Introduction to Modern Tidal Analysis Methods
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INTRODUCTION TO MODERN TIDAL ANALYSIS METHODS B.Ducarme (International Centre for Earth Tides)
By modern tidal analysis methods we mean the methods that have been developed after 1965 i.e. the computer era. After a general review we shall focus on the more recent methods.
1. Preliminary remarks
From the spectral content of the tidal curve we can derive preliminary rules. We have four main tidal families: - LP : longue période - D: diurnal - SD: semi-diurnal - TD : ter-diurnal In each of them the structure is complex. We have also some constituents in the quarter-diurnal (QD) band, typically M4, and in some places non-linear tides, called “shallow water components” in oceanography, are going up to the degree six (sixth-diurnal tides) or even more. Instrument non-linearities can produce a similar effect. We shall discuss essentially of the determination of the main diurnal (D) and semi- diurnal (SD) tides as well as the ter-diurnal (TD) ones, leaving the determination of long period (LP) tides for a more specialised approach. We want to determine the amplitude and the phase of the different tidal waves, but as explained under section 1.1 we have to put the waves in groups assumed to have similar properties and called according to the main constituent. As we know beforehand the exciting tidal potential and as we have reliable models of the Earth response to the tidal force, the goal of the tidal analysis is mainly to determine, for the main wave of each group, the amplitude ratio between the observed tidal amplitude and the modelled one (body tides) as well as the phase difference between the observed tidal vector and the theoretical one. Moreover the tidal loading due to the corresponding oceanic tidal waves cannot be separated from the body tides as they have practically the same spectrum and what we get is the superposition of both effects. The only way to determine the true body tides is to compute what is called the tidal loading vector L(L,) by integrating the oceanic tidal vector over the whole ocean, using the Farrell algorithm. Let us consider the figure 1 corresponding to a given tidal wave with a known angular speed . The x axis correspond to the in phase component cost and the y axis to the out of phase component sint. On the x axis we represent the theoretical tidal vector R(dth.Ath,0) where Ath is the theoretical astronomical amplitude (rigid body approximation) for the corresponding component, dth the ratio between the modelled body tide amplitude R and the astronomical amplitude. For strain, tides in wells and displacements there is no astronomical tidal vector and R is directly computed for a given Earth model.
The observed tidal vector A(Ao ,), rotating at the same angular speed, but with a phase advance . We have generally the relation Ao = d.Ath. In the least square approach d and are the primary unknowns (section 3.1). When there are no astronomical tides we shall define Ao as d.R. The discrepancy between the observation and the model is thus the residual vector B(B,) B = A – R (1.1) If we represent the oceanic tidal influence as a vector L(L,) We can construct a modelled tidal vector Am(Am,m), with generally Am = dm.Ath Am = R + L (1.2)
or a corrected tidal vector Ac(Ac,c), with generally Ac = dc.Ath Ac = A –L (1.3) And compute the final residue X(X,) X = B – L = A – R – L (1.4)
1.1 Separation of the waves
In principle two tidal frequencies can be separated on an interval of n equally spaced observations if their angular speeds differ at least by 360°/n (Rayleigh criterion). It is equivalent to require that their periods are commensurable on the interval T=n*sampling rate T T1.T2 (1.5) T1T2 You can beat this Rayleigh criterion by a large amount if you know the frequencies a priori. The amount depends on the signal to noise ratio. In practice the separation is already numerically possible when you have only 50% of the basic interval. The main implications of this principle are: a) There is a minimum length for the filters to be used either for the separation of the tidal bands or for the elimination of the low frequency spectrum, generally called “drift”. b) We must associate the tidal waves in groups for which we suppose constant tidal parameters for the amplitude and phase.
1.2 sampling rate
It has been shown that a two hours interval between samples is well sufficient for the earth tides analysis but the usual procedure is to use hourly readings, that allow to go reach harmonics of M2 up to of 12 cycles per day. Half hourly readings do not produce any improvement. However the aliasing problem implies that all the spectral content with frequencies higher than 0.5 cph (cycle per hour) which correspond to the Nyquist frequency has been removed from the records before decimating to hourly values. In the past, analog recordings were hand smoothed by experienced operators prior to digitisation, as it was not possible to build electronic filters with a so low cut off frequency. It is indeed the great advantage of the modern digital data acquisition systems to go to high sampling rates that are numerically decimated up to hourly sampling. It is usual to use electronic filters with a 30 seconds time lag and it is then sufficient to use minute sampled data. This sampling rate has been conventionally adopted by the network of superconducting gravimeters of the “Global Geodynamics Project” (GGP), even if most of the stations use higher sampling rates from 1 to 10 seconds. A first decimation is then applied on the original data to get one minute sampled data.
