STATISTICS PRESERVING SPATIAL INTERPOLATION METHODS FOR
MISSING PRECIPITATION DATA
by
Husayn El Sharif
A Thesis Submitted to the Faculty of
The College of Engineering and Computer Science
in Partial Fulfillment of the Requirements for the Degree of
Master of Science
Florida Atlantic University
Boca Raton, Florida
August 2012
ABSTRACT
Author: Husayn El Sharif
Title: Statistics Preserving Spatial Interpolation Methods for Missing Precipitation Data Institution: Florida Atlantic University
Thesis Advisor: Dr. Ramesh Teegavarapu
Degree: Master of Science
Year: 2012
Deterministic and stochastic weighting methods are commonly used methods for estimating missing precipitation rain gauge data based on values recorded at neighboring gauges.
However, these spatial interpolation methods seldom check for their ability to preserve site and regional statistics. Such statistics are primarily defined by spatial correlations and other site-to- site statistics in a region. Preservation of site and regional statistics represents a means of assessing the validity of missing precipitation estimates at a site. This study evaluates the efficacy of traditional interpolation methods for estimation of missing data in preserving site and regional statistics. New optimal spatial interpolation methods intended to preserve these statistics are also proposed and evaluated in this study. Rain gauge sites in the state of Kentucky are used as a case study, and several error and performance measures are used to evaluate the trade-offs in accuracy of estimation and preservation of site and regional statistics.
iii
STATISTICS PRESERVING SPATIAL INTERPOLATION METHODS FOR
MISSING PRECIPITATION DATA
LIST OF FIGURES ...... viii
LIST OF TABLES ...... xiii
1 Introduction ...... 1
1.1 Background ...... 1
1.2 Problem Statement ...... 3
1.3 Objectives ...... 4
1.4 Thesis Outline ...... 5
2 Literature Review ...... 6
2.1 Naïve Interpolation Methods ...... 6
2.1.1 Gauge Mean (GM) Method...... 7
2.1.2 Single Best Estimator (SBE) Method ...... 8
2.2 Other Deterministic Interpolation Methods ...... 9
2.2.1 Inverse Distance Weighting (IDW) Method ...... 9
2.2.2 Single Objective Function Weighting (SOFW) Method ...... 10
3 Methodology ...... 13
3.1 Site Statistics ...... 13
3.1.1 Summary Site Statistics ...... 13
3.1.2 Cumulative Density Function Visual Comparison ...... 16
3.1.3 Kolmogorov-Smirnov test...... 16
iv
3.1.4 Gamma Distribution Parameters ...... 17
3.1.5 Dry to Wet Days Ratio ...... 17
3.1.6 Markov Chain Transition Probabilities ...... 18
3.1.7 Temporal Autocorrelation ...... 19
3.1.8 Probability Density Function ...... 20
3.2 Regional Statistics ...... 20
3.2.1 Sum of absolute differences in Site Statistics ...... 20
3.2.1.1 Regional Mean ...... 21
3.2.1.2 Regional Standard Deviation ...... 21
3.2.1.3 Regional Skewness ...... 21
3.2.1.4 Regional Correlation ...... 22
3.2.1.5 Regional Gamma Distribution Shape Parameter ...... 22
3.2.1.6 Regional Gamma Distribution Scale Parameter ...... 22
3.2.1.7 Regional Dry Days to Wet Days Ratio ...... 23
3.2.1.8 Regional Markov Dry/Wet Spell Transition Probabilities ...... 23
3.2.1.9 Regional Autocorrelation ...... 24
3.2.1.10 Regional Histogram Bin Values...... 24
3.3 Modifications to Interpolation Methods...... 25
3.3.1 Square Root Transformation ...... 25
3.3.2 Dry Days Correction ...... 25
3.3.3 Wet Days Correction ...... 29
3.4 Newly Proposed Interpolation Methods...... 29
3.4.1 Multi-Objective Function Weighting (MOFW) Method ...... 30
3.4.2 Threshold-Based Localized Optimization ...... 32
3.4.2.1 Global Mean Precipitation Threshold ...... 33 v
3.4.2.2 Single Best Estimator Threshold...... 33
3.4.3 Bootstrap Method ...... 34
3.4.4 Fuzzy Logic Compromise Solution ...... 34
3.5 Collection and Validation of Daily Precipitation Data ...... 39
3.6 Determining Historical Site and Regional Statistics of Observed Data ...... 41
3.7 Missing Data ...... 42
3.8 Model Calibration Data and Validation Data...... 44
3.9 Performance Measures ...... 45
3.9.1 Validation of Results ...... 45
3.9.2 Error Minimization Performance Measures ...... 46
3.9.3 Site Statistics Preservation Performance Measures ...... 48
