its output in test cases.
In this chapter, we will validate the I. T. M. in the context of two well- controlled case studies:a rectangular embayment (synthetic dock), and the English Channel and southern bight of the North Sea (Werner and Lynch, 1989). Each case will help us to get a better understanding of:
(1) how to choose the basis functions,
(2) how the method behaves when we introduce changes in the data (small random perturbations), and
(3) what happens when we move the location of the known stations closer or farther to the boundaries.
The main advantage of the two cases used to validate the I. T. M. is that "known" data was obtained from non-linear runs of the TEA-NL model forced by a given set of boundary conditions.Validation could be made for arbitrary boundary conditions; for the North Sea case,I chose the set calculated by Baptista et al, 1989.These given sets avaid all possible errors introduced by external factors, for example, digitizing errors, etc.In this way the I. T. M. tried to fit error-free "known data" at the reference stations so that any differences between the known and calculated data was attributable to the method itself.
The rectangular embayment case tests if the I. T. M. works for a very simple geometry, and, to a first approximation, how the flow field behaves under 23 different combinations of basis functions imposed as the forcing functions.
The North Sea case tests the I. T. M. under a more complex geometry, and illustrates how to choose the best combinations of basis functions to use as the forcing functions.
The results are presented at reference stations in the form of:
(1) time series of elevations,
(2) charts of amplitudes and phases of tidal elevations, and
(3) charts of flow fields and error for elevations.
The error for elevations and the time seriesof elevations are always calculated at the same reference stations.In this way, any changes in location of the known data points introduced to test the behavior of the method are always compared at the same reference stations.The error at each one of the reference stations has been calculated using the following equation (Baptista et al., 1989):
1/2
t Error [ tO{f(Ak,f cos(wf k,f) AC,f cos(wf
Where: N = Number of time steps. T Total time period used to calculate time error. F = Number of tidal frequencies. Akf = Known amplitude at frequency f.
ACf = Calculated Amplitude at frequencyf. = Tidal frequency. 24
k,f= Known phase at frequency f. c,f Calculated phase at frequency f.
Each one of the two cases presented here have all the required data to compare the results between the I.T.M. solution and the reference stations.
3.1 Rectangular Embayment
The rectangular embayment is a channel that has rigid boundaries on the northern, southern and eastern sides.Its western boundary is open and forced by tides. There are no other forcings and the bottom is flat. The finite element grid is composed of 304 similar elements and 180 nodes (Figure 4).This particular geometry was chosen for its simplicity.
The Inverse Tidal Problem was solved using combinations of:
(1) linear and,
(2) linear and trigonometric basis functions.
The wavelengths used for the trigonometric basic functions range between 450 meters and 1,450 meters. The length of the boundary where these functions were imposed was 1,800 meters.
The first guess set the amplitudes and phases at the boundary points to zero. The known data was obtained from a full non-linear run of the model for the same grid and two points inside the domain were selected to be the reference points for comparison (stations 30 and 34, Figure 4). The tidal frequency chosen for the test comparison was the M2 tide.
Table 1 has the values of amplitudes and phases at grid point 30 for the known data and for successive iterations using linear basis functions. Table 2 has 25
Rectangukw Embayment Finite Element Grid
180 Nodes 304 Elements
Figure 4. Rectangular embayment, finite element grid. 26
Amplitude (cals} Phase [radians]
Known Dat.a 0.19380487143(356 0.10000628186546 First Run 0.19377893596946 0.098551942641913 Second Run 0.19380527805611 0.1000064903S993 Third Run 0.19380487160178 0. 100006884S5009 Fourth Run 0.19380487143574 0.10006S8186709 Fifth Run O.193$04S7143656 0.10006S81S6545
Table 1Amplitudes and phases at grid point 30 for linear basis functions.
Real Part Imaginary Part
First Run (L1936E±01 0.2810E-02
Second Run 0.2650E-03 O2S1OE-02 Third Run 0.04GOE-05 Q.7640E-06 Fourth Run .-O.1650E-0$ 0.5770E-08
Fifth Run 0.8770E-11 0.4730E-11
Table 2Values of weights that multiply the linear basis functions to correct the boundary conditions. 27
the values for the weights of one of the basis functions at the same grid point 30.
Table 3 has the values of amplitudes and phases at grid point 30 for theknown data for successive iterations using a combination of linear and trigonometric basis functions. Table 4 has the values for the weights of one of the basis functions at the same grid point 30.
Figure 5 shows the known flow field and the calculated flow fields after the fifthrun usinglinearbasisfunctions and a combination oflinearand trigonometric basis functions.
From table 1 we note that after five iterations the amplitude and phase of the tide at grid point 30 is almost equal to the known amplitude and known phase.Table 2 shows how the correction introduced to the boundary conditions after each iteration is decreasing, and therefore, how the calculated values are converging to the known values.Figure 5 shows the known and calculated flow fields at the same instant (instant = 11 hours).
From table 3 we note that after five iterations the amplitude and phase of the
M2 tideatgrid point 30isstilloscillating around the known values. Combinations of linearbasis functions with sines and cosines of different wavelengths were used, but convergency was never achieved.In all tests with combinations of basisfunctions, the firstiteration gave the best possible approximation at the grid point used as the reference point and the most reasonable flow field.
As we note in figure 5 the flow field for the same instant after successive iterations clearly shows a non-physical behavior that has a tendency to propagate further and further from the boundary as the number of iterations increase. This Amplitude fcrns] Phase [radians]
Known Data. 0.19380487143656 0.10000688186546 First Run 0.19402967223008 0.09527579996317 Second Run 0.19296234813237 0.10680508436959
Third Run 0.19579972788272 0.098634640429191 Fourth Run 0. 19057S2549348 0.10593405166880 Fifth Run 0.194653662163 0.0997234283721
Table 3Amplitudes and phases at grid point 30 for acombination of linear and trigonometric basis functions.
Real Part Imaginary Part
First Run 0.1916E+01 0.3462E 01 Second Run 0.3042E-01 0.4595E-01
Third Run O.4516E-01 0.1908E+00 Fourth Run (J.3552E-01 0.1026E+00
Fifth Run 0.9168E-01 0.5806E-01
Table 4Values of weights that multiply the linearand trigonometric basis functions to correct the boundary conditions. 29
Known flow field
Calculated flow field (linear functions)
,a- a-a------a- a-a- a.;'. a- a--a- ,.' a------,_" a-,- a ------a- -- - - a------_ - - a-' a- a- a-a- -- aa- a-a a a-.- --
a-'- a-i a- ' ..-.. -__ -- - - a, a- a-, - -_ --
Calculated flow field (trigonometric functions)
-: -- -: -- -: -- -: -_
- :
Figure 5. Known and calculated flow fields using linear and trigonometric basis functions for the rectangular embayment at instant = 11 hours. 30 behavior is studied in the next section for the more complex case of the North Sea.
The solution obtained for the case of linear basis functions are as expected and explained by the mathematical formulation. The error decreased after each interation and therewith, the correction to the boundary conditions due to the basis functions.
3.2. North Sea
This case is based on a finite element grid that was developed for the English Channel and southern bight of the North Sea by Werner and Lynch, (1989). The grid has 1,613 elements and 611 nodes (Figure 6). The field data were compiled by Drs. C. Le Provost and G. Verboom and distributed to be used in the solution of a benchmark problem under the auspices of thevithInternational Conference of Finite Elements in Water Resources (Werner and Lynch, 1989).The known data set consists of eleven reference stations (Figure 7).
The Inverse Tidal Problem was solved using combinations of linear and trigo nometric basis functions.The known flow field, phases and amplitudes for the grid were obtained from a full non-linear rim of the model described in Baptista et at., 1989. The tidal frequency used for the comparison was M2.
The combination of basis functions used in this case were linear functions, and linear and trigonometric functions (sines and cosines).The wavelengths of the trigonometric basis functions were chosen to fit within the length of the boundary.In each experiment depending on the number of basis functions, a fewer number of reference stations were used for the solution of the Inverse Tidal Problem.In all experiments the error was calculated at the eleven reference 31
Figure 6. North Sea, finite element grid 32
Figure 7. North Sea, location of reference stations 33 stations.
The first chosen boundary condition used was an arbitrary linear basis function.The model was run, for this boundary condition and its result used as the first guess in the solution of the Inverse Tidal Problem.
The results are compared with the known data at the reference stations for different combinations of basis functions and stations.
Three sets of results are discussed in this section.These sets were chosen among severalcombinations because they respresentthe most interesting experiments. The experiments are as follows:
a) Expel45: It was performed using four linear basis functions as the correctors for the boundary.
b) Expel66: It was defined as the experiment containing the known data.
c) Expe28O: It was performed using twenty-four basis functions as the correctors for the boundary.The basis functions are a combination of linear and trigonometric functions.The linear functions are the same used in IExpel45. The trigonometric functions are composed of ten sines and ten
osines of different wavelengths.
t was performed based on Expe28O, but introducing a andom perturbation in the known data at the reference tations used in the solution of the Inverse Tidal Problem. 'he results of these runs were used to check how the 34
elevations at the reference stations were affected by the introduction of a random error.
Figures 8 through 40 present the time series of elevation of tide at the eleven reference stations for a period of 13 hours.
Table 5 has the values of the RMS-Error for at each one of the eleven reference stations averaged over a period of 13 hours.
From table 5 and figures 8 through 40 we find a very good fit between the known data and the calculated data obtained from the runs of the model. The difference between known data and calculated data for these experiments was never larger than 5.3 centimeters.If we would like to reduce the error at the reference stations, a larger number of iterations would generate a better fitting.
Figures 41 through 46 show the flow fields, the isolines of amplitude and the isolines of phase for the known data and the three experiments.All flow field charts were calculated at instant equal to three, six, nine and twelve hours (one tidal cycle of M2).
All these previous figures look alike, but they are different near to the southern boundary. While the error at the reference stations is within tolerance, the charts of Expe28O and Expe294 show some very distinctive features at the southern boundary. These features are not presented in the known data nor in Expel45.
The reason for these features are the arbitrary basis functions used as the correctors for the boundary forcing conditions.In the three experiments these functions fulfill the mathematical requirements of the I. T. M., that it is to be independent from each other.Linear functions as in Expel45 only generated a 35
Elevations (known dota and expe 145)
6.0
4.0
- 2.0
6
0.0
2.0
4.0
60 0.0 10000.0 20000.0 30000.0 40000.0 60000.0
Tn,,. inecond
Figure 8. Elevations. Known data vs. calculated data (EXPEI45) at station St. Malo for a period of 13 hours
Elevations (known data and expel45)
10000.0 20000.0 30000.0 40000.0 60000.0
Tn,,. in eecndn
9.Elevations. Known data vs. calculated data (EXPE145) at station Cherbourg for a period of 13 hours 36
Elevations (known data and expe 145) Fr.quncy M2 p.iod 13 too totn chrt, 0.6
kno doio
0.4
0.2
S 0.0
0.2
0.4
0.6 0.0 10000.0 20000.0 30000.0 40000.0 60000.0
i
Figure 10. Elevations. Known data vs. calculated data (EXPE145) at station Christchurch for a period of 13 hours
Elevations (known data and expel 45) MZ psnod 1-3 hOO 4.0 k.m dto
2.0
6
0.0
2.0
4.0 0.0 10000.0 20000.0 30000.0 40000.0 50000.0
1.m M cond
igure 11. Elevations. Known data vs. calculated data (EXPE145) at station Dieppe for a period of 13 hours 37
E$evatjon(known doto ond expe 145)
rqo.nq &42 pod 13 ho tio.
- kno..n doto pel46
2.0
0.0
2.0
4.0 L 0.0
in
Figure 12. Elevations. Known datavs. calculated data (EXPEI45) at station Boulogne for a period of 13 hours
E1evotons (known dato antf expe 145)
3.1
2.c
1.0
6
0.0
1.0
2.0
3.0 L . 0.0 10000.0 20000.0 30000.0 40000.0 60000.0
r.n.eineoonds
gure 13. Elevations. Known data vs. calculated data (EXPE145) at station Dover for a period of 13 hours Elevations (known data and expel45) crquny 142 p%o4 13 hou tatio Coløi .3.0-
kno.noto
2.0
4.0
S 0.0
1.0
2.0
30 0.0 10000.0 20000.0 .30000.0 40000.0 60000.0 m
Figure 14. Elevations. Known data vs. calculated data (EXPE145) at stationCalais for a period of 13 honrs
Elevations (known data and expe 145) Fr.quenc MS pellod ii lieu., totlofl WaltO.i 1.5 - epel45o,n40(0 1.0
0.5
S -; o.e
0.5
1.0
1.5 0.0 10000.0 20000.0 30000.0 40000.0 50000.0
Toeee ie seceindo
Figure 15. Elevations. Known datavs. calculated data (EXPE145) at station Walton for a period of 13 hours 39
E1evatons (known data and expel 45)
2.0 -
o(
I-c
C
0.0 -
-l.a
-2.0 - 0.0 10000.0 20000.0 30000.0 40000.0 60000.0
T.md M
Figure 16. Elevations. Known datavs. calculated data (EXPE145) at station Zeebrugge for a period of 13 hours
Ekvations (known doto and expel45) F.qUncyM2 psilod lhaW st000.l &owtoft 1.0 - known doto - wpnI45
0.5
.0 10000.0 20000.0 30000.0 40000.0 60000.0 Tnw n secondn
re 17. Elevations. Known data vs. calculated data (EXPE145) at station Lowestoft for a period of 13 hours 40
Ekvotions (known data and expel45) fqoncy M2 piod 13 hoor ttion Hol voko4lnd 1.0 kno.,oto
0.5
S
E -; 0.0
0.5
I.0 0.0 10000.0 20000.0 30000.0 40000.0 50000.0 i .conOs
Figure 18. Elevations. Known data vs. calculated data (EXPE145) at station Hoek Van Holland for a period of 13 hours
Elevations (known datand expe28O) R.quenc42 p.iod 13 ho tUon St Moo 6.0 known doto
4.0
2.0
S o.o
2.0
4.0
0.0 10000.0 20000.0 30000.0 40000.0 50000.0
Figure 19. Elevations. Known data vs. calculated data (EXPE28O) at station St. Malo for a period of 13 hours 41
Elevations (known data and expe28O) Frnq.oncy k12p.rind 13 howw ,td.o CinDOog 2.0 known doto nnpo2SO
1.0
6 0.0
1.0
0.0 10000.0 20000.0 30000.0 40000.0 60000.0
T.n..n .econd
Figure 20. Elevations. Known data vs. calculated data (EXPE28O) at station Cherbourg for a period of 13 hours
Elevations (known data and expe28O)
F.eqoency 1.12peflod 13 beu. etot,onC*wtstctrn,ch 0.6
known dotg cope280
0.4
0.2
C 0.0
0.2
0.4
0.6 - 0.0 10000.0 20000.0 30000.0 40000.0 60000.0 Tin,c in wecondo
Figure 21. Elevations. Known datavs. calculated data (EXPE2SO) at station Christchurch for a period of 13 hours 42
Elevations (known data and expe28O) o.00n uo.p I.2 penO no.. 4.0
wo.o..oto ..op28O
2.0
e
6
0.0
2.0
4.0 TT'TTiT. 40000.0 60000.0 0.0 10000.0 20000.0 30000.0
Tone .e OdCofld.
Figure 22. Elevations. Known data vs. calculated data (EXPE28O) at station Dieppe for a period of 13 hours
Elevations (known data and expe28O)
coz pence ,j nowe notice 4.0
lu.ewn doto eope28O
2.0
S
0.0
2.0
_4Q1 . -. . 0.0 10000.0 20000.0 30000.0 40000.0 50000.0
Tone .n ,000nde
Figure 23. Elevations. Known data vs. calculated data (EXPE28O) at stationBoulogne for a period of 13 hours 43
Elevations (known data and expe28O)
t.&L peon.Ia flOIJfl Swuon L)c.tr 3.0 - lan. dot. eopo28O 20
1.0
C 0.0
1.0
2.0
3.0 0.0 10000.0 20000.0 30000.0 40000.0 50000.0 in ,econd
Figure 24. Elevations. Known data vs. calculated data (EXPE28O) at station Dover for a period of 13 hours
Elevations (known data and expe28O) Frnqn.ncy &42peeled 13hoI.c*.10*100colOie 3.0 - known dat..
2.0
1.0
C 0.0
1.0
2.0
' a 0.0 10000.0 20000.0 30000.0 40000.0 60000.0 Tine.. in second.