1.3 Causal or non-causal filtering
The tidal signal being purely harmonic we can use non-causal filters. It means that to evaluate some parameter at the instant ti we can use observations performed at any instant ti-k as well as ti+k with k0. It is not the case for example in seismology. To determine the arrival time of a seismic phase at a moment ti we can only use the preceding data. For tidal analysis we can thus build even or odd filters centred around the central epoch which will be associated to the cosine or sine functions of the harmonic constituents.
2. General introduction to tidal analysis
We shall first define several classes of analysis methods and show how they derive from specific assumptions and see if they take advantage from the fact that the spectral content of the tidal generating potential can be theoretically computed.
2.1 The transfer function
The goal of any tidal measurements is to determine the response of the Earth to the tidal force F(t) through an instrument, using a modelling system. In the output O(t) we cannot separate what in the physical system response is due to the Earth and what is due to the instrument. We must thus determine independently the transfer function of the instrument in amplitude as well as in phase. This operation is often called the calibration of the instrument. The impulse response of the system Earth-instrument s(t) is the unknown. According to the Love theory we suppose it is linear. We can also reasonably suppose it is stable and time invariant. As we know that the input has an harmonic form, we can consider the global system as non causal. In these conditions, for a single input F(t) we can write the predicted output p(t) by a convolution integral under the form p(t) s().F(t).d (2.1) In the frequency domain the Fourier transform gives p()s().F() (2.2) As in our case p(t) and F(t) are pure harmonic functions, their Fourier transforms exist only for a finite data set of length T, often referred as data window. In practice this is not a restriction as we cannot produce infinite series of observations. The transfer function s() is a complex one s() s().ei() (2.3) We generally refer to s() as the amplitude factor and to () as the phase difference between the output and the input. The introduction of a rheological model to describe the transfer function of the instrument T() T().ei() (2.4) will give the true transfer function of the Earth s() E() (2.5) T() ()()() (2.6)
2.2 Direct solutions of the linear system (spectral analysis)
If we consider that our modelling system can fit perfectly the physical system i.e. that there are no errors in the observations, we put O(t) p(t) (2.7) Replacing in equation (2.1), the evaluation of the Fourier transform will give us s()O()/F() (2.8) where O() is obtained for a data window [-T/2,T/2].
We can also use the autocorrelation functions COO() and CFF(), defined as follows
T / 2 () O(t).O(t).dt C OO lim (2.9) T T /2
T / 2 () F(t).F(t).dt C FF lim (2.10) T T /2
The corresponding Fourier transforms COO() and CFF() generally, called power spectra, will give us an evaluation of s() as 2 COO() s() (2.11) CFF ()
With these spectral techniques we have no error calculation as we supposed errorless observations. However we can evaluate the mean noise amplitude around the tidal spectral lines and deduce signal to noise ratios
A=/A (2.12) where A is the amplitude of the tidal wave. The error on the phase difference is then
=(/A)rad (2.13)
These spectral methods are generally time consuming and they do not take advantage from the fact that we do know beforehand the spectral content of the input function F(t).
2.3 Optimal linear system solution
If we consider the observational errors we can try to optimise our representation by minimizing the residual power. Let us define residuals as e(t)=o(t)-p(t) (2.14) We want to get
2 2 1 .dtmin (2.15) e limT e(t) lT By introducing (2.1) we get T /2 2 2 1. .dt (2.16) e limT (O(t) s().F(t).d) lT T / 2 The only unknown being s(), the minimizing conditions are e 2 2e 2 0, 2 0 s() s() This yields directly T /2 2 1. 2 .F(tu)dt0 (2.17) limT (O(t) s().F(t).d) lT T / 2
By using the definitions of the auto- and cross-covariance functions we write
T / 2 (u) s(). (u).d C OF lim C FF (2.18) T T / 2 This equation is known as the Wiener-Hopf integral equation
Its Fourier transform is expressed in the frequency domain as
()s(). () C OF C FF (2.19)
Two classes of tidal analysis methods derive directly from the Wiener-Hopf integral equation - in the time domain the so called response method; - in the frequency domain the cross-spectral method.