3.9.4 Regional Statistics Preservation Performance Measures ...... 48
3.10 Ranking of Interpolation Methods ...... 48
4 Case Study ...... 52
4.1 Background of Case-study Region ...... 52
5 Results ...... 54
5.1 Computer Hardware and Software ...... 54
5.2 Homogeneity of Original Historical Precipitation Data Sets ...... 54
5.3 Normalized Performance Measures ...... 56
5.4 Naïve Interpolation Methods ...... 59
5.5 Inverse Distance Method ...... 85
5.6 SOFW Method ...... 96
5.7 MOFW Method ...... 109
5.8 Threshold Method: Global Mean Daily Precipitation ...... 112
5.9 Threshold Method: Single Best Estimator (SBE) ...... 124 vi
5.10 Bootstrap Method ...... 135
5.11 Fuzzy Logic Method ...... 146
5.12 Best Interpolation Method ...... 158
5.13 Best Interpolation Method: Further Analysis ...... 159
6 Conclusions ...... 183
6.1 Contributions of the Study ...... 183
6.2 Limitations of the Study and Recommendations for Future Research ...... 184
7 REFERENCES ...... 186
vii
LIST OF FIGURES
Figure 1: Comparision of cumulative density functions before and after infilling precipitation values at a base station (only non-zero values considered in analysis) ...... 16
Figure 2: Autocorrelation in daily precipitation values at a sample rain gauge in Kentucky ...... 19
Figure 3: Representation of missing precipitation data entries among a network of rain gauges ...... 43
Figure 4: Development of calibration and validation data sets ...... 44
Figure 5: Negligible heteroscedastiscity in a estimated precipitation data set for a rain gauge in Kentucky, USA ...... 46
Figure 6: Kentucky rain gauge stations...... 53
Figure 7: Accuracy of daily precipitation estimates with GM Method at Station 1 ...... 61
Figure 8: Accuracy of daily precipitation estimates with GM Method at Station 2 ...... 61
Figure 9: Accuracy of daily precipitation estimates with GM Method at Station 3 ...... 62
Figure 10: Accuracy of daily precipitation estimates with GM Method at Station 15 ...... 62
Figure 11: Site Summary Statistics for GM Method at Station 1 ...... 64
Figure 12: Site Summary Statistics for GM Method at Station 2 ...... 65
Figure 13: Site Summary Statistics for GM Method at Station 3 ...... 66
Figure 14: Site Summary Statistics for GM Method at Station 15 ...... 67
Figure 15: Regional Summary Statistics for GM Method at Station 1 ...... 68
Figure 16: Regional Summary Statistics for GM Method at Station 2 ...... 69
Figure 17: Regional Summary Statistics for GM Method at Station 3 ...... 70
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Figure 18: Regional Summary Statistics for GM Method at Station 15 ...... 71
Figure 19: Accuracy of daily precipitation estimates with SBE Method at Station 1 ...... 73
Figure 20: Accuracy of daily precipitation estimates with SBE Method at Station 2 ...... 73
Figure 21: Accuracy of daily precipitation estimates with SBE Method at Station 3 ...... 74
Figure 22: Accuracy of daily precipitation estimates with SBE Method at Station 15 ...... 74
Figure 23: Site Summary Statistics for SBE Method at Station 1 ...... 76
Figure 24: Site Summary Statistics for GM Method at Station 2 ...... 77
Figure 25: Site Summary Statistics for SBE Method at Station 3 ...... 78
Figure 26: Site Summary Statistics for SBE Method at Station 15 ...... 79
Figure 27: Regional Summary Statistics for SBE Method at Station 1 ...... 81
Figure 28: Regional Summary Statistics for SBE Method at Station 2 ...... 82
Figure 29: Regional Summary Statistics for SBE Method at Station 3 ...... 83
Figure 30: Regional Summary Statistics for SBE Method at Station 15 ...... 84
Figure 31: Site Summary Statistics for IDW Method at Station 1 ...... 88
Figure 32: Regional Summary Statistics for IDW Method at Station 1 ...... 89
Figure 33: Site Summary Statistics for IDW Method at Station 2 ...... 90
Figure 34: Regional Summary Statistics for IDW Method at Station 2 ...... 91
Figure 35: Site Summary Statistics for IDW Method at Station 3 ...... 92
Figure 36: Regional Summary Statistics for IDW Method at Station 3 ...... 93
Figure 37: Site Summary Statistics for IDW Method at Station 15 ...... 94
Figure 38: Regional Summary Statistics for IDW Method at Station 15 ...... 95