Figure 25. Elevations. Known data vs. calculated data (EXPE28O) at station Calais for a period of 13 hours 44
Eevotions (known data and expe28O)
1.5 --- kn ota
1.0
0.5 S S o.o
05
1.0
1.5 50000.0 0.0 10000.0 20000.0 30000.0 40000.0
Thie 1,, s.CO..d
Figure 26. Elevations. Known data vs. calculated data (EXPE28O) at station Walton for a. period of 13 hours
Eevatioas (known data and expe28O)
p.r.o 2.0 --- knofl _80 oto
1.0
E
0.0
1.0
2.0 50000.0 0.0 10000.0 20000.0 30000.0 40000.0
T,r.e m conds
Figure 27. Elevations. Known data vs.calculated data (EXPE28O) at station Zeebrugge for a period of 13 hours 45
Elevations (known data and expe28O)
Freqoncy 1.12poro.1 (3 hoo tatoo Looe.toft (.0
koon doto oopo28O
0.5
C
E
0.0
1.0 0.0 (0000.0 20000.0 30000.0 40000.0 60000.0
Figure 28. Elevations. Known data vs. calculated data (EXPE28O) at station Lowestoft for a period of 13 hours
Elevations (known data and expe28O) ,. P000 4J flOtfl .mwo flOe. .on fl00000 1.0 - known 4010 oope2SO
0.6
S
-; 0.0
0.5
1.0 0.0 (0000.0 20000.0 30000.0 40000.0 60000.0
7mwn secoodo
Figure 29. Elevations. Known data vs. calculated data (EXPE28O) at station Hoek Van Holland for a period of 13 hours 46
Elevations (known data and expe294)
6.0
4.0
2.0
0 o.o
2.0
4.0
6.0 0.0 10000.0 20000.0 30000.0 40000.0 50000.0
T n *sc,d*
Figure 30. Elevations. Known data vs. calculated data (EXPE294) at station St.Malo for a period of 13 hours
Eevotions (known data and expe294)
M pnaa J 6aiI t 2.0 lm doto
1.0
0
0.0
1.0
10000.0 20000.0 30000.0 40000.0 50000.0
in nconds
Elevations. Known data vs. calculated data (EXPE294) at station Cherbourg for a period of 13 hours 47
Elevations (known data and expe294)
eqoency 142p.r4cd 13 hcos etcijon Ctctmot 0.6 knoon dots
0.4
0.2
6 0.0
0.2
0.4
0.6 60000.0 0.0 10000.0 20000.0 30000.0 40000.0
Tensinseconds
Figure 32. Elevations. Known data vs. calculated data (EXPE294) atstation Christchurch for a period of 13hours
Elevations (known data and expe294) isa penon sa noi,, 4.0 kno.,n dots
2.0
S
0.0
2.0
4.0 40000.0 60000.0 0.0 10000.0 20000.0 30000.0
Tens in seconds
gure 33. Elevations.Known data vs. calculated data (EXPE294) at stationDieppe for a period of 13 hours EIevatons (known data and expe294)
4.0 known dto p294
2_a
6 0.0
-20
-4.0 0.0 10000.0 20000.0 30000.0 40000.0 60000.0 ..,
Figure 34. Elevations. Known data vs.ciculated data (EXPE294) at station Boulogne for a period of 13 hours
EievaUons (known data ond expe294)
Frsqun $42per$od 13 honri station 0oe 3.0 knono data aape294
2.0
1.0
E 0.0
-1.0
-2.0
3.0 0.0 10000.0 20000.0 .30000.0 40000.0 60000.0 m io
re 35. Elevations.Known data vs. calculated data (EXPE294) at station Dover for a period of 13 hours 49
Elevations (known data and expe294) -..' p.oc 3.0
&nOw,o(o p294
2.0
1.0
5
0.0
1.0
2.0
3.0 0.0 10000.0 20000.0 30000.0 40000.0 60000.0 r
Figure 36. Elevations. Known datavs. calculated data (EXPE294) at station Calais for a period of 13 hours
Elevations (known data ond expe294) IVQUCOC)' Mt PC000 O fowl flOLMO WOSCA I.e
known data
1.0
0.5
C
0.0
0.5
1.0
1.5 0.0 10000.0 20000.0 30000.0 40000.0 60000.0
Thr.e I., eco,d
Figure 37. Elevations. Known datavs. calculated data (EXPE294) at station Walton for a period of 13 hours Elevotions (known date ond expe294)
2.0 known doto
1.0
S o.o
1.0
2.0 0.0 10000.0 20000.0 20000.0 40000.0 50000.0 h. ..conds
Figure 38. Elevations. Known data vs.calculated data (EXPE294) at stationZeebrugge for a period of 13 hours
Elevations (known data and expe294) Fr.qu.ncyu2 pedod 13 hoo wtotlo o.ewtott 1.0 known dote
0.6
S 0.0
0.5
-to 0.0 10000.0 20000.0 .30000.0 40000.0 60000.0 Ten, in secoodo
Figure 39. Elevations. Knowndata vs. calculated data (EXPE294) at stationLowestoft for a period of 13 hours 51
Elevotons (known data and expe294) Fsqa.ncy U2 pe.iod 13 hau.0 atation look an ftol$o.at I.e - knoaa. data
0.5
S
0.0
0.5
1.0 0.0 10000.0 20000.0 30000.0 40000.0 50000.0
T.mo in soconda
Figure 40. Elevations. Known data vs. calculated data (EXPE294) at station Hock Van Holland for a period of 13 hours 52
STATION ERROR IN CENTIMETERS EXPE 145 EXPE2SO EXPE 294
St. Malo 5.292 1.311 0.194 Cherbourg 1.599 1.461 0.810 Christchurch 0.983 2.432 1.380 0.790 Dieppe 4.013 3.669 0.019 Boulogne 4.184 3.726 Dover 2.711 2.537 0.270 0.730 CaIa.is 2.230 2.596 Walton 0.168 0.243 0.063 Zeebrugge 0.284 0.980 0.148 Lowestoft 0.002 0.266 0.001 Hoek Van Holland 0.933 0.412 0.177
Table 5RMS-Error at the eleven reference stationsfor frequency M2. 53
Inztoat = 3 hours
Figure 41. Known flow field (EXPE166) and calculated flow fields(EXPE14S- EXPE28O-EXPE294) for the North Sea at instant 3 hours. 54
lnsthat = 6 hours
Figure 42. Known flow field (EXPE166) and calculated flow fields (EXPE145- EXPE28O-EXPE294) for the North Sea at instant = 6 hours. 55
Instant = 9 hours
Figure 43. Known flow field (EXPE166) and caiculated flow fields (EXPE145- EXPE2SO-EXPE294) for the North Sea at instant = 9 hours. 56
Instant 12 hours
Figure 44. Known flow field (EXPE166) and calculated flow fields (EXPE145- EXPE28O-EXPE294) for the North Sea at instant = 12 hours. 57
Amplitudes
Figure 45. Isolines ofamplitudes for EXPE166 (knowndata) and EXPEI45-EXPE28O-EXPE294 (calculated data) for NorthSea. Phases
Figure 46. Isolines of phases for EXPE166 (known data) and EXPE145-EXPE28O- EXPE294 (calculated data) for North Sea. 59 correction for the boundary conditions that resulted in a linear trend along the boundary.In Expe28O and Expe294, a combination of linear and trigonometric functions satisfied the required precision at the reference stations, but it generated a correction for the boundary conditions that was a quasi-trigonometricfunction. This quasi-trigonometric function was the sum of sines, cosines and linear trends. It would never be a purely linear trend, unless the wavelengths chosen for the trigonometric functions are so large that compared with the length of the boundary the functions look like linear functions. This quasi-trigonometric shape function would force, at and near the boundary, the features presented in figures 41 through 44.
Figure 47 shows the differences in velocity between the known flow field and the calculated flow field in each one of the three experiments. This chart clearly shows in Expe28O and Expe294 the influence that the shape of the forcing function had over the resultant flow field.In Expel45 where the forcing function was a linear trend these anomalies were not present in the flow field.
Expe294 suggested that if the error of the known data or in this experiment, a random perturbation of the data, is smaller than the tolerancerchosen during the singular value decomposition as the accuracy of the data, the final result is not significantly affected by this perturbation.This suggestion is important because it allows us to test the value of tolerancerchosen and to check the stability of the results for perturbations in the data smaller than the known error of the data. This is the reason why Expe294 that used Expe2SO as the first guess improved the fitting at the reference stations and reduced the magnitude of the error.
As the conclusion from these experiments with differentsets of basis Expel 66Expel 45
Figure 47. Residual flow fields for the North Sea. Difference between known flow field (EXPE166) and three experiments (EXPE145- EXPE28O-EXPE294) 61 functions we say that any sets of independent basis functions can be used to solve an Inverse Tidal Problem.Nevertheless, itis better to choose lower degree functions to get a better approximation of the actual physical flow near the boundaries.
The I. T. M. is a good data assimilation technique that does not require an excellent first guess, as do other intuitive methods, and allows us to estimate a set of boundary conditions that can be used to force a numerical model to yield a good approximation inside the domain of the region under study. CHAPTER IV
CHILEAN INLAND SEAWATER SYSTEM
4.1 Analysis of Tidal Data
In this section we summarize the tidal data available for the ChileanInland Seawater System of Gulf of Ancud, and analyze the general trends of tidal propagation that can be inferred from these data. The data were suppliedby. the Chilean Hidrographic Institute in the form of amplitudes and phases of extractable tidal constituents in each station.
Figure 48 indicates the location of the seven stations for whichdata are available.We note that there are no stations in the vicinity of any of the boundaries of our region of interest. Hence, neither the ocean forcings nor the link to the Moraleda. Canal (Southern boundary) can be inferred from thedata. This constitutes the most significant challenge for the modeling of the ChileanInland Seawater System of Gulf of Ancud, and has motivated our work on the development of strategies to solve the Inverse Tidal Problem.
Table 6 summarizes the information available for each station, including the length of the record from which the Chilean Hydrographic Institute derived the amplitudes arid phases of the tidal components. We note that, except for Puerto Montt, the tidal records are short (between 25 and 37 days), which immediately suggests potential problems with the accuracy of the tidal amplitudesand phases that we will be using as a basis for our work.Also, because of the differences in the length of the records for each station the tidal frequencies at which amplitudes and phases were calculated by the Chilean llydrographic Institute are notthe same for all stations. 63
lontt
iden
Figure 48. Chilean Inland Seawater System of Gulf of Ancud. Location of reference stations STATION Tidaj Frequency Fa.ro Corona Carelinapu Chacw. TtutU Puerto Moatt Chumildea Castro Amp, (cm) Ppai Amp, (cm)Phrn Amp, (cm) PhaseAmp. (cm) PhacAmp. (cm) PhaseAmp. (cm) Phe Amp, (tm) PP
Ms( 4.30 221.74 8.74 225.43 5.03242.61 15.23216.14 1.89 186.92 7.75 2S1.43 2.11 3.77
11.38 314,27 13.08 326.23 15.86 325.30 11.04 321.08 15.42331,56 13.86 332.70 14.95 329.69
K1 19.73 17.2524.05 5,46 26.41 8.2? 21.13 29.52 22.37 11.10 24.78 357,24 22,33 7,17
51.89 12.2% 87.09 30.96 163.04 40.76 ISS.79 34.36 181.93 42.17164.07 39.99159.18 35.50
S2 25.88 38.7122.58 53.89 48.44 67.13 101.17 73.48 76.52 61.05 60.02 34.68 67.60 50.70
M4 1.95 17.39 9.12 18.34 5.6 8.63 1.70298.85 0.93327.82 0,71 260.51 0,73S6.69
MS4 1.44 63.48 4.11 25.82 3.09 33.28 2.56 340.12 0.93316.00 0.95 255.32 0.48 137.1?
Length o( the liccord (Days) 25 32 37 Uukuuw 370 28 Unknown
Tablc6. I In the remainder of this section, and in the other sections ofthis chapter, we concentrate on five astronomical constituents (Msf, 01,K1,M2,S2),one overtide M4 and one compound tideMS4.These last two constituents are usually called shallow water constituents and they are a result ofthe fact that when a wave runs into shallow water its trough is retarded more thanits crest and the wave loses its simple harmonic form.The shallow water constituents are classified as overtides and compound tides, the overtide having a speedthat is an exact multiple of one of the astronomical constituents andthe compound tide a speed that equals the sum or difference of two or more astronomicalconstituents. (Schureman, P., 1958) In categorizing these data, we found a clear distinction betweenthe behavior of the astronomical and the shallow water tidal constituents. The following conclusions apply: a) Amplitudes There is a well-defined trend that the amplitudes of the diurnal and semi-diurnal constituents increase with distance inside the gulf from itsmouth at Guafo to its head at Puerto Montt. This trend is, or course, to be expectedsince the amplitude of a gravity wave increases as depth decreases, at least up tothe point where the effects of friction begin to take over. This wouldindicate that the bathymetry and geometry of the basin affects the Tidal Amplitude Relative to that of M2 (%) Constituent Faro Corona_) Puerto Mont Tidal Equilibrium 100 100 100 S2 43 42 46 N2 25 24 19 13 13 13 L2 7.4 5.5 4 14 12 58 01 9.4 8.5 41 P1 4 4 19 Mf 0.9 0.2 17.2 Msf 1.3 1 9.1 Table 7Amplitude of leading astronomical constituents relative to for Faro Corona and Puerto Montt. 67 constituents within each of these groups similarly. Table 8 gives the maximum and minimum amplitudes for each tidal frequency and the station at which these values occur.Also, it gives the value of a new parameter called "Tidal Amplification" that measures the incremental change in amplitude that the wave experiences over the length of propagation, and is defined as follows: (Maximum Amplitude - Minimum AmPlitude)} Tidal Amplification { Minimum Amplitude Tidal amplification of the diurnal and semi-diurnal constituents is small relative to that of the shallow-water constituents. Thus, while the amplitude of the astronomical constituents do increase over the length of the Gulf of Ancud, the shallow-water tides grow several times their amplitude between the stations on Gulf of Ancud and station "Carelmapu" on the western mouth of Chacao Canal. This growing is unexpected and it is not present for the other constituents.If we disregard station "Carelinapu" for the analysis of the shallow-water tides, we can clearly note that they behave as the astronomical tides do and that the effect of the bathymetry as a generation mechanism is present. At the station "Carelmapu" we note that an amplification of the amplitude of the tide is anticipated due to changes in bathymetry. This tidal amplification isat most of the order of half of the amplitude for the astronomical tides, but in the case of the shallow-water tides is almost three times for MS4 and four and a half times for M4. One possible explanation for this phenomenon is the location of the station. The station is located on the western mouth of Chacao Canal where it can be affected by strong spatial Tidal Mm. Amp.Stat. Max. Amp.Stat. Tidal Constituent cms. cms. Amplification Msf 1.89 Pert Montt 15.23Tautil 7.058 0.497 01 11.38 Faro Corona 17.04Tautil K1 19.73 Faro Corona 26.41Chacao 0.338 M2 57.89 Fa.ro Corona 188.79Ta.util 2.261 S2 22.58 Careirnapu 101.17Tautil 3.480 M4 0.71 Chumilden 9.12Carelmapu 11.845 MSA 0.48 Castro 4.11Carelmapu 7.562 Table 8Maximum and Minimum known amplitudesand tidal amplification factors in the Chilean InlandSeawater System of Gulf of Ancud. variations of velocity due to tidal currents (Clement et a]..,1988).All these factors, changes in bathymetry, friction and spatial variation of velocity, can generate the unexpected growth in amplitude of the shallow-water tides. b)Phases The tidal temporal propagation is analyzed assuming that the tidal wave can only propagate into Chilean Inland Seawater System of Gulf of Ancud through Chacao Canal (eastward), or along the western shore of Chloe Island (southward), Gulf of Corcovado (eastward) and Gulf of Ancud (northward). (Clement, et al., 1988) According to this interpretation, tides take about the same time to propagate from Ancud to Chacao through Chacao Canal than they take to propagate between the same two stations by fully contouring Chiloe island by the south. This pattern of tidal propagation was based on simple but reasonable considerations on the geometry and bathymetry of the system, and on the celerity of shallow water waves.Unfortunately, the available data set lacks quality to correlate (or not) the proposed pattern. The semi-diurnal constituents show that the propagation is north-south along the western seaward shore of Chloe Island and then it tunis inland toward Reloncavi Sound.The phases for M2 have two anomalies; first, the phase at Chumilden increases compared with Castro and based on the local bathymetry the tidal wave should arrive earlier to Churnilden because of the shallow depths near Castro. Second, the phase at Tautil increases compared with the phase at Puerto Montt and because of the distance that the wave has to travel inside Reloncavi Sound should be the opposite.The phases forS2show the same anomaly at 70 Tautil and Puerto Montt. The diurnal constituents show for the case of K1 the same anomaly in phase at Tautil and Puerto Montt and an anomalous increase in phase at Faro Corona with respect to Carelinapu.01 has an anomalous decrease inphase at Chacao compared with Carelmapu or Chumilden. The shallow-water constituents show for the case of M4 an anomalous decrease in phase at Chacao compared with Carelmapu or Castro.For MS4 there are two anomalies; first, Puerto Montt has an increase in phase compared with Tautil and second, Carelmapu has a decrease in phase compared with Faro Corona. The fortnightly constituent shows the same anomaly in phase for stations Puerto Montt and Tautil. Also the phase at Chacao decreases compared with the phase at Castro or Chumilden. All these anomalies suggest that the reliability of the data is low. The larger values of phase for five of the seven frequencies at Tautil compared with Puerto Montt strongly suggests that the data at Tautil is wrong. This conclusion is supported on the facts that the length of the tidal record at Tautil is unknown and that the length of the tidal record at Puerto Montt is longer than one year. At the same time it is well-known that Puerto Montt has been the reference station for that area for more than 40 years and that the Chilean Tide Tables used Puerto Montt as the main station for the Chilean Inland Seawater System of Gulf of Ancud. In summary, the available tidal data is scarce and poorly located for our intended application. More importantly, the data shows poor spatial 71 correlation, probably as a consequence of the short length of the records at most stations. 