Both of them start from the time domain representation of the tidal input F(t). - In the response method we write (2.19) under its matrix form for a finite data length. We must define the sampling interval of s() and its length n. s() is the “impulse response” defined as a linear combination of the original readings o(t) under the form n s(ti)c j.o(ti(j 1) ) j 1 (2.20) Increasing its length allows more oscillations of the function s() and increasing the sampling interval decreases the bandwidth for which s() is defined. In practice equation (2.19) is solved independently for the different tidal families (D,SD,TD). This method was used at the Tidal Institute in Liverpool and also at the Royal Meteorological Institute of Belgium. The variation of the tidal amplitudes and phases are approximated by smooth functions of the frequency and the response method is not suitable to model the liquid core resonance effect in the D band. - The cross-spectral method was seldom applied and leads to very similar results.
2.4 Harmonic analysis methods
Let us consider the input function F(t) through its spectral representation F() according to the general formula
2it F(t) F().e .d (2.21) which becomes for a discrete spectrum such as the tidal one N 2i j t F(t)1/2 F( j). j N e (2.22) We can introduce this representation into equation (2.19) or directly into (2.17) and obtain for the real and imaginary parts of s() a set of classical normal equations as in the least square adjustments methods. We shall describe later on the corresponding observation equations. This harmonic formulation based on a frequency domain representation of the tidal force is the most widely used. Differences appear in the formulation of the observation equations, in the prefiltering procedure and in the error evaluation. Some authors treat separately the different tidal bands (Venedikov, 1966;Venedikov, 1975; Venedikov, 199?; Venedikov & al., 2003). Others try a global solution and differ mainly by the design of the filters: direct least square solution with a minimum filter length (Usandivaras and Ducarme, 1968), Pertsev filter on 51 hours (Chojnicki, 1967) or long filters with steep slope and high rejection (Wenzel, ????). As a matter of fact the results are very similar but large differences can appear in the error evaluation through the RMS error on the unit weight so. Least square solutions generally underestimate the errors as they suppose a white noise structure and uncorrelated observations i.e. a unit variance-covariance matrix. This second hypothesis cannot be replaced as we do not know the real variance- covariance matrix. The first hypothesis is certainly not valid as we know from spectral analysis that the noise is a so called coloured noise. Moreover there is often a noise accumulation in the tidal band, a so called cusp effect due to the imperfect representation of the tidal potential, non linearities and leakage effects. This first problem has been solved using more detailed tidal potential developments. The methods dealing separately with the different tidal bands as VEN66 lead to more realistic error estimates as they approximate the noise level independently in each band. Another way is to estimate directly the noise by a spectral analysis of the residuals and “colour” the error estimated under the white noise hypothesis (ETERNA3.4, Wenzel, 1997). The “HYbrid least squares frequency domain CONvolution” (HYCON, Schüller, 197?) is studying the variations of the real and imaginary parts of s() in consecutive partial analyses of the data.
More recently (Ishiguro & al., 1981) developed a general method including both harmonic analysis and response method: BAYTAP-G. It allows to determine not only the tidal constituents but it is also evaluating the aperiodic “drift” signal. Moreover additional “hyperparameters” allow to adjust the degree of “smoothness” of the solution. A final version of the program has been developed at the National Astronomical Observatory in Japan.
3 Least squares analysis of the tidal records
A least squares solution requires a complete mathematical representation of the phenomenon through the development of the tidal potential. We can write equations of observation corresponding to each reading l under the form i li vi f(x,y,...) (3.1) where x,y... are the estimated unknown tidal parameters, e.g. the amplitude and the phase of the wave groups, and the vi are the estimated observation residuals. The solution is obtained from the normal equations giving the most probable estimates of the unknowns and the corresponding errors under the assumption that they are normally distributed and uncorrelated. As a matter of fact the tidal records are disturbed not only by accidental errors vi but also by a non stationary noise function g(u,v,…) which is related to the instrumental drift, the meteorological effects and so on. The equations of observation become li vi f(x,y...)g(u,v...) (3.2) However we can reduce the solution of the system (3.2) to the solution of the system (3.1) by using a filtering process. If we apply analogical and numerical filters to the records in such a way that frequencies external to the tidal spectrum are removed we can write l'iv'i F(li) f'(x,y,...) (3.3) and then use a classical least squares solutions.