Figure 39: Site Summary Statistics for SOFW Method (AE Objective Function) at Station 1 ..... 99
Figure 40: Regional Summary Statistics for SOFW Method (AE Objective Function) at
Station 1 ...... 100
Figure 41: Site Summary Statistics for SOFW Method (AE Objective Function) at Station 2 ... 101 ix
Figure 42: Regional Summary Statistics for SOFW Method (AE Objective Function) at
Station 2 ...... 102
Figure 43: Site Summary Statistics for SOFW Method (AE Objective Function) at Station 3 ... 103
Figure 44: Regional Summary Statistics for SOFW Method (AE Objective Function) at
Station 3 ...... 104
Figure 45: Site Summary Statistics for SOFW Method (MRE Objective Function) at Station
15 ...... 105
Figure 46: Regional Summary Statistics for SOFW Method (MRE Objective Function) at
Station 15 ...... 106
Figure 47: Markov chain transition probabilities for best SOFW Method Variant at base station ...... 108
Figure 48: Estimation accuracy at Station 3 for non-corrected SOFW Method with MAE objective function...... 113
Figure 49: Site Summary Statistics for Mean Threshold Method (AE Objective Function) at
Station 1 ...... 115
Figure 50: Regional Summary Statistics for Mean Threshold Method (AE Objective
Function) at Station 1 ...... 116
Figure 51: Site Summary Statistics for Mean Threshold Method (MRE Objective Function) at Station 2 ...... 117
Figure 52: Regional Summary Statistics for Mean Threshold Method (MRE Objective
Function) at Station 2 ...... 118
Figure 53: Site Summary Statistics for Mean Threshold Method (MAE Objective Function) at Station 3 ...... 119
Figure 54: Regional Summary Statistics for Mean Threshold Method (MAE Objective
Function) at Station 3 ...... 120 x
Figure 55: Site Summary Statistics for Mean Threshold Method (MRE Objective Function) at Station 15 ...... 121
Figure 56: Regional Summary Statistics for Mean Threshold Method (MRE Objective
Function) at Station 15 ...... 122
Figure 57: Selection of an SBE Threshold for Station 15 ...... 125
Figure 58: Site Summary Statistics for SBE Threshold Method at Station 1 ...... 127
Figure 59: Regional Summary Statistics for SBE Threshold Method at Station 1 ...... 128
Figure 60: Site Summary Statistics for SBE Threshold Method at Station 2 ...... 129
Figure 61: Regional Summary Statistics for SBE Threshold Method at Station 2 ...... 130
Figure 62: Site Summary Statistics for SBE Threshold Method at Station 3 ...... 131
Figure 63: Regional Summary Statistics for SBE Threshold Method at Station 3 ...... 132
Figure 64: Site Summary Statistics for SBE Threshold Method at Station 15 ...... 133
Figure 65: Regional Summary Statistics for SBE Threshold Method at Station 15 ...... 134
Figure 66: Site Summary Statistics for Bootstrap Method at Station 1 ...... 138
Figure 67: Regional Summary Statistics for Bootstrap Method at Station 1 ...... 139
Figure 68: Site Summary Statistics for Bootstrap Method at Station 2 ...... 140
Figure 69: Regional Summary Statistics for Bootstrap Method at Station 2 ...... 141
Figure 70: Site Summary Statistics for Bootstrap Method at Station 3 ...... 142
Figure 71: Regional Summary Statistics for Bootstrap Method at Station 3 ...... 143
Figure 72: Site Summary Statistics for Bootstrap Method at Station 15 ...... 144
Figure 73: Regional Summary Statistics for Bootstrap Method at Station 15 ...... 145
Figure 74: Site Summary Statistics for Fuzzy Logic Method at Station 1 ...... 150
Figure 75: Regional Summary Statistics for Bootstrap Method at Station 1 ...... 151
Figure 76: Site Summary Statistics for Fuzzy Logic Method at Station 2 ...... 152
Figure 77: Regional Summary Statistics for Fuzzy Logic Method at Station 2 ...... 153 xi
Figure 78: Site Summary Statistics for Fuzzy Logic Method at Station 3 ...... 154
Figure 79: Regional Summary Statistics for Fuzzy Logic Method at Station 3 ...... 155
Figure 80: Site Summary Statistics for Fuzzy Logic Method at Station 15 ...... 156
Figure 81: Regional Summary Statistics for Fuzzy Logic Method at Station 15 ...... 157
Figure 82: Accuracy of estimated precipitation values at base stations for Best Interpolation
Method ...... 162
Figure 83: Preservation of Site Mean for Best Interpolation Method ...... 166