4.2 Definition of Boundary Conditions In the case of the Chilean Inland Seawater System of Gulf of Ancud the Inverse Tidal Problem was solved using twelve basis functions for each frequency. As we can see in Figure 49 the system has three open boundaries which we denote aswesternboundary,southwestern boundary and southeastern boundary respectively.Four basis functions were defined for each boundary:two basis functions for amplitude and two basis functions for phase. The basis functions for amplitude had the phase set equal to zero and the basis functions for phase had the amplitude set equal to a constant.The maximum value used for the amplitude of M2 was one meter.The maximum values used for the other frequencies were a percentage of the 'amplitude of M2 and they were calculated using station "Faro Corona" as the reference station because itis the most representative of ocean conditions. The maximum value used for the phase was one radian and it was the same maximum valueused for the seven frequencies. The constants used for the amplitude in the basis functions for phase were the percentage of the amplitude of M2 used as the maximum value forthe basis functions of amplitude. The amplitudes and phases of the boundary points used for the basis functions were calculated using a plane.This choice was made based on the results obtained during the validation of theI.T. M. for the rectangular embayment and North Sea. The main purpose of "a plane based" basis function was to avoid the kind of behavior for the flow fieldobserved near the boundaries for the North Sea and Rectangular Embayinent (see Figure 47).Basis functions 72 Figure 49. Definition sketch forthe boundaries in the Chilean Inland Seawater System of Gulf of Ancud 73 based on a plane are the simplest forcing functions and would not introduce any strange behavior to the boundary conditions. More complex boundary conditions are justified only when we have data available near to theboundaries that suggest that the real shape of the forcing functions is not linear. Figure 50 is a schematic drawing of a plane used to calculate the values of amplitude and phase for the boundary points along the western boundary. The projection of a boundary grid point over the plane would give us the value of amplitude or phase for that point based on the coordinates of the point and the equation of the plane.The equation of each plane was calculated using the northern and southern most points of the boundary for the western boundary and the eastern and western most points for the southeastern and southwestern boundaries.In each calculation, an extra point, along the same latitude or longitude of one of the grid points was used to ensure a unique solution. The corresponding amplitudes and phases at the nodes of the finite element grid (see Figure 49) for each one of the basis functions were obtained forcing the model (TEA-NL) in a linear fashion with the boundary conditions calculated from the planes. These values were used later to apply the I. T. M. to get the weights for the basis functions and to get a new set of corrected boundary conditions. This new set of corrected boundary conditions was applied to the model to do a non-linear run. The output of this run was used as thenon-linear guess for the I. T. M. (as described in Chapter 2 and Appendix B) and a new set of corrected boundary conditions was generated.This process was repeated until the desired difference among the known elevations at the stations and the calculated elevations was reached (see Figure 78, flow chart of I. T. M.). 74 I lii Figure 50. Schematic drawing of plane used to calculate the values of amplitude and phase along the western boundary for. the basis functions 75 4.3 Analysis of Model Results The results of the model forced with the boundary conditions calculatedusing the I. T. M. are presented in the form of time series of elevations at thereference stations (see Figure 48), charts of flow fields and charts of isolines ofamplitudes and phases for each tidal frequency. Two different combinations of stations were used for theapplication of the I. T. M. to the Chilean Inland Seawater System of Gulf of Ancud: a) all seven stations where tidal data were available b) Fivestations,excluding stationsTautil and Castro,under the assumption that these stations have wrong values of phases andamplitudes due to the uncertainty of the length of the records. Elevations Figures 51 through 57 show the time series of elevations for the knowndata and calculated data at the reference stations for seven frequencies for a period of thirteen hours (one M2 tidal cycle) using seven stations in the I. T. MFigure 58 through 64 show the time series of elevations for the knowndata and calculated data at the reference stations for a period of thirteen hours using five stations in the I. T. M. Figures 65 and 66 and table 9 have the values of the RMS error at the reference stations for the two different combinations of stations. It is clear from the time series of elevations that the fit at thereference stations is fairly good.The value of the averaged RMS error at the reference stations using seven stations was 2.89 centimeters, and using five stations was 2.95 centimeters. In both cases the largest contributions to the error were comingfrom 76 Elevations ot Station Toutil 7 (44fO--4<1 -4.42S2-U444S4) 4.0 kno. at 2.0 6 ; 04 2.0 4.0 o.o 2.0 4.0 6.0 5.0 10.0 12.0 .o in hours Figure 51. Elevations. Known datavs. calculated data at station Tautil for a period of 13 hours using seven stations Elevations at Station Puerto Moatt I known 4ato cac.4atwd data 2.0 6 - ;o.a 2.0 44 0.0 2.0 4.0 6.0 5.0 10.0 12.0 14.0 lin,e in hoorw Figure 52. Elevations. Known datavs. calculated data at station Puerto Montt fora period of 13 hours using seven stations 77 Eievotons ct Station Chacoo 7 fr.qo.nc (Ml-.41-M2-S-si4-44S4) - ko.m dtc - ccutd &ta 2.0 0.0 -2.0 -40 04) 20 4.0 6.0 &0 104) 12.0 14.0 T.i. in hours Figure 53. Elevations. Known data vs. calculated data at station Chacao for a period of 13 hours using seven stations Elevations at Station Chumilden 7 443 - inicn nta - cn4c datc 8 E 0.0 -4.0' 14.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 inhours Figure 54. Elevations. Known data vs.calculated data at station Chumilden for aperiod of 13 hours using seven stations 1I Elevations at Station Carelmapu 7 r.qenctow (taf-Gl -4(*--42-S2-4at-tS4) - known data - cotco4ated data 2.0 ; o.o -2.0 _4 A 0.0 2.0 4.0 6.0 6.0 10.0 12.0 14.0 lkn. In hea.a Figure 55. Elevations. Known data vs. calculated data at station Carelinapu for a period of 13 hours using seven stations Elevations at Station Faro Corona 7 4.0 known data colcototed data 2.0 -2.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 i Figure 56. Elevations. Known data vs. calculated data at station Faro Corona for a period of 13 hours using seven stations 79 Elevations at Station Castro 7 frwqwnnctao (Uat-01 4 - known data - catcoloted data 2.0 e E 0.0 2.0 40 0.0 2.0 4.0 6.0 8.0 10.0 *2.0 14.0 m in kou,w Figure 57. Elevations. Known data vs. calculated data at station Castro for a period of 13 hours using. seven stations Elevations at Station Tautil 7 Irequ.ncin$ 4.0 Imawn data ca4cdated data 2.0 6 0.0 2.0 0.0 2.0 4.0 6,0 8.0 10.0 12.0 *4.0 loan in Figure 58. Elevations. Known data vs. calculated data at station Tautil for a period of 13 hours using five stations Elevations at Station Puerto Monti 40 7 fr.qoedaa (MafOl--4<1M2-52--4#4*tS4) ¼no. data ca4c.ated data 7 0.0 2.0 4.0 6.0 64) 10.0 12.0 14.0 lime in ho Figure 59. Elevations. Known data vs. calculated data at station Puerto Montt for a period of 13 hours using five stations Elevations iniesfi data calculated data 2.0 f 6 -;0.0 2.0 40 04) 2.0 4.0 6.0 6.0 10.0 12.0 14.0 flnie in hou Figure 60. Elevations. Known datavs. calculated data at station Chacao fora period of 13 hours using five stations EI! Eievctions at Station Chumildn / doto coku4o(od dato Jo 4.0'----- I . 2 0.0 2.0 4.0 6,0 6.0 10.0 12.0 14.0 Thk hours Figure 61. Elevations. Known data vs. calculated data at station Chu.milden for a period of 13 hours using five stations Elevations at Station Carelmapu / usquouc,es!,N5t)I 4.0 laosn data oalcutotsd dots 2.0 0.0 2.0 4.0'- 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 l2ma 2n hours Figure 62. Elevations. Known data vs.calculated data at station Carelmapu for a period of 13 hours using five stations 82 Elevations at Station Faro Corona 7 crqonc. (M-01--4(1-M2-S2.-444-$424) 4.0 - ¼no,, cto 4cMt.d otC 2.0 6 S 0.0 -2.0 ..4fl 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 k. Figure 63. Elevations. Known data vs. calculated data at station Faro Corona for a period of 13 hours using five stations Elevations at Station Castro 4.0 . Iaom datd ca4cuøt.d dato _4Ø1 . . I . 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 Sate e I'oue Figure 64. Elevations. Known data vs. calculated data at station Castro for a period of 13 hours using five stations RMS Error at reference stations Chilean inland Seawater System of Cuff of Ancud 10.0 I I I Chocao 8.0 0 26.0 C <3 C I- Castro 2 Tautfi 0 4.0 C,, 2.0 Chumitden Carelmapu Puerto Montt Faro Corona __ I I 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8,0 Station number Figure 65. RMS-Error at the seven reference stations for a period of 13 hours using seven stations 84 RMS Error at reference stations Chilean Inland Seawater System of Cult of Ancud 10.0 Chocoo E 6.0 Castro C C S.- Tautil e 4.0 (I.) 2.0 Carelmapu Chumilden PuertoMontt I IFaro Corona I 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Stationnumber Figure 66. RMS-Error at the seven reference stations for a period of 13 hours using five stations RMS-ERROR (cms) STATION Using SevenUsing Five Stations Stations Tautil 3.83 3.92 Puerto Montt 0.64 0.64 Chacao 8.12 7.97 Chumilden 1.09 1.07 Faro Corona 0.58 0.42 Carelinapu 1.52 1.36 Castro 3.99 5.32 Table 9 RMS-Error at the seven reference stations in the Chilean Inland Seawater System of Gulf of Ancud for frequencies Msf, 01, K1, M2, S2, M4, MS4 86 stations Chacao and Castro.Station Castro is located on the eastern side of Chiloe Island in a very shallow bay and the bathymetric information is only available for a few locations. This lack of information could be the cause that the error is large, especially because the tidal amplification in this area is a very important factor to calculate the tidal amplitude. The RMS errors were large at station Castro and station Tautil when we used five stations because these stations were not used as known data points by the I. T. M. and the solution for the boundary conditions did not try to reproduce the values of amplitude and phase at these stations. Station Chacao is located on the eastern mouth of Chacao Canal on Chiloe Island and is strongly influenced by spatial variations of velocity during the tidal cycle. The spatial variations of velocity and the bottom friction could be affecting the prediction of elevations made by the model because of the non-linear interactions that have to be taken into account due to these terms in the shallow water equations. The bottom friction values were obtained using equations A.4 and A.5 (see Appendix A) and the friction factor used was the Manning factor (equation A.6). The value of the variable "n" in the calculation of the Manning friction factor was 0.25m312 2;variable "h" corresponds to a calculated depth at each grid point. Our choice of this n number is as a first guess. Our bottom friction condition used was based on typical values used by TEA-NL when the model was applied to other areas (Westerink et al., 1988). These values could be changed if the flow field does not show a well-defined physical behavior in some areas of the domain. Flow Fields Figure 67 shows the flow field in the Chilean Inland Seawater System of Gulf of Ancud using seven stations at instant equal to 6 hours of thetidal cycle. Figures 68 through 71 show the flow field of the Chilean Inland SeawaterSystem of Gulf of Ancud using five stations at instant equal to 3, 6, 9 and 12 hours ofthe tidal cycle.These flow fields were calculated combining the results of the seven frequencies used in the simulation.They show, as expected, strong currents in Chacao Canal and through the passages among the islands locatedbetween Gulf of Aucud and Gulf of Corcovado. In the case of the flow field obtained withthe boundary conditions calculated using seven stations in the I. T. M. we notice strong currents at the two southern boundaries. These currents arethe result of the boundary conditions calculated using the I. T. M. These boundary conditions had large amplitudes at the southern boundaries in order to fit the known data at the seven stations.In the case of flow fields obtained with the boundary conditions calculated using five stations in the I. T. M. we do not notice strong currents at the southern boundaries and the magnitude of the flowfield along the seaward shore of Chiloe Island is much smaller than in the previous case.The elevations obtained using only five stations in the L T. M. are much smaller than in the case of seven stations.The largest amplitudes are at the northern most points of the western boundary, instead of at the southern boundaries and southern most points of the western boundary as in the case of seven stations. In both cases we can note that the flows through the southwestern and southeastern boundaries have a more important contribution to the flow in Gulf of Corcovado than the flow along the seaward shore of Chloe Island.Another important feature that we can note in these charts for both cases is that there is a small flow at the center of Gulf of Ancud or inside Reloncavi Sound. This featurecould be a 88 Flow field '1 Instant = 6 hours il*I' II /'ihi IIIY//I/ il/f/f : liii, / / 1I 11,111,1 I I >-';.;-' Ill, 11t1 .ii_ II / ' \ '\'Z'' I I liii! I. I l/ 0 25 SOkrn I'ii 1' I/J-'/ 2.00 rn/s Figure 67. Calculated flow field for the Chilean Inland Seawater System of Gulf of Ancud for frequencies Msf, 01, K1, M2, S2, M4, and MS4 at instant = 6 hours using seven stations Flow field 1 Instant = '3 hours I 'I '2 i'/ _' / S / ,T. ''?J1:."y . '\ , - 4. 0 25 50km /1 Ii'. 2.00 rn/s 'jt1 / ;-_r.V I /-- .