3.1 Constitution of the observation equations
We can write at epoch ti equation (3.1) under the form T T li vi d j.Aj,k.cos(i, j,k j) (3.4) j k where i specifies the epoch j the tidal group k a wave inside a tidal group Aj dj= T is the ratio of the observed to the theoretical amplitude of Aj the group j T Aj,k the theoretical amplitude of a tidal constituent k inside of the group j T i, j,k the argument at instant ti of a tidal constituent (j,k)
j the phase difference supposed constant inside group j
We have to solve the system (3.4) for the unknowns dj and j. In practice it is usual to take as auxiliary unknowns the odd and even components of the wave groups j d j.cos j (3.5) j d j.sin j (3.6) and we get li vi (aij j bij j) (3.7) j where
T T aij Ajk.cosijk (3.8) k T T bij Ajk.sinijk (3.9) After resolution of system (3.7) by various methods (Venedikov, 1966;Wenzel,1997) we return to the original unknowns 2 2 2 d j j j (3.10) j j arctg( ) j
The associated errors can be computed from the unit weight error s0 given by the least squares solution and the elements of weight matrix Q 2 2 ( .Q 2Q Q sd so d 2 (3.11) 2 2 ( .Q 2Q Q s so d 2
3.2 Implicit assumptions of the least squares solution
The following implicit assumptions should be verified by the data set
3.21 The model is complete It means that no energy from deterministic signals is left in the data outside the tidal frequencies. It follows that all harmonic constituents coming from the pressure or eventually the oceanic shallow water components should be included in the model. In practice it is never satisfied for the pressure waves such as S2 and its harmonics. 3.22 The unknown parameters are stable.
It means that the amplitude ratios dj and phase differences j must be constant over the whole recording periods. The changes of sensitivity or instrumental delays have thus to be carefully monitored.
3.3 Sources of perturbation of the tidal parameters
3.31 Noise and harmonic signals inside the tidal bands To minimize the noise in the tidal bands, especially the diurnal one, the instruments should be carefully protected against meteorological effects. As the changes of pressure induce significant gravity variations, of the order of –3nms-2/hPa, they should be recorded as auxiliary signal and included in the model. For gravity meters it is significantly improving the unit weight error s0. Temperature variations can also be introduced in the model. There is no possibility to eliminate an harmonic signal witch has the frequency of a tidal wave. Tidal oceanic loading is the best example of such a superposition. The only possibility to get rid of it is to model these “indirect” effects and subtract them from the tidal vector. A similar problem arises with the planetary pressure wave S2 and its harmonics which have a lower efficiency ( -1nms-2/hPa) than the pressure noise background and is not perfectly eliminated from the corresponding tidal groups. 3.32 Noise and harmonic signals external to the tidal bands The smoothing and the filtering must really eliminate the noise outside the tidal band to avoid eventual “leakage” effects. We call leakage the perturbation of the signal on one tidal frequency of the energy present in another spectral band. It can happen when the side lobes of the filter transfer function are important. The effect could be very dangerous if there is an harmonic signal present in one of the side lobes. Happily, in tidal analysis, we know beforehand the existing signals and can build the filters accordingly. 4. General methodology of data preprocessing
Much care should be devoted in tidal analysis to the preparation of the data and the calibration of the instruments (determination of the transfer function in amplitude and in phase) as well as to the detection of the anomalous parts of the records in order to process only data as fitting as possible to the requirements outlined before in 3.2. Depending on its frequency the noise can be eliminated or at least reduced at different steps of the method, by filtering or modelling. It should be pointed out that a spike or a jump produces noise on the entire spectrum and should be eliminated prior to any filtering. The filtering prior to least square analysis aims at separating the tidal bands and one uses thus generally band-pass filters.
4.1 Elimination of the noise external to the tidal band
4.11 Energy at frequencies higher than the Nyquist frequency (T 2h or 0.5cph) To avoid aliasing effects this part of the spectrum must be suppressed. With modern Digital Data Acquisition Systems (DDAS) we use a sampling rate of one acquisition per minute as a minimum. The electronic signal has to be filtered accordingly. The following decimation steps up to hourly readings require a correct digital filtering with an appropriate transfer function, to avoid any aliasing and smooth out the noise. 4.12 Energy at frequencies higher than the main tidal bands (T 8h or 0.125cph) This part of the spectrum is frequently referred to as “noise” but can incorporate harmonic constituents due to non linearity effects in the instruments or shallow water components, as well as the harmonics of the pressure wave S2. It is easily eliminated by a low pass filtering prior to analysis. 4.13 Energy at frequencies lower than the diurnal band (T 30h or 0.03cph) This part of the spectrum is conventionally called “drift” although it contains the long period tides. Besides the separation of the three main tidal bands the filtering of the data prior to least squares analysis aims at the elimination of the drift. Of course it is supposed that the drift is a smooth function and jumps should be eliminated prior to filtering. 4.14 Energy between the tidal bands It is the most difficult to eliminate in the filtering procedure. Part of it will remain in the filtered data