Figure 84: Preservation of Site Standard Deviations for Best Interpolation Method ...... 169
Figure 85: Preservation of Regional Mean for Best Interpolation Method ...... 172
Figure 86: Preservation of Regional Standard Deviations for Best Interpolation Method ...... 175
Figure 87: Preservation of Regional Correlations for Best Interpolation Method ...... 178
Figure 88: Preservation of Markov chain transition probabilities at Station 7 ...... 179
Figure 89: Preservation of temporal autocorrelation at Station 12 ...... 179
Figure 90: Preservation of Probability Density Function at Station 15 ...... 180
Figure 91: Regional variogram for GM Method ...... 181
Figure 92: Regional variogram for SOFW Method (RMSE objective function) ...... 181
Figure 93: Regional variogram for Fuzzy Logic Method Variant (Best Method) ...... 182
xii
LIST OF TABLES
Table 1: Listing of variables of multi-objective function ...... 32
Table 2: Listing of properties of homogeneity tests ...... 40
Table 3: Results of Homogeneity Tests on original precipitation data set ...... 55
Table 4: Stage I normalization constants for Station 1 ...... 56
Table 5: Stage I normalization constants for Station 2 ...... 57
Table 6: Stage I normalization constants for Station 3 ...... 58
Table 7: Stage I normalization constants for Station 15 ...... 59
Table 8: Stage II Normalization Constants ...... 59
Table 9: Summarized Performance Measures for Gauge Mean Method ...... 60
Table 10: Summarized Performance Measures for SBE Method ...... 72
Table 11: Summarized Performance Measures for Inverse Distance Method ...... 85
Table 12: Summarized Performance Measures for post-corrected IDW Method ...... 86
Table 13: Summarized Performance Measures for SOFW Variants ...... 96
Table 14: Summarized Performance Measures for SOFW Variants with post correction ...... 97
Table 15: Summarized Performance Measures for top ranking, non-corrected MOFW
Variants ...... 109
Table 16: Summarized Performance Measures for top ranking, post-corrected MOFW
Variants ...... 111
Table 17: Summarized Performance Measures for 25mm Global Mean Daily Precipitation
Threshold Method (non-corrected) ...... 112
xiii
Table 18: Summarized Performance Measures for 25mm Global Mean Daily Precipitation
Threshold Method (non-corrected) ...... 114
Table 19: Performance comparison between Mean Threshold Method and SOFW method for identical objective functions ...... 123
Table 20: Summarized Performance Measures for SBE Precipitation Threshold Method ...... 124
Table 21: Summarized Performance Measures for post-corrected SBE Precipitation
Threshold Method ...... 126
Table 22: Summarized Performance Measures for Bootstrap Method ...... 135
Table 23: Summarized Performance Measures for post-corrected Bootstrap Method ...... 136
Table 24: Summarized Performance Measures for Fuzzy Logic Method ...... 146
Table 25: Membership function values for Fuzzy Logic Method ...... 148
Table 26: Summarized Performance Measures for post-corrected Fuzzy Logic Method ...... 148
Table 27: Top Ten Ranked Interpolation Methods ...... 158
Table 28: Homogeneity and Hetereoscedasticity analysis for Best Interpolation Method ...... 163
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1 Introduction
1.1 Background
The study of the spatial and temporal variability of hydrologic phenomena in general -- and of precipitation in particular -- is of vital importance to civil engineers, hydrologists, water resource managers, climate scientists, and related stakeholders. Historical and projected rainfall estimates are integral parameters in the design of infrastructure subjected to stormwater run-off and its management (Akan 1993; ASCE 2000; Rosenberg et al. 2010), such as transportation systems, storm sewers, retention-detention ponds, and culverts. Ground and surface water resources which are managed to provide water to a variety of consumers are, in many cases, directly or indirectly dependent on replenishment via rainfall (ASCE 1996). As such, the management and rationing of this important resource is guided in significant part by reliable estimation of precipitation over the watershed.