--- ,.,ij Figure 68. Calculated flow field for the Chilean Inland Seawater System of Gulf of Ancud for frequencies Msf,O, K1, M2, S2, M4, and MS4 at instant= 3 hours using five stations 0 25 50km 2.00 rn/s Figure 69. Calculated flow field for the Chilean Inland Seawater System of Gulf of Ancud for frequencies Msf, 01, K1, M2, S2, M4, and MS4 at instant = 6 hours using five stations. 91 0 25 50km 1 2.00 ni/s Figure 70. Calculated flow field for the Chilean Inland Seawater System of Gulf of Ancud for frequencies Msf, 01, K1, M2, S2, M4, and MS4 at instant = 9 hours using five stations. 92 Flow field Instant = 12 hours '\ o 25 50km i_ ,1 / 2.00 ni/s Figure 71. Calculated flow field for the Chilean Inland Seawater System of Gulf of Ancud for frequencies Msf, 01, K1, M2, S2, M4, and MS4 at instant = 12 hours using fivestations. 93 result of not considering the river discharges into R.eloncavi Sound. The information of river discharges into the system is not available, but certainly should be an important factor to take into consideration in the areas of Gulf of Ancud and Reloncavi Sound where a salt budget has to be kept.From the profiles of salinity published by Clement et aL, it is possible to see that for some months of the year there is an important input of fresh water into Gulf of Ancud that could generate a net flow from Reloncavi Sound toward Gulf of Corcovado. Figures 72 and 73 present the isolines of amplitudes and phases of frequency M2 respectively using seven stations in the I. T. M. These figures have to be analyzed at the same time to get a better idea of the tidal propagation and tidal elevation in the entire area that can be inferred from the results of the model. The isolines of amplitudes show that northward from Gulf of Corcovado toward Reloncavi Sound the amplitudes increase as a result of the shallow-water amplification. The amplitudes in the oceanic region (west of Chiloe Island) show a very strong influence from the boundary conditionscalculated by the I. T. M. using seven stations and they do not follow a common behavior among the different frequencies.In the case of the phases, the direction of propagation in Gulf of Ancud and Gulf of Corcoval for most of the frequencies is from North to South, contrary to what we expect to find alter the analysis of the known tidal data.Figure 74 presents the isolines of amplitude of and fIgures 75 through 78 present the isolines of phases for frequencies M2 (inland shore of Chiloe Island), M2 (seaward shore of Chiloe Island), 01 and K1 respectively for the case of five stations. The isolines of amplitude show that northward from Gulf of Corcovado toward Reloncavi Sound the amplitudes increase as a result of the shallow-water amplification. The amplitudes in the oceanic region (west of Chiloe Island) show larger amplitudes at the northern points than at the southern points of the 94 )hfUde 2 Amplitudes in meters <0.5 j 0.5 1.0 1.0 -1.5 1.5 -2.0 2.0 2.5 2.5 3.0 - 3.0 -3.5 Figure 72. Isolines of calculated amplitudesfor frequency M2 in theChilean Inland Seawater System of Gulf of Ancudusingseven stations 95 Phases in degrees c:: <0 u::. 0-60 : 60-120 120 8O 180 240 __ 240 300 300 360 Figure 73. Isolines of calculatedphases for frequency in the Chilean Inland SeawaterSystem of Gulf of Ancud using sevenstations !itudes 2 Amplitudes in meters E <0.24 :J 0.24 0.42 0.42 0.60 __ 0.60-0.78 0.78 0.96 0.96-1.14 1.14 1.32 - 1.32-1.50 - 1.50-1.68 - 1.68 1.86 Figure 74. Isolines of calculated amplitudes for frequency M9in the Chilean Inland Seawater System of Gulf of Anc'ud using five stations 97 Phases in degrees <0 0-7 7-14 14-21 21-28 28-35 35-42 -42-49 -.49 - 56 -56-63 Figure 75. Isolines of calculated phases for frequency M2 in the Chilean Inland Seawater System of Gulf of Ancud using five stations (Gulf of Ancud and Gulf of Corcovado) Phases in degrees <300 J 300 - 305 305 - 310 310 315 315 - 320 320 - 325 325 - 330 - 330 - 335 - 335 - 340 - 340 345 Figure 76. Isolines of calculated phases for frequency M2 in the Chilean Inland Seawater System of Gulf of Ancud using five stations (seaward shore of Chiloe Island) Phoses in degrees <280 £.:J 280 288 288 296 296 304 304 - 312 312 320 320 328 - 328 336 - 336 344 - 344 352 Figure 77. Isolines of calculated phasesfor frequency 01 in the Chilean Inland Seawater System ofGulf of Ancud using five stations 100 Phases in degrees <0 : 0-5 5-10 10-15 15-20 20-25 25-30 - 30-35 - 35-40 - 40-45 Figure 78. Isolines of calculated phases for frequency K1 in the Chilean Inland Seawater System of Gulf of Ancud using five stations 101 western boundary. This behavior isstrongly influenced by the boundary conditions calculated by the I. T. M. using five stations and it is the opposite of the behavior obtained from the case of seven stations. The isolines of phases show that the direction of propagation of the tides is from North to South along the seaward shore of Chiloe Island and from South to North along theinland shore of Chiloe Island, as it was expected from the interpretation made by Clement etal. The I. T. M. generated a set of boundary conditions that reproduced the known tidal data at the reference stations within tolerance, except atstations Chacao and Castro. The RMS error at these two stations was larger whenthey were not used in the solution of the I. T. M., asexpected. The I. T. M. is capable of calculating the boundary conditions for the Chilean Inland Seawater System of Gulf of Ancud as shown in Chapter 3, but it will need to have a much better geographic distribution of the data set with stations along the western boundary and southern boundaries in order to reproduce the behavior of the system in the entire domain and not only the reference stations. The right choice of stations is important in. the I. T. M. if we are trying to reproduce the behavior of a large system usingonlya few stations with a poor geographicdistribution. The results obtained using five and seven stations are a good example of how the I. T. M. can force boundary conditions that reproduce the known data at the reference stations, but generate a complete different direction of tidal propagation in the system. 102 CHAPTER V CONCLUSIONS 5.1 Conclusions The following conclusions have been obtained from this study: (1) The application of the Inverse Tidal Method as a dataassimilation technique to calculate the forcing functions of a numerical model is auseful tool. There are some limitations to this technique, such asthe choice of basis functions and the requirement of a geographically well-distributed set of stations inorder to get a good representation of the tidal propagation and flow field. (2)Combinations of trigonometric basis functionsas the forcing functions produced a faster and better fitting at the reference stations than linearbasis functions. The resulting flow field near the boundaries where the forcingfunctions were imposed was, however, affected bythe shape of the trigonometric basis functions and erratic behaviors directly associated with the functions were observed. The flow field far from the boundary did not show thisbehavior. In all cases the wavelengths of the trigonometricfunctions used were shorter than then length of the boundary. (3) The available tidal data for the Chilean Inland Seawater System ofGulf of Ancud is concentrated in Gulf of Ancud, Reloncavi Sound and the westernmouth of Chacao Canal and do not provide a good overall picture of tidalpropagation because of the length of the records and because of poor geographicdistribution of the stations. 103 (4) The result of the combined application of the I. T. M. and TEA-NL to the Chilean Inland Seawater System of Gulf of Ancud gave a credible approximation of the tidal propagation andflowfield in the region, after data from two unreliable stations were discarded. As a first step this study gave us the necessary tools to obtain in the future, with a new data set, a much better approximation of tidal propagation andflowfield. (5) The finiteelementgridhasto be refinedatthesoutheastern and southwestern boundaries to allow a better representation of theflowfield near those boundaries (see Figure 49). (6) A good field survey is the next step required to improve the understanding of the Chilean Inland Seawater System of Gulf of Ancud and to validate and test any modeling results. The field survey is outlined below. 5.2 Outline of the Future Field Survey As we already stated in the previous section of this chapter the main difficulty to model the Chilean Inland Seawater System of Gulf of Ancud is the poor quality of the available data. Additional field measurementswill provide the data to validate and calibrate the numerical model, and to determine the necessary boundary conditions. The proposed field surveys are the following: (1) Elevations:tidal gauges should be installed on the western side of Chiloe Island, Guafo Island, Guaitecas Island, Refugio Island and Gulf of Corcovado.All these tidal gauges in combination with the ones already installed, will give a more complete picture of the system and will make it much easier to calculate the boundary conditions.The new tidal gauges 104 located near the boundaries will force the I. T. M. to generate aboundary condition that will reproduce the known data at these stations aswell as the known data inside Gulf of Ancud and Reloncavi Sound.The length of the records should be at least one year to take into considerationall the possible tidal frequencies. (2) Currents: currentmeters should be installed on ChacaoCanal, northern bound of Moraleda Canal, and between Talcan Islandand Chauiinec Island (Apio Canal) to validate the flow field.The length of these measurements should be at least thirty days to take into considerationpossible differences between tidal cycles. (3) Vertical Stratification: measurements of density should be made in Gulf of Corcovado, Gulf of Ancud, Reloncavi Sound andMaullin Bay. These measurements should be made at least four times during a yearto register the seasonal changes in vertical stratification.This information will help the implementation of a future three-dimensional model. (4) River Discharge:measurements of flow discharge should bemade at Reloncavi Estuary in a continuous basis, for several years, toknow how much the seasonal river flow is into Reloncavi Sound. These measurementscould be correlated with the vertical stratification in ReloncaviSound and Gulf of Ancud. (5) Bathymetry:acoustic soundings in the entire system should be made to improve the bathymetric information used to generatethe finite element grid. 105 5.3 Outline of Next Steps in Modeling After the field survey is completed and the resultsof the simulation using TEA-NL has been validated and proven to give a goodapproximation of the tidal propagation and flow field the simulation canbe improved developing a 3- dimensional model. The new model would take into accountthe vertical stratifi- cation and would allow a much better approximationof the flow field.For this model we will require information on profiles of currents tovalidate the results at different depths. At the same time that this 3-dimensionalmodel is being developed, a transport model can be implemented using theresults of the flow field from the 2- dimensional model.Actually, there is a 2-dimensional transportmodel already implemented to use the circulation obtained withTEA-NL as the input for the flow field (Baptista et al., 1984). As an example of one of the features of the transportmodel, figure 79 presents the paths of a few particles placed indifferent points of the domain alter being advected by the flow for six months.Note that displacements are quite small, as a result of the model being driven onlyby oscillatory tidal forces. The model canalso be used to provide more conventionalrepresentations of contaminant plumes, subject to both advection anddispersion. 106 eIoncav1 /Estuary Figure 79. Pathuines of particles obtained from the flow field calculated using five stations superposed with an hypothetical river discharge at Reloncavi Estuary 107 BIBLIOGRAPHY 1. Baptista, A. M., Adams, E. E., StoLzenbach, K.D., 1984. Eulerian- Lagrangian Analysis of Polutant Transport in ShallowWater. Report No.296,R. M. ParsonsLaboratoryfor Water Resources and Hydrodynamics, M. I. T., Cambridge, Mass. 2. Baptista, A. M., Westerink, J. .1., Turner, P. J., 1989.Tides in the English Chamiel and Southern North Sea. A frequencydomain analysis using model TEA-NL. Adv. Water REsources, December1989, Vol. 12, pp. 166-183. 3. Bennett, A. F., McIntosh, P. C., 1982. Open OceanModeling as an Inverse Problem: Tidal Theory. Journal of PhysicalOceanography. Vol. 12, October 1982, pp. 10044018. 4. Brattström, H., Dali, E., 1951-1952.General Account.List of Stations. Hydrography. Reports of the Lund University,Chile Expedition 1948- 1949. Lund University, ARSSKR., N.F., Adv. 2,Bd. 46. Nr. 8. 1-86. 5. Clement. A., Neshyba. S., Fonseca. T., Silva T., 1988.Oceanographic and Meteorological Factors Affecting the Cage SalmonIndustry in Southern Chile.Seminario Internacional Organizado por elDepto. de Recursos Marinos, Fundacion Chile, Tecnicas de Cultivo yManuejo del Salmon: Desarrouos Recientes. Tomo 1, October 1988. 6. Daily, J. W., Harleman, D. R. F., 1966. Fluid Dynamics,Addison Wesley. 7. Dronkers, J. J., 1964.Tidal Computations in Rivers and CoastalWaters. North Holland Pubi. Co., Amsterdam. 8. Forsythe, G. E., Malcolm, M. A., Mover, C. B., 1977.Computer Methods for Mathematical Computations.Prentice Hall, Inc., Englewood Cliffs, N.J. 9. Ippen, A. T., 1966.Estuary and Coastline Hydrodynamics.McGraw-Hill Book Co., Inc., New York. 10. Lanczos,C.,1961. Linear Differential Operators. D. Van Nostrand Company, Ltd. 25 Hollinger Road, Toronto 16,Canada. Chapter 3. 11. Pond, S., Pickard, G. L., 1986.Introductory Dynamical Oceanography, pp. 260. Pergamon Press. 12. Schureman, P., 1958. Manual of HarmonicAnalysis and Prediction of Tides. U.S. Department of Commerce, GovernmentPrinting Office. 13. Van Dorn, W., 1953. Wind Stress on anArtificial Pond. Journal of Marine Research, 12(3). 108 14.Werner, F.E.,Lynch, R.L.,1989. Harmonic Structureof English Channel/Southern Bight Tides from a Wave EquationSimulation. Adv. Water Resources, September 1989, vol.12, pp. 121442. 15. Westerink, .J. J., 1984. A Frequency Domain FiniteElement Model for Tidal Circulation. Massachusetts Institute of Technology. 16. Westerink, J. J, Connor, J. J., Stoizenbach,K. D., 1988.A Frequency Time-Domain Finite Element Model Tidal CirculationBased on Least- Squares HarmonicAnalysis Method. InternationalJournalfor Numerical Methods in Fluids. Vol. 8, pp. 813-843. 17. Wu, J., 1969.Wind Stress and Surface Roughness atAir-Sea Interface. Journal of Geophysical Research, 74(2). APPENDICES 109 APPENDIX A GOVERN1G EQUATIONS The following description was obtained from "A Frequency Domain Finite Element Model for Tidal Circulation," Joannes J. Westerink, 1984. The equations which are used to describe tidal wave propagation can be obtained by depth averaging the Navier-Stokes equations and by making the following assumptions: a) Hydrostatic Pressure Distribution b) Constant Density Fluid c) Constant Pressure at the Air-Water Interface d) Negligible Eddy Viscosity After these assumptions the resulting equations are (Dronkers, 1964): + [u(h + [v(h + =0 A.1 + (h+q) (h+q) ut+gii_fv_4. +rb +UUx+Vlly=O A.2 'r.b(h)+uv+vv=0 A.3 where u(x,y,t) =is the component of water velocity in x direction. v(x,y,t) =is the component of water velocity in y direction. ij(x,y,t) =surface elevation relative to mean sea level. t =time h =depth to mean sea level g =acceleration due to gravity 110 p = water density f coriolis parameter = the applied surface stresses = the bottom stresses Bottom stresses are quantified as (Daily and Harleman, 1966): = v2)h/2 A.4 2)1/2 = Cf(U V A.5 where DW Darcy-Weisbach Cf Friction Factor = I Chely A.6 Manning h1"3 The fully non-linear friction terms can be approximated by linearized terms as follows: b b,lin [_h 1.7x=u A.7 [(h+q)] P p b b,lin [(h)]P = A.8. 