4.2 Elimination of the perturbations inside the tidal bands.
As already stressed in 3.2 we should avoid or model any coherent or incoherent energy besides the true tidal spectrum. Coherent energy will mix up to the tidal spectrum. If it has the same frequency we should correct the results by modelling the effects. It is the case of the oceanic tidal loading effects. If there exists additional waves such as shallow water components it is necessary to introduce them in the model. If not they will corrupt the corresponding tidal group. In the last versions of VAV some options have been introduced to solve these problems. There exists also steady waves of meteorological energy such a S1 and S2. If these waves are seasonally modulated it will create sides waves in the spectrum. For example the annual modulation of S1 corresponds to the frequencies of P1 and K1 and the semi-annual one to PI1 and 1. As these modulations are changing with time it is practically impossible to model these effects which are directly affecting the determination of the liquid core resonance near the K1 frequency. Here the only solution is to protect the instrument against these thermal influences. Incoherent energy is mainly produced by the pressure and temperature variations. It is often possible to determine “efficiency coefficients” for a given source of perturbation and to eliminate its contribution to the considered frequency band. The noise level is well characterised by the RMS error on the unit weight s0. The diminution of s0 is generally a good criterion to evaluate the real impact of the perturbation on the data.
4.3 The leakage effect
We call leakage effect the influence of one tidal frequency of the energy really present in another spectral band. Due to unfavourable window functions coupled to the side lobes of the transfer function of the filters, perturbations from even distant frequencies could leak into the tidal bands where they could bias the tidal parameter estimates (Schüller, 1978). For the 48h filters designed by Venedikov in 1966 there is a maximum leakage from each frequency band situated at 7°.5/h in angular speed. A numerical simulation showed that the O1 group gets a maximum effect from the frequency 21°.44/h (16.8h) and 6°.44/h (55.9h). However we know that there is no coherent energy at these frequencies in the tidal spectrum. Venedikov himself considered, in the VEN98 method, the problem of the leakage of one tidal band into the other by introducing the diurnal waves as a group in the evaluation of the SD groups and inversely, showing that the effect is negligible. To avoid the leakage one has to smooth the window function (ideally a Hanning or Hamming window should be applied in place of the usual square box), avoid gaps and remove side lobes by designing better and thus longer filters. It is the basic philosophy of Wenzel in ETERNA. However long moving filters are spreading out spikes and jumps and the loss of data becomes important in presence of gaps. It is the main reason why data “repair” becomes necessary.
5. Data repair prior to analysis
In tidal data analysis it is customary to interpolate small gaps (less than one day), to remove spikes and to adjust obvious jumps in the curve. It is a controversial matter as we cannot “re-create” missing or spoiled information. In fact the smoothing of digital data is useless when the applied band pass filters are short and not overlapping as in VAV. It is then sufficient to eliminate the corresponding perturbed sections from the filtered numbers in the computation procedure. It can be done automatically. An intermediate solution is to weight the filters in function of their internal noise as done in NSV98. The problem is worse when people want to use very long moving filters to get steep slopes and avoid side lobes, like in ETERNA. The number of data lost on each side of a block can become very large in “gappy” data and gaps are affecting the transfer function of the data window, by creating side lobes. The main concern is here always the possible leakage from adjacent frequencies. It explains why data interpolation is so popular even with hourly sampling rate. With high rate sampling for tidal data it became necessary to decimate the original data to minute and even hourly data. Decimation filters are built to avoid aliasing and are generally rather long. A typical filter to decimate from minute to hour covers xxx minutes. A missing ordinate will thus create a gap of xxx minutes. Moreover, as the minimum filter length for tidal analysis is 24 hours we shall loose a minimum of 24 hours of data. It is why interruptions of a few minutes should be interpolated. Concerning spikes and tares the moving filtering will spread out the perturbation on all the adjacent readings. This effect will also increase with the length of the filter. It is a reason why it can be necessary to suppress spikes and jumps. However in some instruments the curve is slowly going back to its previous value after the jump and the correction of the jump is in fact producing an artificial drift. Correction of jumps can seriously affect the determination of LP waves and long period gravity changes. Spikes and jumps should be corrected on the original data prior to decimation as the decimation filter will spread out this kind of perturbation. A last category of perturbation usually “repaired” are the portions of the curve perturbed by large earthquakes. In fact most of the signal is harmonic and with a high frequency compared to the tides. It will be filtered out during the decimation process to the hourly sampling. However some instruments can show an offset during earthquakes and some repair can be necessary. As a conclusion we should say that each philosophy has its advantages and drawbacks. Either you want to use short filters to avoid loosing too many data in “gappy” records and you are exposed to leakage from the side lobes of the filters and window transfer functions, or you are using long filters on uninterrupted data series with optimal transfer functions and you have to repair the data from spikes, and tares on one hand and to interpolate the gaps on the other, “creating” artificial data. The decimation procedure requires to interpolate gaps smaller than the final sampling rate to avoid enlargement of very small gaps.