Aspects of climate science also rely on long-term statistical trend analysis of precipitation such that critical changes in micro- and macro- climatic conditions can be identified or predicted.
The collection and analysis of precipitation data for this application is taking on increasing importance as climatologists explore the magnitude and implications of a changing climate (Dore
2005; McRoberts and Nielsen-Gammon 2011).
While in recent years there have been advancements in the collection of precipitation data, the oldest, and most common tool for the collection of precipitation data are networks of rain gauges. Rain gauges represent point- - dispersed within a watershed or region of interest. Rain gauges also serve as the only tool 1
available for the direct measurement of rainfall (Cheng et al. 2008). As it is impossible for a rain gauge network to cover every point within a region of interest, a neighborhood of rain gauge readings are often used to interpolate the precipitation at a point within a region for which there exists no recorded rainfall data for a given time interval. Neighbor-based interpolation is also employed when a rain gauge, due to random or systematic errors at the gauge, may be missing precipitation data.
Various rain gauge data driven methods have been proposed for the infilling of missing
historical precipitation data. Interpolation methods fall into broad categories such as temporal
interpolation, distance-based weighting methods, non-linear deterministic methods, regression
and time series models, and stochastic variance dependent methods (Teegavarapu 2009).
Temporal interpolation methods are particularly useful at relatively fine time-scales (less than one
day) as for such intervals, serial autocorrelation is significant; however, at coarser intervals,
temporal interpolation is no longer useful and spatial interpolation techniques are relied upon for
estimation of precipitation data. Commonly used distance-based interpolation methods used for
infilling rainfall data include the Thiessen Polygon Method (ASCE 1996; Tabios III and Salas
1985; J. A. Smith 1993) and the Inverse Distance Method (ASCE 1996), both of which, despite
their popularity, have limitations (Grayson and Bloschl 2001; Vieux 2001; Sullivan and Unwin
2003; Brimicombie 2003) that may preclude such methods from being recommended for spatial
interpolation (Grayson and Bloschl 2001). Non-linear deterministic methods include the Normal-
Ratio Method (ASCE 1996; Xia et al. 1999), Gauge Mean Method, Single Best Estimator Method
(Xia et al. 1999; Eischeid et al. 2000; Teegavarapu 2009), as well as variants of mixed-integer
non-linear programming methods (Teegavarapu 2012). Regression and time series models
include Global Polynomial Interpolation (Wang 2006), Local Polynomial Interpolation (Loader
1999; Regonda et al. 2006) and Thin-Plate Spline Methods (Chang 2004); however such methods
2
commonly involve functional forms for which there is not universal agreement. Stochastic methods such as Kriging and Co-Kriging, which provide a measure of uncertainty along with estimates, have been investigated for the interpolation of rainfall (Dingman 2002; Ashraf et al.
1997; Seo et al. 1990a, 1990b); however, such methods are generally computational intensive and involve the selection of parameters for which there is not universal agreement. Despite their popularity, complexity, and/or simplicity, spatial interpolation methods have generally not been evaluated for the preservation of site specific (rain gauge scale) and region specific (rain gauge to rain gauge or watershed scale) spatial statistics, and such statistics are an essential component of previously mentioned applications involving precipitation data.
1.2 Problem Statement
Rain gauge measurements are prone to both random and systematic errors which result in missing values in the precipitation data set at the rain gauge at various time intervals. It is not uncommon, given a lengthy data set, that upwards of 20 percent of precipitation values may be
magnified when multiple rain gauges within a network suffer from missing data.
Because of the relatively low temporal autocorrelation amongst precipitation values recorded at a single rain gauge for time scales exceeding one day, time series interpolation cannot be reliably employed to infill missing values at a temporal scale of a day or greater. Instead, observed precipitation data from a neighborhood of rain gauges in the region of interest may be employed to interpolate and infill missing entries. However, spatial interpolation methods involving a network of rain gauges used for estimation of missing precipitation data at a site are seldom checked for their ability to preserve site and regional statistics. Such statistics are primarily defined by probability distributions describing historical observations at a rain gauge 3
site as well as by spatial correlations and other station-to-station statistics in a region. Maintaining site and regional characteristics is essential for incorporating precipitation data sets into hydrologic and climate change studies, as precipitation trend analysis often depends on such statistics.
1.3 Objectives
This thesis evaluates the efficacy of traditional deterministic interpolation methods for infilling missing precipitation data in preserving site specific (rain gauge scale) and region specific (rain gauge to rain gauge or watershed scale) statistics. Furthermore, modification of traditional interpolation methods, as well as new interpolation methods intended to improve preservation such statistics are proposed and evaluated.