111 where A = Cf U = linearized frictioncoefficient U = representative flowvelocity for tidal flow with only one frequency present andthe linearization being performed on an equivalent work over a tidal cyclebasis,) has the following form (Ippen, 1966): 8 "- 'MAX - Cf A.9 where UMAX= RepresentativeMaximum Velocity During a Cycle. Although the linear friction does not characterize thefully non-linear term in spreading energy to other frequencies, it can approximatethe magnitude of the actual non-linear friction term quite reasonably. Thislinearization allows us to decompose the non-linear friction term into a linear termplus a small non-linear term. Linearized friction is helpful in the fullynon-linear scheme as in iterative stabilizer. Wind stress can be represented by theempirical formulas (Van Dorn, 1953; Wu, 1963): CD . U0 . cos O A.1O = "air 112 CD .UO . sinO 4 = '°air where U10 = wind speed measured10 meters above the mean sealevel = density of air 1'air Cd = wind drag coefficient 9w = wind approachangle The applied wind stress term in the governingequations can be simplified by assuming that finite amplitude effects will beunimportant, and approximating 1/(h + ,) by 1/h, a very reasonable assumptionwhen considering the empirical nature of the formula used to represent the termand the inherent limitations of a depth averaged model in simulating wind-drivencirculation. The coriolis parameter is calculated as: f=2çsin A.12 where = rotation rateof Earth = degreesof latitude. Finally equations A.1, A.2 and A.3 become: + (uh)+ (vh) = (un) -(vu)=P' A.13 x y x y V. 113 2 21/21 nl (u +v) I A.14 (h+)ju_(uUx+VUy)=Pu 2 21/21 ni (u +v) I (h)jv_(uvx+y)=Pv A.15 where ni P = Non-linearfinite amplitude loading term Pnl U non-linear (fraction and convective acceleration) terms in the x and y directions floading P,1 Equations A,13, A.14 and A.15 can be reduced to their harmonicform by assuming that the responses as well as the right hand side can beexpressed as Fourier Series (Nf iw.t i.e. A = ReEA.e' A.16 J J Nf or A= L(Al cos(w.t) .) A.16a j=1 where lii A represents i,ii, v or P -ii. A.17 is a complex quantity A=lAjie Nf is the number of frequencies required to represent thesignificant constituents of the tidal spectrum is thethfrequency of the spectrum 114 is a phaseshiftrelative to a reference angle. If we replace equation A.16 into equations A.13, A.14 and £15and we take the time derivatives we obtain: Re{.)[ij+ (uh) +(h) =0 A.18 pth}ult} = A.19 Re{.[iw3üj+ + +fâj =0 A.20 Re{I[iwj+(g) + where =complex elevation amplitude of theJthcomponent of q(t) =complex velocity amplitudes of thethcomponent of u(t) and v(t) respectively p1 thharmonic component of continuity equation non-linear finite amplitude loading terms p1 p1 thharmonic components of the momentum equation non- linear friction, linear friction and non-linear convective acceleration loading term in the xandydirections. i (_1)1'2 It can easily be seen that the above system of equations issatisfied if: +(jh)+(h) P fori =1, Nf A.21 -fr ii= for j=1, Nf A.22 iw. +(g) + 115 + (g3) +fiiJ for,j 1, N A.23 + Equations A.21, A.22 and A.23 are the final equationsthat will be solved by the frequency domain model. The boundary conditions associated with the governingequations are elevation prescribed and normal flux (or velocity)prescribed, conditions which are respectively expressed as: q(x,y,t) q*(x,y,t)on A.24 and Q(x,y,t) = Q*(x,y,t) On. A.25 where = elevationprescribed boundary = flux prescribedboundary as shown in figure 80, elevationprescribed boundaries are usually associated with open ocean boundaries and fluxprescribed boundaries are usually associated with land boundaries (including river discharges). Since the governing equations A.1, A.2, and A.3 arebased on first principles (with the exception of surface stresses), their ability tomodel the circulation in embaytnents or estuaries depends on whether the assumptionsmade in their derivation are satisfied. The depth average iii of theequations preclude the accurate modeling of strongly stratified estuariesand/or wind induced circulation in deep water bodies. The constant density assumptiondoes not allow density 116 Figure 80. Definition sketch for typical elevation prescribed (rn) and flux prescribe (rQ) boundaries. (From Westerink, 1984) 117 driven currents to be simulated and the hydrostatic pressure assumptionrules out the modeling of short or intermediate length waves. The importanceof eddy viscosity has been deemed as negligible in most estuaries (Drorikers,1964) and hence only empirical support is required for the surface stress term. 118 APPENDIX B INVERSE TIDAL METHOD (I.T.M.) A) LjjfSymbols k = Basis Function Index Ik ,h1() =Amplitude and phase at station r,at frequency n n(X)=Amplitude and phase at station r,at frequency n, obtained 'pr' from a full non-linear run of the model (xr)=Amplitude and phase at station r,at frequency n for the basis function k, obtained from a linear run of themodel =Amplitude and phase at any node of the grid, atfrequency n, for the basis function k, at frequency n, obtainedfrom a linear run of the model. =Amplitude and phase at any node of the grid, atfrequency n, obtained from a full non-linear run of the model. =Amplitude and phase at any node of the grid, atfrequency n. =Unknown weights that multiply the linear output to correctthe arbitrary set of basis functions in order to get thedesired amplitudes and phases at each one of the stations atfrequency 119 = Frequency correspondingto frequency index n. A(kfl() Amplitude and phase at any node of the grid, at frequency n for the basis function k obtained from a linear run of themodel. a(11), bf(r) = Constants obtainedfrom and where a(r) = A(r)COSk,n(r) br) = Ar)COS A(), n() = Amplitude andphase at the known stations (data) at frequency n. a,b' = Constants obtainedfrom A(c) andn() where = An(xr) Sqn(xr) b = An(xr) sin n(cr) AFs), Fm() Amplitude and phase at any node of the grid, at frequency n, obtained from a full non-linear run of the model. F,n F,n A (), B()= Constants obtainedfrom AFfl(s) and where F,n cos A () = AFfl(s) F,n B () =_AF,fl()sin AFn(r), Fn(cr) = Amplitude andphase at the known stations at frequency n obtained from a full non-linear runof the model. 120 B" = Constants obtained from AFfl(r)andFn(r) where AFu(r)COS B11 _AF,n(r) sin complex values obtained from the linear runs A"k = Matrix that has the at frequency n, at station r and foreach basis function k. Dimension R x k) = Matrix that has thevalues of the non-linear guess minus the known data at each one of the stations at frequency n. (Dimension R x 1) B) Definition of n(xj N iw.t t) = Re > e } where i() is a complex number. i. j(x) e =ti(cosj+ i sin = ReI (cos + I sin.)(cos wit + i sin L'j=1, or I N jwti5 Re E i. e e (j=1 N = i(cos w.t cos q. - sin w.t sin.) j=1 J .1 J Finally (x,t) = .cos(.t .i=1 J 121 where Amplitude at Frequency j = Phase at Frequency j C) Our first goal is to prove that the following equationis true at the known stations: Fn() CIi qfl() ib() k=lkr= a) - Proof: We know that: Re{.j()e3}. Ifj() is complex then: = j. () +ii.. (s) 3 ,Jreai Jimag and (dropping the i's) = rea1) + ''imag() sinwt 'rea1 wt 'imag So at given data points: q11(x,t) = a'(x) cos t + b() sinwt 122 where a'(x)=rea1) b()= imag Therefore: ,,h1(x) a'(x)-ib(x) or = a(x) cos wt -ib() i sin wnt (x,t) = a(x) cos w1t + b) sin wnt. F,n Now for (x): Fn (N F,n iw.t F,n(x,t)is of the form ' (,t)= Re () e '7r r } where F,n is a complex number. .1 Then F,n F,n Fn COS Wflt Sin Wt r(x,t)= Ar() + Br () where F,n F,n Ar (r) ()=real 123 F,n. F,n B (x)= (sr) imag F,n F,n Fn 'r(x)= Ar(x) iBr' ) nih F,n F,n F 11t w11t iBr ('s)i sin '7r"(j,t)= Ar () cos F,n F,n F,n t + Br() sin '1r (,t)= Ar () cos Finally, knowing that: F,n F,n F,n (x) iBr '7r = Ar() () and = a(x) -ib(x) we can rewrite and rearrange ourfirst equation: (F,n Fn K n a(x) -ib'() Ar(x) iBr' (x) C }k=1k Known from Known from Known from Data Non-linear runs Linear Runs where Ck is the only unknown. Then: qFfl(x)+ k=dk k 124 and with this equation we can get the new set of boundaryconditions using the coefficients Ck obtained from the equation at the stations. D) Calculating Coefficients If we start with the following equation: F,n F,n =a'(x)-ib(x) {Ar() iBr()} and we define 71(xr)=A11k and ( F,n F,n a(x)-ib(x) Ar(x) iBr(x)J dr then LAK. n C n=drn k=1rk k Dropping the superscript n and writing the equation in matrix form: A.C=d 125 where the dimensions are ARx K CKXl xl and C is the matrix that contents the unknown coefficients. In order to get the values of Matrix C we will do a singular value decomposition of matrix A. A=USV* Therefore the system becomes: USV*.C=d and C = VS4Ud where the superscript * denotes a transposed and conjugated matrix. Now to get the corrected set of boundary conditions. F,n K T () = q () + E Cq"() k=1 k where all the terms on the right hand side are known. After getting ij"() we have to transform this complex number into amplitudeand phase to run TEA-NL. 126 ,1t1()= Real Part +Imaginary Part But '(x)= Amplitudecos (Phase) + iAmplitude' sin (Phase). Then: Real Part = Amplitude. cos (Phase) Imaginary Part = Amplitudesin (Phase) Amplitude =..J(Real Part)2+ (ImagPart)2 Phase = arc tugllmag Part'l LReaIPart] For the next iterations the steps are the same,but the values for F,n F,n Ar () and Br (c)have to be recalculated using the last results obtained from the full non-linear run of the model. E) Fortran Codes Used f.I.T.M. This section contains a brief description of thecodes used to solve the Inverse Tidal Problem for the Chilean Inland SeawaterSystem of Chiloe. The given examples of inputs for the differentcodes were obtained using data provided by the Naval Chilean HydrographicInstitute (LH.A.) and from the outputs of TEA-NL for several basisfunctions. Figure 81 is the flow chart that illustrates the stepsthat have to be followed to solve the Inverse Tidal Problem and to obtainthe final set of boundary conditions to apply to TEA-NL. The mathematical formulation of these codes isexplained in chapter II and in the previous sections of this appendix. 127 Figure 81. Flow chart for theInverse Tidal Method. 128 The singularvalue decomposition and some other mathematical operationsinvolvingcomplexmatriceswereperformedusingIMSL subroutines version 10 of April, 1987.The description of these subroutines can be found in the math/library manuals of IMSL on pages 282-283-284-285, 984-985, 1052-1053-1054 and 1141. All the codes work for complex matrices, but converting real matrices into complex matrices by defining the imaginary part equal to zero is trivial, and allow us to use these codes for any situation. F) Description The codes described below will solve the following problem: A. C = d (calcA.f) A usvT(svd.f) C =vsuTd(getc.f) n F,n K=12 (getbc.f '\ () = () + C) getfinalbc.f) k=1 1)calcA.f: Will use the known data at each one of the tidal gauges (known stations) in combination with the results from the model for the first guess (nonlinear-data) and the basis 129 functions (linear-data) to generate matrix A and matrix d. 2) svd.f Will perform the singular value decomposition of matrix A. This code only has to be run one time, because matrix A depends on the basis functions and they do not change during the successive iterations. 3)getc.f: Will Calculate the values of the weights for the basis functions (matrix.C) using the matrices obtained from the singular value decomposition of matrix A and matrixd obtained from calcA.f. 4) getbc.f: Will use the results from the model for the first guess (bound - ni) and the basis functions (bound - linear) at the boundary points in combination with the weights (matrix.C) to generate a new corrected set of boundary conditions in complex form (matrix.BCcomplex). 5) getfinaibc.f: Will convert matrix.BCcomplex from its complex form into a matrix that will have the new corrected setof boundary conditions in the form of amplitude and phase (matrix.BCFINAL). This new matrix will become the input for TEA-NL. After each run of the model (TEA-NL) the error betweenthe known 130 data at each one of the stations and the results fromthe model will be calculated.If this error is less than certain definedvalue the iterative process will end. If this value is not reached, anotheriteration will be done using the results of the previous run as the first guess(bound-ni and nonlinear-data). Code to calculate rnitrix.A and matrix.d caicAf I':i 1 phi (41-39.90 c This program generates th. input (or codes avd.t pbs (S)'30.96 o and g.to.t pha (01-12.21 pha)7)-35.59 do 999 i-i,1 phi (1) -pipha (1) /150, 999 continue double precision A(7),B(),tmagd(7),r.sld() o calculste elements of matrix d doubl, precision pi,p o non-linear run + dtx double pzeoision unk(7),3unk23(S4),imagA(7,17) double precision t,apreal(04), teinpimaq(S4) do 500 1-1,7 double precision ampFH,phs(7),r.alA(7,12) r.ad(l,$) unk(l),A(1),5(l) character'dO inputl,outputl resld (il-amp Ii) 'dcos (pha (1)) -A (i) 'dcoa (a (I)) wrtt.(,')'snt.r nan. of tile with n-i data' imagd(i)-amp(i)°dsin(phs(i))-A(l)'doln(a(i)) write)id,109) r.ald(i(,imaqd(1) o tile nonlinear-data is on enampla (or inputl 500 continue C read (°,$$) lnputl o calculate elements of matrix A ep.n(unit-1,status-'old' til.-inputl) o linear runs open (unit.13, status-' old', (il.-'t.st.$V) do 402 -1,S4 o tile linear-data is an example (or test $2 r..d(13,') junk23(3),teapraaiU(,tenpimagU( 402 continue writ.(',')'enter name (or output (matr&ad.$2)' 0-i read)', 99) onopatI do 501 i1,7 op.n (unit-id, status-' unknown', (ll.-outputl) k-i 0p.n(unit-1, status-' unknown', tile-'matrlxA.KZ') do 502 j-m.I4, $9 (ormat(a40) write)',') i,j,k,unh23(3(,tempreal(j),templmag() p-l.00000nOO realA (l,k( -tempreul (( 'dons (tempinag (I)) pi-4'datan (p) lmagA (1, hI -tenpr.sl (fl'dsln (tsnpimag (I)) write(',120) p1 wriOe(17,ill) i,h,realA)1,k),lmsgA(1,h( k-k +1 These ire the known values at the stations 502 contInue the phases are in d.gre.s, in cue. that rn-mw 1 the phases are in radians remove 'do 995' 501 continue smp(l(.'l.6579 120 Foroat(t25.l0) arp(2)-1.1193 109 ro:inat(2f2s.16) snp(37-l.6304 ill (ornst(212,3x,e25.16,3x,e25,16) smp(4)-1.d40 close (1) amp(S)-Q.*709 close (13) amp(d)-0.5?$9 close (16) sop (7)-i 591* close (1) phi (11-34,36 end phs(2)-42.11 pha )3)-40,76 I' I. 60211714402471004+00 0. 0.1349V773742$S000E+01 447 59934625$14$39004+00 0. 4749232902$14200C+00 0. 794 $0341327945321004+00 0. 0.710437954133140U+00 707 HO04+00 9733 1507727$ 0. O.t44SO2724$4919OO+O1 414 75544347341606004+00 0. O.1SU2124O1DiOOO+Oi 347 0.7511095737ft40004+00 0j674310410034100E+O1 222 3700E+00 43544495 75 0. O.16114SI37I$73QOt+O1 1 J LOJJSL non1inear-dat f calcA code in inputi for 1e Exam 7. +00 02 80 902 612167755 7 0. 0.11764610525831001-01 222 9000,00 4886 374 $34 53 .7 0 0.11612373069984001-01 1 0,20211314924011000+01 Q.19410296616922001+o0 897 67435000+01 63 69 19049 0. 0.31050211476219001-01 794 0,101$7155504926002.01 0.15420213233224000+01 67 40106110000+01 .16162* 0 0.I$549D9703404000+00 707 0.82705542602056010+00 0.76881?97900862010-01 796 0.21609216331076001+01 0.10232611281011000+01 414 0.56301647016336000+00 0.22701056916316001+00 707 .216401533$2100000+01 0 0.18917401467055000+01 347 0.12649020936611000+01 0.17825344913182001+01 414 21676543111$17001+01 0. 0.11191116791022000+01 222 6,12901535211004000+01 .1822S346499069000401 0 147 0.21431210623222000+01 0.1166053S206234001001 1 0.12793773057808000+01 0.20663131530073001+01 223 0.97030450666915000.00 0.951032$1766381000+00 687 0.12761239562133000+01 0.20447601212331001+01 1 0.74526643924147000+00 0.6O30549664904000-01 796 0.71640247440926001+OQ 0.13861050301272001+ol .61 0.36622793613962000+00 0,1'71$2050109949001+00 701 0.45585507113910000+00 0,75151212006429001-01 796 0.11122191271971001+01 0.10601255204633001+01 414 0.1993343717543000+00 0.21500284421265001+00 707 0.11344264442013001+01 0.11443061936234001s01 347 0.S7684111903181000+00 0.15996979601673000+01 411 69071211097351010+00 0.11115041209790000+01 O,12569866202926000+01 222 0. .1721130709$639000+01 0 341 6000401 0.tU$490t3416 0.12421743040430000401 I 0.69086600667127000+00 0.11$2026041090S001+01 222 0.62479447454270001+00 0.23236756437069000+Q0 887 0.687474108551$1000100 0.16322023262630000,01 1 1000+00 43 1602 54 0 0.97445663577260000+00 16143 0.69821160410044001+00 796 0.20274743006416002+O1 667 0.14137$79812390001+00 0.71067*12310171000+00 6.07204117228966001+00 707 0.19952071605924001-01 796 0.11349613646469001+00 0.02000214064166001+00 0.36433234600763001+00 0.30506416853761001+00 414 707 906000+00 90 6119 90 402 0. 