5.1 preprocessing techniques
In the early stages of the tidal research it was necessary to have uninterrupted records of at least 28 days to perform one tidal analysis with separation of the main tidal groups. The situation did radically change with the use of least square techniques which are only limited by the filter length. It became possible to use data with only two consecutive hours (Usandivaras & Ducarme, 1967), two days (Venedikov, 1966), 51 hours with Pertsev filtering (Chojnicki, 19??, Wenzel, 19??) or more. Very general methods have been developed for unevenly spaced data (Venedikov, 2003) and the notion of gap has thus disappeared. However it has always been customary to interpolate a few missing hours. In the early days smoothing was performed manually and simple filters were designed to identify bad portions of the curve, such as Lecolazet. Specific filters allowed to discriminate 5 spikes and tares (Z½ , DM47). The new preprocessing techniques are based on the “remove-restore” principle. A model of tides (tidal prediction) and pressure effects are subtracted from the observations (“remove” procedure) and the corrections directly applied on the residues. Interpolation of missing data becomes a simple linear interpolation between the two edges of a gap. Corrected observations are then recomputed (“restore” procedure). The quality of the tidal model is essential as the interpolated data will in fact be replaced by this model during the “restore” procedure. Of course the degree of automation is critical. Completely automated procedures such as PRETERNA can be dangerous. It is why the “T-soft” software (Vauterin, 1999) has been developed with a high degree of interactivity. Generally there is no “a-priori” model and the model will be derived from the analysis of a less perturbed part of the data set. As a first approximation it is also possible to get “modelised” tidal factors using a body tides model and oceanic loading computations. The model can be improved in an iterative way. To apply safely the remove-restore principle it is necessary to remember that - The tidal factors of the model should be computed without application of the instrumental time lag; - The calibration used to compute the model should be the same as the calibration applied on the data in the preprocessing phase. Here also T-soft can provide useful options. It is possible to compute a linear regression between the tidal prediction and the raw gravity data to determine automatically an apparent sensitivity, without any knowledge of the real calibration. The global fit replaces the tidal prediction and the residues can be directly corrected. The corrected data are obtained by summing up the global fit and the corrected residues. The modelling can be improved by adding to the regression a pressure channel. It is even possible to take into account an unknown instrumental time lag or timing error by using the time derivative of the theoretical tides as auxiliary channel. This procedure is only precise for time offsets of a few minutes.