The objectives of this thesis are as follows:
1) Identify the site statistics and regional statistics of interest to be preserved when infilling
missing precipitation data at a rain gauge or a series of rain gauges.
2) Evaluate the performance of traditional deterministic interpolation methods in preserving site
and regional statistics.
3) Modify existing interpolation methods such that the efficacy of preserving site and regional
statistics is improved.
4) Develop new interpolation methods with innovative mathematical programming formulations
specifically intended to preserve site and regional rainfall statistics.
study region in which infilling of missing daily precipitation data is required. 4
1.4 Thesis Outline
The chapters of this thesis are organized as follows:
Chapter One: Introduces the necessity of infilling missing precipitation data, the problem of evaluating the preservation of site and regional statistics, as well as the objectives of this study.
Chapter Two: Provides a review of literature on traditional deterministic spatial interpolation methods for the infilling of missing precipitation data, as well as the advantages and limitations of such methods.
Chapter Three: Details a step-by-step framework and its relevant concepts by which various precipitation infilling methods may be evaluated and ranked with regards to preserving site and regional statistics. Critical site and regional statistics that are to be preserved by any given interpolation method are identified, and new interpolation methods for infilling missing daily precipitation data are proposed for investigation in this study.
Chapter Four: The methodology mentioned in Chapter Three is applied to a case study scenario
for which the infilling of missing precipitation data is required. Background information on the
application of methods to the case study region is explained.
for infilling missing precipitation data for this region is selected.
Chapter Six: Presents conclusions, contributions, limitations of this work, and recommendations
for further research.
5
2 Literature Review
To spatially construct rainfall fields or to estimate precipitation data at a point in space, various deterministic weighting and stochastic interpolation methods have been proposed
(Teegavarapu 2009; Wei and McGuinness 1973; Simanton and Osborn 1980; Tung 1983;
Krajewski 1987; ASCE 1996; Vieux 2001). Traditional weighting and data-driven methods are commonly used for estimation of missing precipitation (Teegavarapu 2009) and report superior
interpolation techniques have been recommended (ASCE 1996) for the infilling of missing precipitation data, interpolation techniques used in the past possess limitations of varying significance. Furthermore, interpolation techniques generally have yet to be evaluated for the preservation of historical site and regional statistics when such interpolation methods are used to help produce a serially complete rainfall data set containing observed and estimated rainfall values. The following is an overview of existing interpolation methods for estimation of missing precipitation, their advantages, and shortcomings.
2.1 Naïve Interpolation Methods
Naïve interpolation methods are generally not recommended for the estimation of missing data; however, these simplistic methods provide a performance benchmark that may be used to evaluate the usefulness of alternative methods or newly proposed interpolation methods. More sophisticated interpolation techniques are expected to exhibit superior performance than naïve methods at least with regards to minimizing estimation error. Naïve interpolation methods include 6
the Gauge Mean Method and the Single Best Estimator Method. Details of these methods are given in the next two sections.
2.1.1 Gauge Mean (GM) Method
The GM Method estimates a missing precipitation record as the arithmetic average of observed precipitation at all neighboring stations at the time of interest and is given by the following formulation:
(2.1)
Where is the precipitation depth estimate at the base station for a given time interval, ;
is the observed precipitation at station at the same time interval; and refers to the number of rain gauge stations excepting the base station.
The GM Method is not an ideal method for infilling daily precipitation records when the neighboring rain gauges are relatively distant from the point of estimation or are not significantly correlated with the base station. Furthermore, the GM Method does not capture the variability in contribution of neighboring stations to precipitation at the base station; rather, every neighboring station is assumed to have the same relationship to the base station with regards to precipitation estimates as every other neighboring station.
The aforementioned limitation on GM Method may be mitigated using a method proposed by Paulus and Kohler (1952) in which the estimated precipitation record is taken as the average of a local selection neighboring stations, and the selected neighboring stations each have an average historical precipitation within 10 percent of the historical average of the base station. The formulation is given as follows:
7
(2.2)
Where refers to the number of neighboring stations selected as a subset of a global network of rain gauges. If the difference in one of the selected neighboring rain gauges exceeds
10 percent, then a normal-ratio method is recommended (Paulhus and Kohler 1952) and is given by:
(2.3)
Where are the observed precipitation readings at each of the selected stations at a given time, ; and are the long-term mean annual precipitation values at each of the selected stations. is the long-term mean annual precipitation values at the base station.