0.3651*150602292000+00 0,11667114637174000+01 0.23461811674126001+91 414 347 61001+00 061.10572 476 0. 8.34429601325011000i00 222 0,11604260922573000+01 0,25417114324391001+01 347 6'7010151421139008+00 0. 0.2913413$332934000400 1 0.11909629110725000+01 0.272,0931649636000+01 222 -0.31298200728594000+00 O.1961303$804596000+00 0.11679405127244002+01 0.27011109203316000+01 1 III 6000+00 400 9019374 .123 0 0.1414667661$499001+Q0 796 -0.22562460766564000+00 94416011412108000+O0 0. 861 66000+00193221230196 0. 0.12266947123137000+00 -0.46161131042708000+00 0.010472143$1164000-01 796 707 -0.I$35969$262798000+00 0.28966712214411000+00 414 12151997010510000+00 -0. 0.16$34936433120001+00 707 +00 00 90 217832 46 92 59 2 -0. 0.32093044299241001+00 347 -0.79917216702S16000-01 0.10S46728691504001+01 414 9000+00 63 6244 371$ 208 -0 0.312902$3299219000t00 222 -0.56962106921317001-01 0.11632118146472002+01 347 116001+00 37675176 314 -0 0.31856569109672000+O0 1 -0.11600902864340000-01 0.12546416357$77001+51 222 -0.30183320170266000+00 O.17232951444175001+00 807 -0.75406080798706990-01 0,12406115015609000+Q1 1 001+00 9374 436 27 .24114 0 0.19619844180116000+O1 796 0.16700558066097000+00 0,81142153766313001+00 607 0.32736641234416001+00 0.10240600766939001+O1 707 621330375001+00 0,75164 0.0)71211110*264000-01 1*6 -0,174U44%'U3330001#60 0.247.3646U51038000400 414 0.36154970298933001+00 0.16475535234930000+00 707 37494532299494000+0O -0, 0.3134130U51171000.00 347 0.11062031017710000+01 0,1Q1$4411910994000+51 414 1+90 90675067700 20604 -0. 8.29523197957704001+0O 222 0.11294072903893001+01 0.10132227124771001401 347 -0.21438751493844001+00 0.29337037017870000+00 1 0.t1,1,26611641669000401 0.11703910104401000+01 222 0.22712260627103001+00 0.539901937$1411000-0l 0.11089273458058000+01 887 0,11570509362360001+Q1 1 -0.33129079414219001+00 24426671024374000+0O 0. 0.71912567966379002+00 796 0.146130167S2495910-02 887 47600+00 .0.218141568491 O.22316796128$93001+O0 707 0.11150410326026000+01 O.10069473300759000-02 700 917610001+00 04 4116 16 0. 51126390031770000+00 0 0.722094$36695000-01 414 0,322$02171$4206001-01 701 56346457146917000-01 -0. 0.S47656S2635163010-01 347 0.77651131184461001+00 O.10446937496046001-01 414 0.10910116654722000+00 0,66i35631127326010-01 222 0,7668729,912302001+00 0.10263934900234000-01 347 0. 992$l71121449500-01 S$564026317381001-01 0, 1 Li linear-data calcA.f code in tesLM2 for Example crzl'1 9D1 .1 0O+00P1006600511000 '0 00431069000000006P5)'O 0 0 t0-319000000000000)0'O C P 00-3900t0SIPLL66,ttL'0 0 0 0O+3000t000000000001O 00+200I01001fl60P6,rO- 0043L60tO0tIOlt0LL6'0 0042t6SOt990tctPtSP'0 5 0 OO+3900000600000660 0 0O+3U01000010009000'O P P tO+20$tLPt0PcttLtVO 0 0+39 0600 000900 PC506. ' 0 006. OO+200000066)0000050'0 0 0 ro+or.ocnco,apccparo 000 tO-35000Ufl1005001rO- 0 P 00+30tLI0St0P0ttI0 000 OO+3t6t6900611.9900000 OO4210010505140I65000 t0+9P06a06sU.L0Vo 00+30560600061001U0'O tO-20090060009100400'0 I P 00+ 0000109 09066006,0 0L 00-3St,00CL,00059010'O t04300000000500600000 00+20090000000006055'O Ott OO+30005006669100006'O *0 Ott oo+,'c,6cot,otootL.o CL to+3L,9u5905,00I,cro t0+290500500860500rO t0-PtC9ttPt,5tPtt".0 0O-000000690061tOL'O to+zpououucopoocro oO+300U0050P*P$000lO Ott oo+acpoLsLzptfts,oro 0043 0069 LU0000600' 90 00430000660050601000O 6 0 +*0,0010t000 0 06. tO-30006USU0060IIrO- 0o+ttu'o,'sfteo,.o 00+30006000 PP050161.6 '0 00+29050t6P00010009P0 a t OO+2556U510P009P001'0 PC oo+LDttocoott,otro to OO-300015001006099000 00-30000090000069PSL'O C C 00+ ta tgco coostit o OO+300U005009000ftc'O O0+3t,009000PZU6$0 OO+390P050Pft0660000'O- 0 C 0- tO-3L00000000$050009.0 0I. 0 0 to-'aa ttouo tcPoor 00+ 30 50 POOL 06 0005601 00+3001P0909199900600 tO-tLLtt010ttLOttVO tO-3000000U06500906'O 00 o 000 oo+3)000ft500001000rO oo+tsotospppPp,a000ro 00-30 tO000000 0060 ocr a 0O+30001090U1000000'O C 0 to-9taLUot6t90ttco IT, t0-3100000006t9tIPOIO- to-c,otpcuc.otro 1O-a00900,050010fl'LrO oo3;010up100000,oro- tO+3099000990P000000'O 0 C tO-66000000U001I.00 009 0 C to-u6000Ltp,ot$tooro OQ-36000t,000699005.P0- t0-3050006600P000001'O o o-, coos to to taco ocr0- t0-3000U00000001600'O 61 00+00006906606609C95'Q 010 9, to+360u00000n0901r0 000 0036C0C00600000 ocr a 0, To,aosttotuo,toppt.0 tO*36610000000PP0900'O tO-3000000666L060006'0 v0-tlflbocsaac000ro t0+30509000990100000'O 010 PCI 0- 9 I 0+ 30 Pt PS oo.oncottcr 0 00-39ICR CO 0909CC 00 0' 0 tO-300001000000TL tO"3051900090506066*0- tO+39900t050900P150t'O 6 0 0tO00 0 100) 0 09 t0-tcc000L09900L600rO 00-3016 06.5 TO +306000 0050 00 09 000 O0+30500000090000010'O a 0O+300P0000000CP9000,O 00 O0+30000900LtL609009'0 0O-300000010PtU50090 00-30060000000000001'O C 0 OO+000S50500000000O C, 0 0 00-300 oCO Ccctc o so r a to43*c9t0*fl9cR60 OO+3Ott9O6Pt0t6C$t0'0- 004 03 0 0 00 60 L010000100O 10 TO *35 1 tO £9 OP fltlV 0 OO+305990060P0500105'O tO-306CCPc6LCPt0506C'O- OO+3000ft*CU0006001'O 00430906 LOP 06 0 00 P 00 0 ' 0 00+i,l6CotLcao.ogtit'O p 0 O 10 00+ ,9tCO000090000'O OO+390ft0606)0006I6t'O tO-369900000116001,)0 '0- oO+t0000ctosvooalaoro C 0 oo+o,,cttpucucuoro tot 0 0 oo-R00009060L000Pro Ott Oo+390559000006069tr 5- O0*31006050000600000'O oo+tp0000t9060000000'O 0O+30066006P00000000'O 10-300005000069009000 5 0 0100 60 O0-3000500511ft60006 '0 OO+300996000t0090000'O- 00+36CS6 99 060 09 tO+30000000065U1U1 '0 O0+30100019600000065'O OTT oomc6toscos0000aro 0O+200965010000000'O 66 t0+30006950000099000'O 001 oo-00000,stoocc.6)oro OS tO+3O6O0555POSCOOUVO Ott 00001000,09000009r0 'S t0+3000000000U00000 '0 tO+36L606P6009006001'O e0000t6otcccecutoro to-3p00600IScn6905ro tO-2tLLtg09656tOI)t6 '0 t0410006100500600000O 0 1 0 ' 0 so 0O-tct000050sLCft100'O 00+300900tO P0)06009 00+30001050600000005'O I 0 00300t*cLP)Oe90ta9s'0 PS 00 +20)00000) P0 000000 '0 £ I OO+3009P100000000009'O cc 0o-0061619ptppepsp6L '0 00-39050P0601P1009POO 0O+t000ft6005190100t'O 00 000 0000000000 000 '0 OO+30P01605050600006 .0 OO+200000006900050510- I 0 05 S I oo+octuotoototou'O 0O+30)00000600000996 15 t0+300C00600 500011.0 00+ILPO6000016,t0000'O 00+310090096100000ttO 00+30000000005000000'O P1 tO'30166Cp090101090P'0- top 0O+3IPII000006IP9PU '0 tO+30106PP0006600'O oo+ts0000nt,otoe,to'o I0PL90L0L$PO$V 0- 0041000000090,3000000 C 0 to+5065000001.,000Vo 01) OO+3,P060,,,00Pf,,10.0 0 I t0+3060019600000000'O 00) t0-200UOU0960PIPOP 0 1O+30000001600006000,O 004 tt000Lt90060 t0-3010000160000600S'O 1O3056P5t0*0)CPt5$'O I I 000050000ttcpt,a0- 10+30010P1009000tOOVO 4 L .n Jvoio opo3 1.uo.1J indin C;' 21OS6923$S45 0. O.246O6U474S3S -O.23S7O134$134S7O -O.00626157339S O.O112I72$3l1.Tl3 O.O62S7$flO2235 O.19$29O312*OaH -O.0024a23oE74$254 O$O2o,51e6140a443 Q,o7o13444$UUz7 O.i32S46SO*71I73 -O.O6S4SO22$4443 O.3S1496591O562 matrikd,M2 :,9OJO1'27 calcA.f code from Output o,s 3 Singular Value DecomvosUion U, vd ', 4 u.. 01This .atrlx proqrs. A Obtalnoddo.. the *I000LM sung cod.VALUE caicA.f DIC4PO30TION 0C *****1 CALL UKACN 42, 600?) Print n.ulta ao coMputesFO* 19459.A C0040LtX sydv.r.lon I4ATRIX 01 1,0o astris April using19$? ZMSI. ubrautln.s CALl.WRITSdo pwncRN('U',URA,#M,U,wU,O) 77' (500?,') 1-1,1 '29.165-' ,IMNK do 75 3-1,7 UT(i,3)-u(3,i) o debl0doubi.000SLOdaciats pracisionco.plu C45LX v.ti.bi., UT47,74. A(LDA,$CA), o,y VT3,12),Ss(na,nou),*4NCA) 04 Psxua.t.r (NRA-?, RCA-la, Z.DA-N0A, LOU-NM, LDV-$C) 0,II$A), V(LOV,NCA) 777$ 40continue 00 i-1,7 dcostinu. 01 3-1,7 open(unit-lI,stutua-'untnown',Opafl(Oflitll,itOtOa'Unk0000',jil..'MattI*VT.H2')Op(Uitl0,It5tu9.'Unkcowfl',ti1...'matXUT.942') ti1a-'IetrjnU.HV) 9$91 continua continuewrit. (1$,kS, 107) 1,3,0(1,3)1, j,UT(i,DWRCPJI 3) ('O',l, sp.n(unit.20,opan (unit-U, statue-' stutus-' unknown' unknown' , tiie'mutrtxS tLl.-'s.utrjxV.)42') .942' doCALL $7 1-1,12 do 50 3.1,12 PNRCP.N('V' VT01(1, (I, 3)-doon3g (Viii, 3) j)v(3, ,NCA, SCA, V,WV, 0) (I$CA, 0, 1, 6) o eutrio.Afunctionther.sd valuesmatrix is A slreadyat by each ooinnna, writtanstation each outforSilo.. column theby rows,sane has tbltraryfirstto hays 07NO do 05continua 1-2,12 doconticu. 04 3-1,12 oa o on.12 of the arbitrary (unotions A is an 7x12 matrit at. th. V.1st. 01 station 533 jot u3 9$04 docontinue $2 L-t,1 wtite)17,107)wiit.)30,307)docontinue 63 3.1,12 i,3,v(1,3)(,i,VT(i,3) o do 0$ i-k,? do $0 3-1,12 WtitS(',l07)AU,read 3)-a' (9,107) (l,0)+y iuflkl2,3unkl3,x,yi,3,A(t,3) 10,19 Is (1,ala.itCi.eq.3) il-U' (1,0)+0' then (0,1)1. (1, 3) -o 4 3) 0 440$ oontinu. continua nba. ($3 6)02 do 9*continue 1-1,7 contin.do 51 3-1,12 ndit oO CONDOrS009. 4.110., ALL SINGULAR the Vt.CIOU - 12 toI.r.nc. u,.d to d.t.zrnjn. wh.n 3231 aloe.contiss.alas.close (1$) (1$)(17) continue wrlt.42O,107) 1,3,5.41,3) Oo cilstiv.inw sikgui&this spacingci,. ii has (as. bean ubroutina set to DiQ.CH) b. Lb. kurgsst009.CALL DUVCR(NP,A,NCA,A, WA4 IPATH, TOt, IPANK, 3,0, LOU, V, WV) - IODACKf4)volt. is negligibLe 107 160fornst0105.420)olos.(19) (2i2,3*,52$.1$,35,52$.l6) Ou ut from code svd.f, matrix with Singular Values of matrix A 1 21 0.00000O000O00OOOO+OQO.9593304212049944t+Q0 O.00000000000000001.00O.0000000000000000E+O0 niatrixS M2 4104 431 9 0.0000000000000006*0000,0000000500500000E+OO .0000000000000000gioO 0.0000000000000000+00O.0000000O00000000+000.0O00000000000Q00+000 .000000000000O000+OO 1iJ J 1 3 4 0.0O000000O000OO6o+Oo 0.0000000000000008C+00 413 0 .00000000600006000+60 2 21 2 6 0.0000000000000000t+50O.000OOOOO5OQOOOOQ.QO0.0O000000O0O6OQOo0O O.0000000000000000t+000.0000000000000600+000.0000000000000000E+0Q 5 21 0,0000000000000000E+000,0000000060000000+000.00000000000000000+00 0.00000000000Q0000+00000000000000000OO4000.0000000000000000t+00 110t1 9e7 0.0000000000000000t+O00.0000000000000000t+0O0.0000000000000000C+oo0.0000000000000000c+oo O.0Q000Q000O000000+000,0000600000000000C+0OO.0000000000000000c+OoQ.0000040000000060E+OO 5$ 453S 6 0.0000000000000000+0O0.60321725730677120-610,0000000000000000t400 O.00000000000000O0+O00.0000000000000000t4000.090000000000000004000.0000000000000000t+00 1121112 2 1 0.1$362$29470U344E+O1O.000000000000000ot.000.0000000000000000t+0O0.0o00000000000000+oo O.0000000000000060E+00O.0000000000000000L+O00.0O0006O000O00O00+000.0000000000000000C+0O $ 7531$10S5 9 00000000O000O00000+000.0000000000000000L+000.000OOQOQ00O000O0+OO0.0000000000000600s0O0 .60000000000000000400 00.000b00000000Q000+Q00.00000000000000O04000.0000000O0Q000000400 .00000000000000007490 27 302 1 54 0.0O00000600000O0Q+0o0.00O0O000OO000QOo+OO0.800000060000000ot+OOO.0000000000000000C+0O0.00000000006000009+OO 0.0000000000060000C+0OO.00000000000000001+0O0.0004600000000000C.0O0.0O00000000O0O000s00O.00000000000000000.000,05000000000000000+00 406513 3142 0.00000000000000006..0O000000000000000000+000.00000000000000000+oO0.0000000O000000000+OOO.0000000000000006E.Q0 O.0000000000000000EsOD0.0000000000000000g+ooO.0000000000000000E+000.00000000000000D0+000.0000000000000000+00 2122112102 8 9 0.80000000000000091+QO00.0000000000090000tsOOG.6004000000400000L400 .000000O009O0Q00Q+Q0.00000000000000Oo+øo O.0000000000000000t+0O00.00000000000000000+0o0.00060000o000o00+0o0.0000000000006000c*00 .00000000000000000.O0 $46 7$9 40 0.06000005000000001+O00.00000000000000000.000.4i11942Q03304S32-600.00000000000008050,000.00000000000000000+00 0.6000000000000000esOO0.00000000000000000+000.00500000000000000+OO0.00000000000000000s000.0000000000000000t400 3 13 3a 4 0.0006000060000000t+000.0000000000000000g+000.103063311394$305g+010.OoO0000600000000c+oo O.0000090000090000g+oo0.00008060000000000.0O0.0000000000000000+00 7010 31012 11 2 0.00000000000000000+0Q0.00000000000000000+000 .00000000000000061+00 0.000000000000003004000. 0000000000000000g+oo0,00000000000000000+000.00000000000000000.00 3 $9857 0.000000680000000o+oQO.0040004000400400E.0O0.0000000000000000t+000.0O0000O00000000osoO0.0000000000000000ts06 0O.00400000000000000+0O0.0000000GOO600000+00O.0000000000000000c+0oO.00000000000000000,000.0000000000000000g+0O .80000090090000000+0O 7 $37 4 4 0.00000000000000000,00O.00000000000000000,00O.00000000000005000.000.00000000000000060+00 0.00000000000000000+000.00000000000000000+0Q0.00000000000000001+000.00000000000000000+O50.00000000000000000+oo 4312311310 12 0.0000000006000000t+oO0.0000006000000000E+0000.0006000000000O00+00 .0000600000000000t+00.0000000O0000000g00 0 .0000000000000000C+O00.0000000000000000E+000.00000000000000001+000.0000000000000000c+oo 7337167 *79 0.000000000000000004000.00000000000000006+004.21123404702422$?6-0200.00000000000000001+00 .00000000000000003+00 0.00000000000000000,000.00000000000000000+0000.000O00000O00000OtOO .00000000000000000,00 4 434 957 0.0000000006000%3g4000.0000000000000000c+OoO.00000000000000060+QO0.00000000000000000+oo0.241H4'522002435E+000.0006000000000000C+OO 0.00000000000000600+0o0.0000000000000000E+00O.Q0000O0OOOO0006O+5O0.0060000600000006t4900.0000000000060000C+OO0.0000000000000000C+00 712 0 .000000o00oo00o00+O0 0 .00000000000000000s00 -1 J. Output from code svd.f, matrix with Singular Values of matrix A ______0. 470644312334342fl+001534001496fl13fl1+00 -0. 14370111S10237640.464142106274666.-01 1+00 matrpU1MZ 7 437 -0.10414043215215106-010.51S3351003$706710+000.247U349375123426+00 -0.51053163222430010+00-0.500211002147$7016+000.510010914051Sl11E-01 ____ 0. 9243043$2fl0427g+00 -0.3303$215574122SS1+00 7 -0.0307500442744$276-03 -0.54016417007601421-03 j -0.-0, 1$46$1$4S4173Q6I+00U642603$737440U+00447304615I337345+000. 7373*$160S6$1$0L+00 -G.12316711$21300541+00-0. 17112324ft1066510+000.0.35033106152745311+00 36170111207500316400 0.400229159991800S0-02 0.44169374664321770-03 -0. 60606565619642426-02-0.-a.-0, 4460077U$2326006-01S34160244330S64I+0O0. 492$66O0136434$0+0O 0.0.41641463006251460-01 213430177307110926470-0L65636775362527151-05 52361 43$ 62 661,00 -0 .3-0, 1331050712 101050.77060424771436420-01 6S 0.45572113042123126+0O1303 620006*00 45 2640-01 6U$12 1703 04 40 50+00 0.41171266771230200-020.4O.15110612$16171316+00 12620.2$447330540530311+000.1l753730Z04$2$26+000, 4731206151I0$6S60+00 64 66713370 51+00 -0.1233117$l$4341411*00-0.3111704$736113$1tQ00.62031314113301011-010,0.1154260111335ft01-01 4111426437011191I+000, 42062144332154100*00 -0.-0 .$046$703364500311-01 31663727$117l241+00-0 .43$650743644$2S6+000.0. 30004611137141006'01 647063404 535*16+00 -0 .11743073316603011+00.300500450142$0001-01-0.66243702713772656-010. 401420.134314fl$32124301*00 17 27 42115 4 6-01 -0.77ft31$1305l2231+00-0.2103$S11604452401-01-0. 5 675042316040.10160311564012441+000.0.5245.7250330143040-0I 32$63135115162146400 61231+00 -0.-0.12565311736345510-010. 75645101603712116+00 4173602372217420+a00.27104133$47710000+O000. 737 503070074 00 67 25254 14127306+00 57$ 41-02 -0.14717277710036161-0'.0.73507.1201$1221771-010,112S6770351707166+000.0.17131043310401411-01 2421212477462043C+00 -0.53617623356173611+00-0.16131771543267630+000.45002114SS5721060t0000.20012341051406741*00 .3174013$SS66$41E-01 -0.-0 .146 231$70174S6110771+0020577026144240136-03 5; 47770.173105$32)1540676+000.5001$436071714451-02 110.62017134207511741-01 631320+00 -0 25377-0. 22611154336413730+00 120665650.35046644911415130+0000.16417437400721046400 .35644453$11320731+00.S1S10170744021421-01 44 50.00 -0.2$421$402206631010,3li6S120S40161660+00-0.131S1247710113611+000,16000321021105001+000. 10254034116240626-01 C,)4-. 60+00 65 4121390150001 -0. 0.7700424710434421-01 .0,23$6191$1130S041+0O 0 17300+005146520654 1. -0 21-03 6064 0417007 4$ 5 0. 61034054421445271-02 -0 -0.10204034116240520-02 -0.22615244330492331+00 -0.4001$436071U4450-0l 0.2S95341O5140741+00 4.