6. Monitoring the sensitivity changes
The calibration factor C is expressed in [physical units] per [recording units] i.e. for a gravimeter connected to a digital voltmeter in nms-2/V. The instrumental sensitivity s is the inverse and will thus be expressed in V/nms-2. The usual way of calibrating an instrument is to apply a perturbation with a known amplitude K to the instrument and observe the amplitude of its reaction d. The calibration of the instrument is then C=K/d (6.1) and the original observations should be multiplied by C prior to tidal analysis. Its sensitivity is then given by s = d/K (6.2)
If K is constant the variation of the sensitivity is directly proportional to the variations of d. The calibration procedure is generally a long and tedious procedure and, what is worse, is perturbing the tidal records. It is why during the recording period there are only a few calibrations available and nobody knows how the sensitivity behave between two calibrations. The best we can do is to compute an instantaneous calibration value by linear interpolation between two successive calibrations. However we can follow accurately the changes of sensitivity using the tidal records themselves as we suppose that the tidal parameters are perfectly stable. If we have a good model for the tidal amplitude factors and phase differences, we can generate a tidal prediction in physical units and fit the observations to the prediction on let’s say 48 hours blocks. The regression coefficient d’ is thus expressed in [recording units] per [physical units] and is thus an expression of the instrumental sensitivity. This option has been implemented in T-soft under “moving window regression”. Auxiliary channels, such as pressure can be incorporated in the multilinear regression. We can thus follow from block to block the sensitivity variations. Due to noise and perturbations it is often necessary to smooth the individual values to get a continuous behaviour. However one has generally for block i where a calibration was performed with an amplitude d
d d’i. As we know that the calibration value must be the same we can write C = K/d = K’/di’ (6.3) With K’/K = d/d’i= fi (6.4) or K’ = fi .K (6.5) To determine a mean value f we have only to compute d’ on the days where a calibration has been performed and average all the d/d’ values. Then all the successive d’j values from each block will provide a continuous series of calibration factors
Cj = f .K/d’j (6.6) If we are using non overlapping filters of the same length as the blocks we can directly multiply the filtered values by the corresponding calibration factor. This possibility has been widely used at ICET with the VEN66 method. More generally we can linearly interpolate between the smoothly changing values of Ci and multiply each observation by the corresponding value.
7. Recent tidal analysis methods
As outlined before, in the least squares approach, there are two main families of methods: - moving window filtering and global evaluation of the tidal families following T. Chojnicki. The most popular tidal analysis method along this line is “ANALYZE” from ETERNA package by H.-G.Wenzel (1999). - Non overlapping filtering and separation of the tidal families following A. P. Venedikov. There is acontinuous lineage of methods from VEN66 to VAV03. Numerous tests on the same data sets have never shown any difference in the evaluation of the tidal parameters at a level exceeding the confidence intervals. However, as outlined in section 2.4, the RMS errors determination can be very different, due to the coloured noise characteristics of the tidal data. The separation of the tidal families allows to approximate the noise in each frequency band, providing directly realistic error estimates. The BAYTAP-G method (Tamura & al., 1991) proceeds from a different philosophy.
7.1 ETERNA
ETERNA became the most popular tidal analysis package due to its versatility. It is very well documented. The associated data format became the official transfer format adopted by the International Centre for Earth Tides (ICET) One can use different sampling rates from minute to hour. It is valid for all tidal components: potential, gravity, tilt, strain, displacements. It can use any tidal development from less than 400 waves (Doodson) up to more than 10,000 (Hartman-Wenzel). It is possible to include up to five auxiliary channels in order to evaluate a simple regression coefficient between the main channel and the perturbing signals. It includes a tidal prediction program (PREDICT), a tidal analysis program (ANALYZE), a preprocessing program (PRETERNA). As auxiliary facilities you can prepare time series from IERS polar motion data to compute polar motion effects on gravity and evaluate tidal loading vectors from any oceanic tidal model. The data can be separated in blocks. On each block one can apply a different scale factor and time lag for the main tidal signal. It is also possible to apply a bias parameter on any channel at the beginning or even inside of a block. The most usual way is to apply a band pass filter for the evaluation of the tidal families from diurnal up to four-diurnal. A wide range of filters is available from the simple Pertsev filter on 51hourly values up to very long filters. In the unfiltered option it is also possible to model the long term “drift” by a polynomial representation on each block. This option should be used to evaluate the LP tides and even the polar motion signal. The drawbacks of the program ANALYZE are: - The least squares adjustment suppose a white noise on the data, hypothesis certainly not verified as geophysical noise is coloured. The RMS error on the unit weight is thus a mean value on the tidal bands and the computed errors on the estimated parameters are thus under- estimated in the D band and over-estimated in the TD one. In the last version ETERNA3.4, Wenzel improved the situation by taking into account the noise in the different tidal bands to “colour” the error estimation. - To avoid leakage effects it is necessary to use very long filters with the problems already mentioned in sections 4.3 and 5. - The program is computing the residues on the filtered data and producing a list of the residues larger than the statistical 3 level. However the rejection of the corresponding data would produce gaps with the problems outlined before. The only solution is the reiteration of the preprocessing to correct the corresponding portions of the curve. - The only way to take into account the sensitivity changes is to create blocks with constant calibration. It is why it is recommended to multiply the data by their calibration table before the analysis, using at least a linear interpolation between successive calibrations or even a smoothed calibration table as explained earlier in section 6.