While these localized average methods may improve results when compared to using a global average, in the literature, these modified averages were recommended for the infilling of monthly precipitation records and may not be ideal for estimation of precipitation data at finer time scales.
2.1.2 Single Best Estimator (SBE) Method
SBE Method estimates a missing precipitation record at a time instance as the precipitation
related to everything else, but near
(Tobler 1970) is not necessarily valid when considering the Euclidean distance of neighboring stations from a base station, especially when differences in elevation between nearby rain gauge stations allow for orographic effects to influence readings. To circumvent this issue, the SBE station may be selected as the neighboring
8
station with the highest positive historical correlation in precipitation values with the base station.
The formulation is given as follows:
(2.4)
Where is the observed precipitation value for a time, , at the station with the highest correlation in precipitation values, , with the base station.
The efficacy of SBE Method is entirely dependent on the strength of correlation between the SBE station and the best station. As such, this method may provide poor performance in study regions where the maximum correlation with the base station for a potential SBE station is less than 0.50. It should also be investigated whether correlation between precipitation values should be the sole criteria for selecting an SBE station or whether other shared statistics should play a significant role.
2.2 Other Deterministic Interpolation Methods
Deterministic interpolation methods provide an estimate for a record without any indication of the error or uncertainty associated with the predicted value (Li and Heap 2008) and are generally not computationally intensive.
2.2.1 Inverse Distance Weighting (IDW) Method
IDW Method estimates a missing precipitation entry using a linear combination of values at neighboring rain gauge stations weighted by an inverse function of the Euclidean distance from the base station (Li and Heap 2008) and is given as follows:
9
(2.5)
Where is the Euclidean distance between station and base station ; and is an exponent applied to the distance; depending on the context, can be between 1 and 3 inclusive
(Simanton and Osborn 1980).
One of the shortcomings of the inverse distance method is that is relies on the assumption
(Tobler 1970) which is not always valid amongst rain gauge
ent- artificially high rainfall depth values being concentrated near gauging stations participating in the interpolation.
2.2.2 Single Objective Function Weighting (SOFW) Method
The SOFW Method estimates precipitation at a missing station as a weighted, unbiased summation of observed precipitation values at all neighboring stations. Nonlinear mathematical programming formulations and their minor variants may be used to determine the appropriate weights for each station involved in interpolation (El Sharif and Teegavarapu 2011). One method to determine station weights is to select weights such that an error measure (a single objective function) between observed and estimated data sets is optimized. Such objective functions may include Root Mean Squared Error (RMSE), Correlation Coefficient ( ), Mean Absolute Error
(MAE), Mean Relative Error (MRE), or Absolute Error (AE).
The formulation is given by:
10
(2.6)
or
(2.7)
or
(2.8)
or
(2.9)
or
(2.10)
Subject to:
(2.11)
and
11
(2.12)
Where is the total number of time intervals; is the weight assigned to station ;
and refer to the lower and upper bounds of each assigned weight respectively.
One of the strengths of the SOFW Method is that it allows for the direct minimization of estimation errors and lends itself relatively well to automation using programmable solvers using a solution space search method such as Generalized Reduced Gradient (GRG) Method. However, it should be noted that commonly used computational GRG solvers may converge on an inferior local optimum solution depending on the initialization of feasible values of the solver, search direction, and available computational time. It is also possible that optimization of some of these objective functions may minimize estimation errors at the expense of the preservation of site and regional statistics.
12
3 Methodology
Spatial interpolation methods used for estimation of missing precipitation data have not been thoroughly evaluated for their ability to preserve site and regional statistics. Site statistics are primarily defined by the parameters of statistical distributions that can be fitted to observed precipitation data at the missing station. Regional statistics are primarily defined by spatial correlations and other site-to-site statistics in a region in relation to the missing station.
Preservation of site and regional statistics, along with the minimization of estimation errors, represent a means of assessing the validity of missing precipitation estimates at a site.
Furthermore, maintaining site and regional characteristics is essential for incorporating precipitation data sets containing estimated values into hydrologic and climate change models, as precipitation trend analysis often depends on such statistics. What follows is an overview of site and regional statistics to be considered for preservation when estimating missing precipitation data.
3.1 Site Statistics
3.1.1 Summary Site Statistics
Three summary statistics related to estimated precipitation at a missing station include mean, standard deviation, and skewness.