$01$34330004fl231+40 0.234314$1$33224301+00 -0.1564700)404$35111400 12921394993309011-02 0. 649072231001+0 $2 12 -0.4 200S4+00 .31339450712 -0 .1115232494100911+04 0 0.1137395141411IJQEsOO -0.$100109440510100-02 .0.104S4432$S20$10I-00 0.2194002$S40169661+00 0.3510044499405534+00 -0.34322124774120434+00 42724023724317424+04 -0. 0.2769395163951*ZU+00 024370271311USI-01 -0 0.31*21137$1175.241+0O 41fl4fl4374944410$00 o 13300412124072391+0O -0. .0.1030124702044100e00 -0.36170110070030+00 -0.11442403$73744fl0+00 0.S110*3222430090+00 024)354003$700710+00 0. 620110342$7011740-0l -0. -0.2S377420l5044S0+00 -0.111300433940141.1-O2 ,11104033447740005+0O 0 0.2103S$19404653401-01 0,40143fl27452$540-01 554321-01 *4103204 $04 0. 41330000-01402 11542 0. 90926410-00 302173015 -0 44S5071454232450t-01 -0 .3003)losl$2140311s00-0 -0.44730440513373441+00 O.9002001314757011400 0.2470434437$U3420s00 5701740010774+0O 222 0. 4001U40417lbOUtOO 0.4 7300012029921177$-02 -0. 443171343410 423 .0. -0.*260323$214262941+00 .0$7430733$s43o9o+00-0 67230140530224+00354 -0. -0.0237I1S4340400t00 -O.2174133$143163664+OO -0.1il4000244330U4L*00 0.3303395574132504+00 0.9243043$233042I+00 04324934l$1J3414+00 fl$10I1O74403)43I-0 0. -0.17310412311540611400 .5340740331107341L400 -0 -0.1136677035170170+0O -0.15644Sli*374Z01ls00 -0.55417010330043041-01 -0.30400045004341U1-00 11153120204026120+00 -0. 0011541030340244g-01 -0. 4503*135302027951-01 -0. 44044$$4044424U-02 -0. 14fl0271490237641+00 0. 15315014110113Ut+00 0. Z*421140253214$31-01 0. 440111133073t+00 3044 0. 20511626144240220-03 0. 0.35740$$lO4$4-0 0. 1 7 147212719003690-01 0. 7370007Z03$4I1S4-0 -0.44559374664021770-02 0.40022915919916050-02 420U4433$054$U+00 0. 0.2641,437450729040+00 0 7 3S064112*31141014-01 -0. -0.16050329021905050+00 45119266775330214-00 -0. 4S$73fl70fl23U+00 0. O.1485047779263232440O -O.1613*?71543207030+00 5 7 0. 4 7 1-02 5145 300*5 4104*4 -0. 4021$$00130434U+O0 -0.101603$1564012441+00 0.503010514941213S0+00 -0. 470044202334342U+00 0. 0.4430607426445250+00 -0,31121046,73697311+00 3 7 46406fl02251401-02 fr12 matrixUT matrix svd.f, code from ut Ou 2'O/Q7 I jt7 fr!. A matrix of Values Singular with 0.1103909246897306-02 -0,874012205439488fl-01 4 3 644338-0 .76*27395722 -0 946856-026056605075 .4 0 4. 037306-01 716188 57562 -0 -0.29$S$131103564$S8-01 3 $ 0.11249078350115236+00 -0.93116143433823246-01 4' -0.41953371336030376-01 -0.2630405832031948-01 3 $ -0.2$424$705$6200406*00 0.10*36833230432686+00 4' . -0.21214$15318633236+00 0.130677102$$201368+00 3 $ -0.14037164604024346+00 46.02 24 224962240105 -0 4$ 2116+00 44 4483 0* 4404 2 0. 0.6965S603597933868-01 712 0.$6043343515160418+00 4209*6+00 8037 61 21793 -0 44 *00 18 2029 91 42 50 64 202 -0. 0.22928054349464608+00 711 00 46* 66 5667 53 104643 .3 -0 40371331240536+00 .147 0 43 *00 25 2076 94247 60 0.160* -0.1573$915661$22738+00 710 0.51130037146012405+O0 -4.3441711094094*235+0o 42 -0.9762980130699669-01 -0.260$2222436273846400 1 7 -0.$02$510411264$178-01 0.4)424242919172016-01 41 0.7237H7973900I05+00 O.42629*21455321035+O0 I 7 0.14127237315633$I8-03 -0.2712791545033*34i-03 322 311 -0.44114169259063436-01 0.39391601616390796-01 7 7 -0.14)899$5356$91931-03 O.4$0406)S99461418-03 -0.60726318503729466-01 0.39060329104$11378-02 4 7 _0.92178011054256436-04 -0.9621616*141376176-04 310 -0.30376302037801271-03 -0.51837305141745446-04 76-01 333708 9805 92 .185 0 -0,1160$7$6512280176-01 S 7 3, 6-02 6 009192 609 51561 -0.2 -0.232154410621969'70-02 4 7 -0.51610926049572318-03 -0.397842113*2447916-03 2$ -0.64175534294191736-03 -0.92396)l$13464349-03 3 1 -0.4949158042894338-03 -0.1I0*043060982766-02 3, 0.43042022344304036-03 -0.355825693605$0915-03 2 7 0.44309324321*64058-01 -0.4*833466715631586-01 26 9616-01 217 21 93 9 -0 .0.16044000277411646-02 0.19354280151472696-02 1 7 0.1545614611164541$+00 $9 931 3$ 0.1290918440497138-01 0.342135367631*1486+00 612 305923107148+00 13 314 -0 0.610$021414111125+00 34 -0.32623531341199566-02 0.71019726162$599*6-01 611 -0.200462626687231)6+00 0.412009905493)3416+00 23 626+00 97 713 22804 124 -0.3 -0.120706330*733949-01 610 0.204086)7295455966+00 -0.43112690989920906+00 32 0.3007317271291886+00 0.11459131677697276+00 $ 6 0.20743133*88341996-01 0.89074413694163616-01 21 6+00 943 9707 42 94 26 133 0. 0.4316152357$082438-01 8 6 0.101I1$S$22964640t-04 6-09 614 *7 47 6)8969 215 0 212 *00 06 94 6962 284 94 647 2 -0. -0.785S61980164$7998-01 1 6 1355041136-05 1950146 -0 -°.1,fl72'3'$7,36136-0' 211 210 -0.$$8346$2971970826-01 -0.40249243495303076-01 6 6 76-06 1391664 3493064 -U -0.317$1741$$40262$6-05 0.334 21 16+00 23 1683 673$ 96 9 .37 0 -O.2S$246381667677$6-01 $ 6 -0.14471200714*13175-04 6137120035*2*5-05 -0.12$$5010047847776-04 6-01 8 913070 4144 6617 .7 0 -0.4131$$96125094$1C-01 4 6 -0.3*386353730fl1$7t-04 2$ 0.27349437135759596-02 -0.3006762690013$5$t-O$ 1 6 0.3231$711$O4lIOlfl-02 0.4371$370$01993116-03 27 6-02 98 60099 35 1124645 0. -0.499041711102723$8-01 2 6 0.42454$*3741$13606-03 6-023274354 43 9913 90$ -0 26 42161456-03 314165423 0. 31 60106+0004124 64 004 2 3 -0. -0.1209710$994915656+00 1 6 -0.U76$802$S$11*7$6-01 16*00 6004 20341* 13614 0. 24 0.29*96793349744156*00 -0.25$923212$79402$8+00 $12 -0.4$42636930663566-01 33 0.2272821049$20$$26+00 -0.22152094527033028+00 $11 355948+00 18649312620 0. 30743116+00 $3 21 14104 -0 77 3 0. -0.9164$76730390$968+00 310 0.376668$3$74562191+00 -0.44143963411213446+00 22 9106+03 69 01200134 21 0.33*67613700424515+00 -0.047701553649772ft+O0 $ S 0.51676424493456146-01 O.11131$31$31419046+00 0.0000000600000000+00 113 -0.1549083666362698+00 0.2216794220S$00196-02 $ $ 0.00000000000000008400 .00000000000000006+00 o 111 -0.12*205203762999+00 0.1197041031099128+00 1 5 0.0000000000000000c+00 0.00000000000000006+00 210 215-01 894 55 665597 310 -0 Q.16200003729019105-03 6 3 00000000000000005*oo 0. 0. O.00000000000000006+00 10 6*00 2 27 8048 151191106 0. -0.323411373*8926436+00 $ $ 00 00000000000000006+ 00000000000000006+00 0. -O.320740O0627S0$6-01 3$ 0.47169913367737516-02 _06574p133656191435L0j 4 9 0.41410$350470$10$t+00 37 6-01 8 19024291437 44 17 -0 -0.30455$23373460268-01 1 $ 0.00000000000000006+00 0. O.$10412$2074140I6+00 16 -0.465468906*473649-01 -0.24516169962297$18-01 2 $ 00000000000000006t0Q 00 6+ 32642163141762 30 0. 1$ -0.22159250146399068+00 0.13$966123005535$E*00 1 5 0.00000000000000005s00 803135064631765UtOO -0. 14 S267768-03 97 1816510 -0. 0.3991881$171335566-04 412 00000000000000000g*00 411 0.0000000000000000E+00 O.$26$7O73$233)00t-01 13 16-03 2 21912142 $93 .340 0 -O.2$442$5931953292&-03 U. 411H3$23706100 303 1$ -0. 12 -0.1394377299094726-04 0.6871405270350388-04 410 00000000000000006+00 0. 99t-01 30420721201664 0. 11 0.1903929767537478-03 0.207331314373539)6-03 9 4 00000000000000006+00 atrixVM2 ofmatrixA Values 8inular with matrix svd.f, code from Ut Ou '-4 '--4 O0+210061t609*00010- 00+3006001006U01600'0- 0111 6O-26000L0L1609101690- 00+36t069000Ut66009I till Z0-2009t6091.000500060- to-s010tu09005ftosro 0111 0093060660006001ttCt.O 10-lot)91000006U90 6 It 00t7066601166006100t0- tO-S0016Ottt09009LEO it 00420006090 ocoro 00+69090tPP6Sr0- 0It 004100116IOU 90600 P 0 0°+21600100959606900'°- , tt O0-2000UL1U60000100- t0lH60150060000069'0- Oit t0-PPSU010t609610 P0 tO-26t69050t,600LLtrO- 0II 10-3060U0,t°t,fl66r°- tO-SOU6I0000IOOL090I- O It tO-26019L000109969000- 0O-SS006911,690060rO- 0 It O0+$6009L000660166L00- 00426fl906U6510L0L0'O o to t0-010tL*0600D6101t0 to-lt000toto,otsocoro- etot OO-301001.U06000600 tO-lOUIlCft0090006I'O toot tO-201.O6L6900t0000tL0- 103t5100060099090t0 0 tot £0-50081ft$P&tt60006 t0-fl9005006I.601666t'O 01 10-S09906001L61160600- tO-206606.S011StttOOt'0 01 0o#2t0u'totoct0000rO- 0O+lC$t0000t090060000 1. 01 oo.zKtcnnutu ro- 00+3901109100U600010 O 01 00+200100000660t1.P0- 00+SttSL)IPLUtLt9P0 o 01 00-SOLO 9 006509 06 961. 0- 00+3Ctttftt00t06000'O 0 01 10-2096 009 006046 6Sr0 tO-26096t0060fl006010- 0 01 00+SttPt601L906100010- t0-260000ftlOt96000V0- o 01 oO+S0011100000s1001ro- 0O+20000tftC$t506000'0 1 01 OO+SOLO$Ut00969006t0 00+3tftct996lLtOSItI- 016 T0-390006U00 9006060 00+3t9006tt0tL001'0- 100 tO-S99010t090t6t61900- tO-$U669956UP156'0 006 t0-S006UOII0055060PO- 004361t56500001t,060'O 66 00+SOt0000000050096t0 0O+SLI.t0500900605050'0 56 t0-20t69.01690600$r0 tO-3Lftt6tt596I6S9Cr0 06 tO-3S0600t66556t.tL6i0 tO-SO 060000090ClSOt'0- 90 00+SS0060ft00691100PO- 0042969 56 LL0000 60 9000- 0- 0- 96 10-30909069I000000LL0- oO+U096L004t91060600 £101 t0-36S65lLtt66t9Lc) tO-SSt6PH6669100000' '0 06 oo+3068469t960polt0 00+39000tO 0000 110- 1101 tO-S0656I00004I1500P0- l0-300000650056tt06b tO-SSOUOOtOftOtttOO'O 10-200910060S0960900 £6 to20t6.LLE0969t0r0 10-SO 409tt66000900 0101 I 16 1O-5L006000POt60090tO tO-S pcLcsoL0l6t1ro- 601 t0-3696CLLCLt9O59PI O04SI600ttS690tOt090 oo+ntoo,co,ttott,oro toaiottt066900ctlLO- ot 00+2010696000t05691I- 0O4SL660Lt001.5t600t'0 £10 oo*S0014t,0101000c6rO 0o+n9totO,0110966r0- 0 01 00+SPIU000100900100I- t0-3069t090910H.t160'0 110 0 tO-3600U05000510900'0- 010 oo+So601ten40094coro 0O+S9161006166t06161'O 0 O0+30060166000610000 .0 69 19000 0000096O- lO-2690660060900000r0 601 IO-2000000L016609r0 00+S,t0006005090t001 to-I '0- 59 t00990000t690SOStVO ,ot 00+2t1660t0000000tOt'0 00+20009000000t9000t to-310u00611699000ro 1.9 tO-3I0L9900St00019910 to-sot000p600uto,6r0- C £1 00+20L6065009L6000000 OOs3tUOOt0000000001I '0- 10000$ '0 9* l0-I0000000060t06010O- l0590909600009069000 0 t0-ft1Pt9006601t601 i0-SS*90L09 '0 00+lt900005LOttStSOt'0- 15 0O+5006S9SI0000L6900'O- O0+506660906690660910 I 01 OO4S69019ftIP900000t V XUW JO Sflj 1flU1 tLUMXLIW 'JpAS O3 WO1J 2ttcIttO 'I O-UCPftULUtLtHL 0 CO-It00-2961669L0900990990)0-ICLCOIL900t09059r IL 6 00191 60 L6C0- 0- CI'5C, tO-300L0096PIIPLI9ILOCo-3,tsS600609L9009roeo-300L1196e000100ltCO-316L1LL95096691L60- 0- 90-fl666000UUftflP'O-Z0-IL6ICIOI666LCCrOt0-3LU0106100CTOOLrO-to-2C6tOtOfttttOPLOrO- o9C5 P9 P O0431LCt11Z1t01U0oo*apotc9,6,tc0tro-OO+3OLtt9tfttI6tObOL6tLLOt9IUtOO0+o000OOOOOQ0o00O0Oo-3tcUoU60tftco 004300040043t61t9t0901tL96600-to-uoftH00606t900ro- 39tO-1L59C3t0ft61619tC0 9909CC 69 09 0966 9 LOSt CO tO 99 65 Ott 00- OLtttOIL1* OO4I0000000000000000OtO-3190ft0095509Z999OG0+361L,L69000lPtt'00043,609 11000099 '0- OO43001CIC9690900000'00O+2t90095LPt09P990V0oo+30L1616991U05019'Ooo+2uoftCCoStoftueo-t0-26t006#OOLC000060'0 tICICP PP9 tO-tteO6c60ttt,90oO+Lftft,Lc,0L0r0to-tt9aO,s9o0S9ecr o- 00439004tO-6L0699009ft0L10-36 0009 LOS 100609 tO 99 0000ro L3960LtOtOOL6tt 0 09 16 06 C 0 0 0- CL tO-391LgSLtStUtft9VOtO-I0951L66999CC'0tO-30940069L0606U50tO-IPOSCICIIOI91069VO '0 tO-IOftOStOttCt66660'0t0-209000659056ttCOP'0to-ltts.uott600LoorO-tO-390HILLSLCOCL6CVO- tIC010*6 C0 oo+tcz,Ltozc0flVOoo+nt0oceLo60tt.OO+O0969,U,,0r0 0 ca-zucu000060n006CtO-39ZLCICCOCOL96ICO. SC 66 9090 1.0 9 9 CC 9 0 CLOL tO-3tCtSIP6CPtftLtO9Oto-ioticos,tntoscuo10-2101tLtPt$9P9LWO90-36 UCLSC ILC,60 CLV 0- t0-IPC0960LCLZ050900'0-tO-3969090599ZOL5000'0C0-I6PCC9PCIO&696016'0- I CP0L C oo+ooQ000000000000o.0oo+o6ttt0)ot0pcorO6tOIt0OtLtCtV0 00- 004319606061t0699t0-004390990L90909096L9OO4IOUI060PCNOLOUOOo+iteotootstiotcstr0-It 0- tt,ItCII019 oo+IP,00;PIOIOCflttWO-00430013619L09909019669t'O00,30000000000000000 0 COLt 606WO 00+20606066006600CtP'O0O+20909ttLttL009L9tOt0-2006L9C960L0L6906000439*tpL090fl160690'0- C CI C 00+39tt5Ltt9t9tlttOO0+3t09t0t0LtV0IO3COOL6tL00096C0110to-290flLtO0StCflLO'0to-"6610100960tt600.0 606 CL 6 90 90 90-35tO-S0069ULCO6609SCCtO-1L0C0C06191690060-C 660-ICC OttO 1.95 69 56 0$ S ICO COOS 0060EO 06 r o LII,0*69 0045t10011Tut0*000t.0tO-26P*9C090S569990I0-3L066t6010LI600St' S C0-I00969II96600069WO-tO-IPOSL9ILPLOP099CCO-to-at,n&000,op,00,ro10-000L901000IP*C00t0- 06IllOtt 0 00066009"OLOIncVo10-30to-21D6069L600,901rO 06 60 tO CCOOCP 00- Co-3665ttctpC11650WO-004219C660U0C,9000t010-3656t0-2,cLt,SCC6P6LHP tOOt 506019 We 0- 991$09'9 ?0-3000t000901U6ftP'Doo-nssto.otionotoro-tO-100905PPCCISC9109O-10-IL0000901IL000091'0t*-Is,u00105691.toPO t0-319&9C09669t6t9910tO-I501LCOLIILIP0666'0tO-3160909U96119100-t0-ftStt0000600600U0- 9*C CC 00+30000000000000000OO4OCCLLOCP460000ILrO10t90t600CL190Co-260061(.tZOftOULOO 00+100900+30lo-3696959L05L0600600-tO-269L66CL91900CL06 69 LOOt 0 09 1.6 *6 CS CO U U 69090 19 9 0 019ItoCOO 00.200000000000000.0O043$L90OLtOft999LtOOO439100601CL6610600O0+I06010066L000t969'0- o- 00+3OLCUI0611PL940IO00+IPPCICIL9096061PPO-004306006609069CLICPO00+3000960660ULIIIC'0 Cp Ct1C 0 C oo4lu,6LoLoLftOtcorO00tO-ILIOLCC09016C0509 ft tO 96 6 C 0- oo+It600toItotUtcPrO-t0-3L9000CL999L0011I'0 66CO*0 0O.I006IIO1LLC$POOU0D,s,,o,L00096C906LrOoo.3uOftuou6L6Poro 0 OO4ICOCOIOLL$I9000LWOoo+166IILOLC096LLOLrOOO4I0699L506ft06909V0 IIICII6011 1 OO43Ot00060969LL006t0043010- 410 9600906t611910-tO 9 9 r 0 l0-319566109LS0L0066CO00+306010-359L9995096C9000 0- OP 00000999010-SOC 1t600C 69S6CO60 00+itttt*60061ftUICoo+iotopzIpopIpootro$0-I99LIPLLC000P609tOtO-2ft960LLCC000996P o '6- 0O.3901089001LL9001'0oo4usLOttOftCIIttOt'000+30906900L0t99650t'0004399ft6OU601L600t'000-I6CL,1O00$CP6t6W0 0C 1I6 I 00410000000000000000.0OO.1t6SP9ti*PtIftttOt0-669'LCOt9696t09.0-tO-]SCSOLO6t56660PCCO- tO 660t0109LttO oo+3coouepopCc9oroO0-099LftCC999IPICO10-ICIOPCSL1SCOPLLIPtO-3C900000t6909Iitt'O 00- 9"09616toZIP 10-2u,.u,106UPLOrO-to-Ict.Pfl1L,0506tOrO0O+IS066006I069195UtO-IOLRPCOPPCI4LO9O 0 0 00+3POSI9SLOL09106LVOtO-219U9969LL9PL066'0t0-II9CLt6IOCIO)9P660 10C9 1 I1 U t0-ILL0606006D6060t0tO-l6169tLtOS"L99.0 V XUWW JO SflA 1r1nUTS tO-IStfl09996911109C0-00+CI0506000690V ZW LIAXUMU I't3M X13UI 'flDAS jDOO WO1 00+30000000000000000 0 to-zup,,totIztOopOro tLOfO6i 1504051149450328+00 -0 -0.$12SS31792613404+00 1112 0704451-02 911 4900554 0. 0.03*$91270O6$31fl+00 1111 0.490779"37,33950-03 0.S930300049)140205-01 1110 029513000-01 -0.5393004 -0.10021431$3005410g+oo 9 11 0.3013Q50$9300l149+00 0.091II4710241490-03 5 11 97302950+00 5042 20244 0. 0.3212095634ft0400g+go 2 11 0.3212353134799900-0I O.71019730143991$L-01 0 11 99200920+00 227252904 -0. -O.231920ft52103302g+oo 9 11 50-03 349593275121072 -0 -0.25442$59319532921-03 4 11 0.1491995$3551$1931-03 0.410406959940101$E-03 3 11 0.1990745195501U31-0$ -0.19013303U3li61fl-00 3 33 0.00000000000000005+00 0.0000000000000000400 1 11 -0.303096$2SS$9311-01 0.42U009$fl0O43SQ-01 1012 0.53040070710911321-02 0.21$4517400001313g-01 1011 55-02 197947 122 9 34 12 .7 0 -0.1003023S313S$1fl-01 1010 0.70191510$4V7$44fl-01 0.I5514$25S5459035E-01 9 10 3727993300H4H25*O0 -0. -0.074322272N930330-D1 0 20 0.160U094247207121+00 -0.1933191900152218+00 3 10 0.112423004fl39825+Q0 -Q.12070U3013329411-01 0 10 9101+00 09 99350134 177 -0 -0.S$64$7J30790$96+0O S 10 9901.94725-04 943772 13 0. 0,10114093103903$30-04 4 10 0.52S710110543$*435-04 -0.li31010014137$1,g-04 3 10 75-00 44 0010 13 53054 .34 0 -Q.213$9141$S4029350-05 2 10 .00000000000000005+00 O.0000000000000000+00 1 10 -0.12033934413192275-01 -0.011519740133$448-01 913 4545+00 09 40 113753 133 -0. -o.05294297ol949414g-o1 911 90329117414fl$771-03 0. 0.199910S7942$301U-01 910 9.2017904099649548-03 0.792019034259$319U00 9 9 1 99107*72372101-0 340 0. 0.171343$40$0$414+00 S I 9742IS0130ft90Ig-01 0. -0.20053222430332$4+00 3 9 St+00 43 91 30073172212 -0 0.1140fl37776913fl+00 0 9 -0.334l4930$424S1+00 -0.14730$$$904$3720+Q0 9 9 0.77109003151000600-01 0,29404161400706625+00 1213 -0.1903929975214735-03 0,*03131314333S399-Q3 4 9 0.13709109119244110+0O -0.235102250010$9035+00 1211 128-03 307302037S0 0. -0.$1S313051417$S44-04 93 -0347154202041714S5-01 -0.40105104730300411-01 1210 0.144112007I413175-04 0.