7.2 Evolution from the VEN66 to the VAV03 method
Initially the main advantages of the VEN66 method was that it was very cheap in computer time due to the non overlapping filtering on 48 hours blocks. Moreover the separation of the different tidal bands lead to more realistic error estimation than the other least squares estimates. This version was until recently in use at ICET which extended the program to strain computation (MT71) and adopted the Tamura tidal potential (MT71tam). Auxiliary programs were used to detect the bad portions of the data and the corresponding blocks were easily removed. For instruments with variable sensitivity it was also possible to produce smoothed calibration tables following the procedure described in section 6. The last version takes into account the systematic difference in tidal response between the waves m m 1 2 derived from P2 and P3 and refines the group separation in order to separate the P3 and P3 waves depending on the Lunar perigee (8.847 year period) and the nodal waves associated with the 18.6124 year Saros period. This program is now obsolete due to the fact that it is not possible to include auxiliary channels to correct for pressure or temperature perturbations. One important step in the evolution is the VEN98 variant which, besides the improvements of MT71, included a lot of new options. - The influence of pressure and temperature was evaluated in amplitude and phase, separately for each tidal band.; - It was possible to weight the even an odd filtered number according to their variance. - It was possible to reject bad intervals on the basis of a given confidence interval e.g. 3; - It was possible to introduce non tidal frequencies besides the tidal spectrum; - The Akaike criterion was introduced to judge of the real improvement of a solution when the number of parameters increased; 1 2 - The P3 and P3 components were treated as specific groups mixed up with the normal 1 2 P2 and P2 waves. Finally the VAV software was developed since 2000 and will be extensively presented here. 7.3 The BAYTAP-G approach
BAYTAP-G (Bayesian Tidal Analysis Program - Grouping Model) is a general analysis program for earth tides and crustal movements, which includes Bayesian model in analysis. BAYTAP-G (original version 85-03-26) has following utilities.
(1) estimation of tidal amplitude factors and phases (2) determination of trend and estimation of its spectrum (4) interpolation of missing data and estimation of step value (5) rough search for abnormal data (6) calculation of ABIC (Akaike's Bayesian Information Criterion) which shows goodness of analysis model
BAYTAP-G determines several constants minimising next equation with hyperparameters D and W which are chosen to minimise ABIC.
n M K 2 n 2 2 yi AmCmi BmSmi di bk xik hzi D di 2di1di2 i1 m1 k 0 i1 M 2 2 2 W Am Am1 Bm Bm1 m2 where m is a group number of tidal constituents, Cmi and Smi are summation of cosine and sine parts of each constituents in m-th group respectively (theoretical values), Am and Bm are tidal constants to be determined, di is drift (trend), xi are associated data, bk are response weights, h is step value and zi is a step function whose value is zero till step exists and varies one after step. The summation order K fixes the degree of the impulse response for the associated data. Entering K=0 is equivalent to evaluate a single efficiency. The hyperparameter D is a coefficient referred to 2nd order difference of trend (optionally it is possible to select 3rd order difference). If the value of D is large, the trend will be determined closely linear. In opposition, if D is small, it will be determined bending close to original data. We say that the trend is rigid in the former case and soft in the latter. Hyperparameter D is automatically chosen to let ABIC minimum by BAYTAP-G. It is assumed in BAYTAP-G that the trend is only changing slowly with time. So we can treat complex trend which cannot be expressed as a polynomial function or a periodic one. This is one of the most interesting features of this analysis program. The hyperparameter W expresses the smoothness of the tidal admittance curve. This condition is not made at the boundary of tidal species (D,SD,TD..) where there is a gap in the frequency distribution of the tidal constituents. In the previous equation, this condition is not implemented to clarify the analysis model. A special option allows to represent the liquid core resonance using a theoretical model based on a resonance frequency at 15.0737deg/h with Q=1,150. It is optionally possible to suppress the harmonic expression of the tidal potential and the program becomes a simple response method. If one prepares associated data sets and does not include theoretical tides in analysis, BAYTAP-G works as a response method. Though the standard analysis model of BAYTAP-G is an harmonic one, the use of the response method can be actually useful in the analysis of non-tidal data or heavily disturbed data. To determine tidal constants by the response method, it is necessary prepare a time domain representation of the theoretical tides in each tidal band as associated data sets.