$240233120035$211-00 2 9 -0.19030013015590731+00 -0.11901709613354110+00 9 12 o.0000000000000000r+oo 0.00000000000000000+00 1 0.66552104294490140+00 0.10191$57041309075400 0 12 -0.30013031000430121+00 0.3$31374$6211101fl-01 912 -0.21401006452442110+00 0.19$5S103S93933$ft-01 1 12 0.123ISS2S9113Sft05+O0 0.S74434$41034-01 $11 -0.129191S4404977320-01 0.24213530839414S5+00 0 12 0.05491417170930441-01 0.110U11909,35$3g-01 $10 -0.29190793345744115+00 -0.25$923313$7940250+00 9 12 -0.1$434134777400101+00 0.2I352SU2I50171+00 9 0.151051097527$7315-03 0.2991SS15171335560-04 4 12 -0.141S7037301399111090 -0.1o30ft33s104$034+oo $ -0.14937297335033155-03 -0.1712791005039150-03 3 12 -0.71370t0I739QS40It00 0.4202902$490311031$90 3 I 0.101S195$22940400-04 0.21509S9194111U4L-0S 2 12 -O.13320944i93019430+0* 0.43$01$23i7$0S243-01 0 0.00000000000900000+00 Q.00000000000000001+00 1 12 0.154901300$)02$131+00 0.22197942309001fl-02 S y I A matrix of Values Singular with matrix svd.f, code from Ut Ou L:jO.9 o Code that calculates the weights g that multiply the Basis Functions J221NV(j, 0) -rulS' (3,0) tainagi' (0,1) c thatThis multiplyprogram sash cslgule0io one at the the Oslo.. basis 01 0000tion. the weights 903993 $2219V(3,i)-(O,O)got..ndit 993 (MitThe eabroutin..eultiplicetio ((Kit version 1.0 1997) at complex metrios. is done using 3336 continue continua endit * c parameterthisinteger parameters lda,ldb,lda,nce,ncb, (are-?, nSa.?,are sly lda.7, (or ncc,nre,nrb,ncqUT'd nrb-) 1db-?, nra-i, naa-1, ldc-7) , ncb-L, (3m? and lxi) 999 do 999 i-1,12 do 999continas 3.1,3 wcite(20,107) i,3,32315V(&,3) doubledoubledoubl. somplem samplespreolsion bT)3(7,7),d(7,,U?224(i,l),V2)(13,12) 110VU732d(12,l),C2)(l2,l),13219v(1),?) maguT,realU?, reald, error 999 docontinve 49 00,12 dO 4) 3.1,12 read (14,10?) 3unk$, 3unki,reulV,mimsgV(0,3) o datavalue.Thedoobl. trror precisionthat I. er. tO. smaller re.W, arit.ri, xtag thanused V. tO. miesgd, to knownn.gl.ot real, error singular simagi at to. 4645 continue Continu. VI) -r.olv' (1,0) +ximagV' (0,1) r.ad(5,')wrte(,e)'Tnter error oriteri. to n.gl.ct value,spa.op.nopen(unit-14,status.'old'Opma(unit-13,etatu.-'old' 01 (uuit-13,(unit-iS, main. eiatua-'old'status-'oid',1' ,tile-' tile-'mstrlxUT.M2')tile-'smtrixd.w2')tll.-'matrixV.K2') mainix$.K2') callcompute DKCRCR the (0RO,NCII,UT22, muitiplioetioiu IDA, ot MiD, two OLCD, complex 5, (DO, mutnicee SiC, 04CC, U7d UT224, LOC) do 34 i-i,? do 33 3-1,?openspan (soft-3D,(unit-i?,(unit-U,(unit-l9,.tetus-'unknown', etitus-'statue-' statu.-' unknown', unknown', til.-'IatnixOZUVUTd.K2')tile-' t1l.-'uatnix$IMV.M3')tile-' metrizuTd.M2') r.etniec.K2') 110 97 continuedo 97totmut i-1,7 (l2,3x,129.l4,3m,135.l6)writ.(1S,l10( i,UT22d(i,1) 107 3433 continuetormat (202,3*, 131.10,3*, 125.16)aontinu. Sillread (i, (12, 3)-reals?' 107) 3unkl, (1,0) 3vnkZ, realS?, xLmagUT +alaagU?5 (0,1) o callcompute PMCRCO the multiplIcation(12,3,32210V,l2,7,1,OT22d,1,12,1,922INwT32d,12) 01 two complex matrlcs. S'(-l)'UTd 109 44 do 44tormatcontinue 0.1,? (3121.10) read(23,105)d(i,1)-isald r.aid,*lnaqd (1,0)vximagd' (0,1) 59 40 continuaII 0-1,12 wnit.)19,llO) i,822INV5T33d(i,l) theelementsTheTh. Zo31ouin'StrOm inversion sO with tO. to n.ro.s, densmaine tnv.:ting sad tilling only tO. diagonal linac are tO. inversion 01 Matrix C. up the neat 01 c callcompute DO4COCO the sultiplicotion (12, i2,V22,l2, sO 13, two 1,$32I9VUT22d,12, complex matrices 12,i,C23,12) V'2(NVUT4 40Matrix 37 1.-i,? is 7013, matrix i-inverse I. 1207 22 continuedo 33 1-1,12 wnit.(17,1(0) i,c22(L,l) do 20 3-1,13 r.ed)15,107) 3unhl,3snk4,real$,ximags sloeenditresin-SOit(reali.lt..rror) th.n ciouecloseclose (13)(12)(14) (17)(15) itU..q.3)elsegot.it(k.eq.3.and.r.ail..q.0) 991 .1501227J1V(3,then 1) -)i/r.ali) then (1,0) +ai.egl' (0,1) end 4-, lnd3n0 woij po O . .; : ; '0 00+3t0000t000000tOOt 00+"000000000000000.0 S. C 00+20000000000000000'0 00+30000000000000000.0 00430000000000000000'0 '6 00420000000000000000 S. P 00+200000000000000000 00+30000000000000000'0 0 00430000000000000000 '6 00+20000000000000000 S. C 00420000000000000000'0 oo+0000000000000000'o C0+20000000000000000'0 0 00+30000000000000000 6. 0 00+20000000000000000'0 00430000000000000000'O 00+30000000000000000'O 00+30000000000000000'0 S. C 00+30000000000000000'0 O0+300000000000000000 '0 00+20000000000000000 '0 00+20000000000000000 S I 00+30000000000000000'0 00+30000000000000000'0 0 00+00000000000000000 '6 00410000000000000000 0 6 00420000000000000000'0 00+300000000000000000 '0 00+30000000000000000 0 00+30000000000000000 0 C 00+300000000000000000 00+200000000000000000 00+306S.1000000S.500CC'0 '0 00+30000000000000000 S 0 00410000000000000000'0 00+20000000000000000'C 00+"0000000000000000 00420000000000000000.0 0 0 00+20000000000000000'O 00+a0000000000000000.0 0 00+20000000000000000' 00+30000000000000000'0 0 0 00+300000000000000000 00+0000000000000000 0 00+20000000000000000' 00+200000000000000000 I S. 00+20000000000000000'0 00+30000000000000000'0 0 00+30000000000000000 00+30000000000000000'0 0 6 00430000000000000000'0 00+30000000000000000'0 00+10000000000000000.0 00+20000000000000000'0 0 0 00+200000000000000000 00430000000000000000'0 0 00+20000000000000000 0 00+30000000000000000 6 C 00+0000000000000000O 00+00O00000000oo000.0 O0+30000000000000000'O 0 00+10000000000000000 6 0 O0+0O0O000000000000O 00+30000000000000000'0 oo+2oL0000t0006uoLro 00420000000000000000'0 0 6 0O+200000000000000000 00+300O000O00O00OG000 00+10000000000600000.0 00+30000000000000000'C 6 0 00400000000000000000 00+000000000000O0000 0 00+20000000000000000' 00+30000000000000000'0 C S. C0+300000000000000000 00+3000000000000000b0 0 00+20000000000000000' 0 00+30000000000000000' 01 1 00+30000000000000000'O 00+0O000o00000000000 0 0O+20000000000000000' '0 00+30000000000000000 Ct 0 OO+100000000000000000 O0+300000000000000000 0 00+30000000000000000' OQ+30000000000000000'O 00 C 00+0O00O000000000000 O0+30000000000000000.0 0 00430000000000000000' 00430000000000000000'0 01 0 00+I0O000000000000000 00+0oo00000000000000 00+30000000000000000'0 00+10000000000000000 01 0 C0+00000O0000000000 00+oo000o000000Co000 t0+l000001tt0100LttP'0 0 00430000000000000000 Ct 0 0O+200000000000000000 00+30000000000000000.0 0 00+10000000000000000 C0+30000000000000000'0 01 S. 00+300000000000000000 O0+300000000000000000 0 00+30000000000000000 0 00+30000000000000000' It I 00400o00000o000000o0 00+o00000000000000o.0 0 00+30000000000000000' 0 00+20000000000000000 It 0 00.00000000000000000 00+300000000000000000 0 00+20000060000000006' 00+30000000000000000'O It t 00+0o00000000000O000 QO+300000000000000000 0 00430000000000000000 0 00+20000000000000000' 11 P 004000000O0000000000 00+300000000000000000 00+100000000000000000 004200000000000000000 It 6 O0+200000000000000000 0o+000000000000000ro 00+00000000000000000.0 00+20000000000000000'0 11 0 00+a00000000000000000 00+30000000000000000.0 0 00+I0000000000000000' 00+30000000000000000'0 It S. 004300000000000000000 00+l0000000000000000o 004200000000000000000 0 00+10000000000000060 11 1 004300000000000000000 00+30000000000000000o 0 00420000000000000000' 0 00+20000000000000000' 01 0 0O+300000000000000000 O0+300000000000000000 0 00+20000000000000000 00420000000000000000'0 01 C 00+300000000000000000 004200000000000000000 0 00+20000000000000000 00+30000000000000000'0 p 00+0000000oo0o00000 00+300000000000000000 00+300000000000000000 0 00+20000000000000000' o ooi00000000000000000 oo+00000000000000000 00+20000000000000000'0 0 00+20000000000000000' 01 00+300000000000000000 00+30000000000000000,0 00+300000000000000000 0 00+20000000000000000' 01 00+300000000000000000 00+300000000000000000 00+200000000000000000 0 00+20000000000000000' 0 00+30000000000000000' 0 00430000000000000000' 0 00420000000000000000 0 00+20000000000000000' 0 00430000000000000000' 0 00420000000000000000' 004uo.c0uuctnctt0-3ftfl000t0Itttst .0,0 0-0 t0-z,tu,t,,,tot.cgzo-n,LsLtLttpp&gsro .0 0 t oo.utLpoucuft0O+20UflL0,tcft$01o0+aIoutco*co,ttt 0 oo4toc9olttlsp65ttst00+UttOULOPL)L0c,to-ucLtcs,ctLnc o. '0.0 -.4.-- o po thoij ndrno t0-1Itflflct0iu 'i -" LOf96: ; 0 tO-IOOflOtOttLZ600LOO 0 CO-2UCItOOCI300560S C t02000t0611tOLtftCt'0- .0 t0-2U0tc6t601tt660 C t0-UCfCdCCtNtOO tO-2OCttOUttLOt6tO 0 0- O0+I0P6C0$$0t6ILI90 '0 t0426116LCLLOL100tOt 0 '0 00+I0000000000000000 0 00420000000000000000 0 0042000000000000000.0 '0 00430000000000000000 I. 0 00+000000000000ObO0 '0 00400000000000000000 6 '0 O0+00000O00000000OO .0 OO40OO000OO000O0O0 6 004100000000006000 .0 O0+0000O0000000O000 ot 004000o000000o00000'0 00+200000000000000000 Ot 00+000000000000000'0 0 00410000000000000000 et 00*16000000000000000.0 0o+o0000000000000o0'b N- '-4 t0cttLL0,ItV0 U tO-stCtcto6,LflftO- U OO+1S6LOLU;6gttt 0 00+3tItI*I0LLtOttO 0 1O-3ft6St.tttfttO o O to-3ILpLctotu,OL,o tOIStOtLOLtOS 0 0O00CtIO0tS6tI'0 L OO+*OtttL9O*Ot 0 to-aCacotILftst0So- 9 OO+3ttO6fl4UtZO '6 to.,ut000u,ucro c oo+1,cuflssst .0- 0042L99000PL0tLIt0- 9 00+1tC60LStt000tt00 0 to-ooctc9c9ototoeoco- 00+I4t006I0t,tt0t0t0 0O+$0f.60ftftttU 0 oO4!0LcD9#9nTflt 0 t 4LOfU6 O1 pO3 uioi; indin Code that calculates the NEW BOUNDARIES , in complex notation -fr . ' 0 o DC)C43)-3)29 (20) -1121 OoC andthIs phasespragram at usd9 the boundary the £01.. polItebound-r,1.K2 ebCin.d tramwIth theOh. amplitudesBQI.'11054 a 0C121)-0041DCBC 121)(29) .1142-1140 o bait.boundaryrunThe i.d tanotlon,. bousd-lia..:.H2sulttpltostton point, obtatnsd at complexwith tram Ohs Ohs macrIse. amplItude. lInear 0.runs done Bnd sung acIngph.... Oh. 0)401 at the c a doread 10 tIle 0-1,21 bodnd-nL,H2 r.ad(1, a) bour,d,onpb000ds,phae. o o subroutInes10 wilt convert (0)431 th.asv.rsin LIlsa1.0 1967) Into a complex tar. baIb 10 QonUnd. $0 9500(1B0UHP500(i,1)-ampi100d.'dca.(pha.e)'U,O) )-BQU9(D900(3.,1)+au$Otud.'dsin(phase)*(0,1) 0ao tins)and thanset at .ulttply boundary Oh. Qondlt1on tottowtng matricesi(29x12) to get XOh. £11201) 0 non-linanu In soap).. doom. B.C. 0a 101 read(urlaco)...)I) tIle (04, bound-1i040r.M2 3o, .25.24, 3o, .25.50) o o doubt. omp1.s B0000000 (29,1) B0UNDLIblUR (29, 1) ,1),$LC238(29,t),ICrIlI,2 29,1) S 12 do 15 0'1,12 do 12BOUJSPL090AR)j,I)-onp'dooslpha)'(t,O)+anp'deln(pho)°IOtr.ad (2,') bound, amp, pha t,29 - daub).doubleIntegerdouble pr.oieionoosplsx BC(29)praoieio, CUllaap0itude,phae.,a.p,pha t.alC,laagC a to cant Lou. contInue op.n(sntt-4,st.tus-'uotnown',tI.1e-'Lattiat.IUtARC.142')sp.nlunit.3,open(unit-Z,.tatu.-'old',t1l.'.'boOnd-ttn.ar.K2')op.n(untt-1,00stus'apen(unLO.5, statue-'otd', atatu.-'knon', old', tIle-'ttl.-'bound-nt.HZ') eatrtzC.)42')111.-' mat rtxBCoompl.s.N2') 0 a dore&d 20 Ott. 0-1,12 matrinC.HZ C225U,3)-rmalC')t,00stinsgl'read (3, 1,10) (0,1) a, tee IC, lbgC oa BC(2)'4021BCboundary (1) Is -1020 the point.vector that ha. the grid paint sweDen t the 11020 tormat(L3,lu,a25.t6,3*,e2%.16)contInua BCBCIO)1023 1$)(0)(3)BC -1030-1032.0023 (3) -103) ault(p1Icat0ocall DIOCP.CB)29, 01 B000DOI$txa'C3U 12, 90069)460kB, 00, 12,1, C225, 12,29,1, BLCZ2I, 29) BC (6)BC(13)-2049BCBC(l0)103$ -1033 (11)(9) -2034 '-1020 : 30 addcontInuedo 30 BIC22S 0-1,29 wOOl, 90009500 wttt.(4,100) BC(i),B0.V22911,l) BC (15)(14)(13)BC (11)-1064'4064-1059 -1019 40 oia..(41contInuedo 40 0-1,29 wrtt)1,109)aCrIwxi.(0, DC(t),BCFIIIAL(t,1( t)900NDSOOIO,1)09I,C225 (0,1,) BCBC(1$(-101t (20)(10) -1103-1090 and010a.(1) .0 oo+I600p990L00000s0- 0O43$L0tt0tPOPT0L0IO- 0011 O0+occfto91scctLuro. oo.OcutLfl00001000ro- toll Q0410tt09909tIO- o0+306tt01000001.Lt,rO- Oott 0042tflb6coft0.0sor0- 0042,o,,tt,,S,,0or0 6011 0O*200lC0*O6900tttLO" O04ZLltILlO0LIt0G 1001 004300L9t0t00Pt60C0- 0043006ft00069LIIOOVO LIlt 0O+3tLtOtUOtUtILO 00430006980c,00L,ottO- 9111 o0+scosoom.00teo- 004 PL9occo.00toro colt 00+2ZP0600Lt90690tC'O- 0O4100Lft9S*969000000 0011 00+2ltOLOt1606lOftI90- 004n0.6c00060000ttLro- 0001 0043tLt9ftL9teuo10- oo+attoc,,osococtooro 1801 o0.06tOH6$flcULc6'0- oo.n000TLLL009000tro- 0601 0ooflbcocot&iccporo oo,166lttutcuuuo 000t 00+100016Ot0000ftOOtO- OLOI 00+3OLpOftStttLpftic0 0041,PO)tS0t,C,O0tt0 9901 00+60t6900t0t000t9C0- 10-1L00690010600060- pool O0,01*PLc0OlIpLC0tO oo.lPG000Dctt19000rO 0101 00+30600990ftttt66000- to-ofteot86010toouro 000t 10-160 0 0 00 00tOO'0 tO-l10009tUOt009ttOO- 0001 tO-lfttlHtttootOLls.0 tO-lOLft1101totL0108 1001 tO-10000060100Lft6t0.0 004394000010t1t0601V0- 0001 tO3lt086C0t0t06LIP'0 OO+3606fl0000060000!'0- 0th tO-2OL000000100000010 oo+300t,961006000101'O- 0001 cO-I0660LS00000t00000 OOflttOft0000t$tOltt'O- 0001 tO-101001.P,t0560tZU0 OO4lOtlt060t0000810tO tool tO -300 06 p to oo cr0 00+3000L00006ttLOOtt'O 0001 to-166090001ft,00ue o- tO-Z00009600080011900 0001 00416000 00tOtO £61001 0- t0-1S$t00Pt008tILP'O 1001 _i,004200060110t1.$Ottt.t0. to-OI6006IPOULOL008'o 0001 yoq poo woj 3ncirno boundaries to be used in TEA-NL (Amplitudesand Phases) Code that calculates the final Set of ge1cf otoo.(t) 4 cto..(2) C 4 tOts proqtoa 1ooh;ts. the inptttid. and phase it c nih on. it t)i boondarpoin itnq the cople* C VitU.s obtiLn,d £4Q* 004. qetbc.t and .tt.g.d to C IUttIX$C000PtC*.142 C tntegeec doibi. PCSCt.ion $creat,aCtnag,140p,phi doibts pteot.ton p,pt open(untt-1, stiti.s-'otd' ttla-'aatriaBCc*.pl.x.N2) open (int4-2, otitn-' unhnowo , ttte.oitgt$CFIN)J..NV) tend oatttCgnp)Sa.N3 p-1.00000+00 p-44atan(p) wrtte(,12Q) ph wttta(2.e)' 2$ Ml' do 10 t-1,Z$ t.ad(1,1OS) OC, OCoe;1,eCtsag OipdE(tt CtU1BCtilt+0Caq'BC1q) phi-dat.n)BC.ag/$Cr.a1) phi-dabs (phi) tt(ICr.ot.gt.Q.ind.aCtioig.tt.0) thin pha.l'pi-pha .1.. .r,dtt it)$Cr.a14;.0.and.$Cloaq.1t,O) then phn-phl+ph she. ndit tt($Cteal.1;.0.and.$C.toag.gt,0) then phi-pt-phi use ndtt vait.ge,.)wttti (2,tQs) BC,smp,pha $p,phi 40 conttnu. 105 torast(44.la,.35.40,3a,e2$,h6) 110 £otisat4t2$,10) F-' 01 F-' t0+0E0tU0c1LopT0to- oo.no:znoocpsotvo- olt t0420t0Ifl00StP0Lt0 OO43OCtstftft00000,0- titt t0sotOtto,6V0- to-2otUtsftusottco- Out oo42$0,czHo,Tz900rO- 00+3ZOOC00001CLOCtZLD OUt t042t06t0t60ZIPV0 003t0Ct000O6O0tLt0- SlIt 00420S0S$Itfl90Uf000 00+30000000LL601IOU0 CUt iO4$t00c01.0006tttt0 tO-300000010OttLP00L0 ,ttt 00410 00tOQUOftO0 0- 0041001040000400000 0011 00+3000UPt tI.LPft OtO0- 0042000t0601100C1&0 0011 t0+3t toCi 00 U ,00tt 0o+iotoo,ctttp000to coot 00+lPUt000cELtttttOD 00+ Ott00000000000 1601 oosi:,iuoctoccstioro- 004361000tSft0061LtrO 0601 oo+iouupt,o*ti,o.0 0o+24u1p,t1000DBrO OCOt O0+390000t100000000t.0- 0O+OtttOtLt40000466t6 SLOt 0043LPtt$t C.tIL$0 oo4l,COPC,oz000utcro 0001 00+310000000tOt&IItP0 00+26tStittItt6tt)i0 0901 0043Lc.00S0H6UtiL0 Oo+t,000Oc,.ozcoso 060t 00+395P0IOtSO0L6Lt00 $001 0o+l000900t051tOtO 00+1P6ttt000tC9t6Pt$ $001 00+3tItt0L0PCCtL0iV0 oo+u000,tptlocccsoro 5000 o0+3tct0t0ttt6L;tt0 oo+IICCOU190660L6,V0 0001 oo+3tssit06ot$ctaot0 00+I0050L0600LOOCI'0 0001 t0OPC,ttLiOiOOiPClL0 10-314 IP000000016004'0 0001 tO-3OtOOItOPt6Lt0000'O t0-30006t6100tOt0000'O 0001 oo+i9,co,ttttotticiro 00+306600t000069.IOS'O 0001 O0+3000ECCOI000tL0000 oo+lPtDooc00600Pooro clot oO4lL000tfl0000tCtoC0 0O+lt00500UOo000060'0 1101 oo+3OLOCt001ttOOLctt0 00+lOLt0310000016000'O 1001 to-3t,t,oIctttLztooro- 0O+3Sttt0tC66C660t0'0 0001 ;'oqi xiotj ndnO j t:js 0,40445300304472908+01 0.56953091073422466+01 2142 44200310247230251+00 0. 0.17610012033032016+01 1141 0.46714092064701406+01 0.15901190911050216.01 1140 0.59S51150179230061401 0.1426707S317210036400 1529 0.47011595271663306s01 14127043409000258+010. 1126 0.55150009305113036+01 0.94533461441960496400 1111 0.47650035621094126401 0.1,3336410199S32551+01 1116 0.$5649729002919576+01 0.04905007414495156+00 1109 0.55646014052833456+01 0.94909661130001636400 1104 0.46441069391925118+01 0.11709139240041618.401 1103 0.5584S1320774005'76401 0.05053691170034001400 50*1 0.4*?799951,1731006+01 0,97004941493217746+00 1090 0.169476073200*0978400 0.13500254034643518+01 10'0 0.51U507,3,13.4626+01 o,90414633654a32566+oo 107* 75502927153609996+00 0. 12130000331020942+o2 0 1000 0.54644726995451666+01 0.60059606069043066+00 1004 0.09249101115SZ5L+00 0.11636131643314456+01 1050 0.$1931920461061426+01 0.4095691194022706t+0O 1046 G.$49900213213002o6+00 0.2124S3254$991S216+00 1036 b.1034021031531926+00 0.2542346276902$556+00 1035 0.00290129103003216+00 0.25305211ft9330416*00 1034 516956600 16101154 0. 1700059l322I0540+00 0, 1033 61301656100640361+00 0. 0.12603419024I09fl+00 1032 0.61363924247602+0Q 5922243l454e3-Q1 0. 1030 00397410700201540+Q0 0. 1531609054030513C+01 0. 102 0.5054636S943245S3t+O0 O.9205165S5242242B+O0 1023 0.36O3O7O1444S661+00 O.I1131535754713t+00 1032 1922271212405435+Q0 0. 0.7139077363005?dlt+O0 1021 0236314571642246t+01 0. 0415985503366591L+0O 0. 1020 62 25 M2 matrixBCFINAL 31 t8 TEA-NL for Boundaries of Set final the contains file this getfinalbc.f, code from Output