Mean, , is given by the simple average of precipitation values at the station of interest and is given by:
13
(3.1)
Where is the -th precipitation value at the station of interest, and is the sample size of precipitation values at the station of interest.
Sample standard deviation, measures the spread of data of a given sample and is given for precipitation at a station of interest by (Gosset 1908):
(3.2)
Sample skewness, , describes the asymmetry of the probability distribution of a random variable (a sample of precipitation values) and is given by (Cramér 1946):
(3.3)
These summary site statistics are relatively easy to quantify from a precipitation data set and provide an efficient means to assess the validity of estimated precipitation values when the summary site statistics of a historical observed data set are compared with the summary site statistics of the historical data set with values replaced by estimated precipitation values. The formulation for the preservation of these summary site statistics are presented as follows:
14
(3.4)
(3.5)
(3.6)
Where , , and characterize the preservation of the rain gauge mean precipitation, standard deviation, and skewness at the base station after infilling. and refer to the mean precipitation at the base station before and after infilling respectively, and refer to the standard deviation of precipitation values at the base station before and after infilling respectively, and and refer to the skewness of precipitation values at the base station before and after infilling respectively.
15
3.1.2 Cumulative Density Function Visual Comparison
Figure 1: Comparision of cumulative density functions before and after infilling precipitation values at a base station (only non-zero values considered in analysis)
Comparison of the Cumulative Density Function (CDF) plots of two data sets is a visual representation of whether the two data sets are sampled from the same distribution. This visual test allows for a qualitative assessment of the magnitude of deviation between the distributions of two data sets. When dealing with precipitation data, it may be of interest to additionally perform a visual test considering only non-zero precipitation data as shown in Figure 1.
3.1.3 Kolmogorov-Smirnov test
A two-sample Kolmogorov-Smirnov test is used to determine whether two sets of data are sampled from the same distribution; the test is sensitive to differences between data sets that may not be apparent through comparison of summary statistics (Wilcox 2006) or by visual comparison of cumulative density plots. The null hypothesis is that the distributions for the two data sets are equal, and the alternative hypothesis is that they are not equal. When dealing with precipitation data, it may be of interest to additionally perform this test considering only non-zero precipitation data. 16
3.1.4 Gamma Distribution Parameters
The Gamma distribution has been used in meteorological rainfall models (Encyclopedia
Britannica 2012) and has been characterized as being able to reliably fit daily precipitation data
(Simolo et al. 2010; Bridges and Hann 1972; C. Jones et al. 2004; R. Bradley et al. 1987; Wilks
1995; Groisman et al. 1999; Nicholls and Murray 1997; Dunn 2004).
The Gamma distribution is defined by a shape parameter, , and a scale parameter,
(Encyclopedia Britannica 2012). Ideally, these two parameters should be the same between observed and estimated daily precipitation data sets. The formulation of these site statistics are presented as follows:
(3.7)
(3.8)
Where and characterize the preservation of gamma distribution shape and scale parameters respectively at the base station after infilling, is the gamma distribution shape parameter at the base station before infilling, is the shape parameter at the base station after infilling, is the scale parameter at the base station before infilling, and is the scale parameter after infilling.
3.1.5 Dry to Wet Days Ratio
Overestimation of wet days is a common problem in the estimation of missing precipitation. Comparing the ratio of dry to wet days is a means of assessing the magnitude of
17
overestimation between observed and estimated data sets. The formulation for this statistic is presented as follows:
(3.9)
(3.10)
(3.11)
Where characterizes the preservation of the dry days to wet days ratio, , at the base station after infilling; and are the number of dry days in the base station precipitation data set before and after infilling respectively; and and are the number of wet days in the base station precipitation data set before and after infilling respectively.
3.1.6 Markov Chain Transition Probabilities
Markov Chain transition probabilities refer to the probabilities of the following scenarios:
-zero precipitation entry) provided that the day before
: dry (zero precipitation)
: dry wet
:
18
These transition probabilities are commonly represented as first order, two-stage Markov chains (Schoof and Pryor 2008).
3.1.7 Temporal Autocorrelation
Daily precipitation data have a characteristically low, yet measureable, temporal autocorrelation when the time scale of autocorrelation lag is one day or coarser. The relatively low serial correlations are exemplified in Figure 2, in which the correlations for observed precipitation values for a rain gauge located in the state of Kentucky, USA with data lagged by up to 13 days is presented.
1.2
1
0.8
0.6
Correlation 0.4
0.2
0
-0.2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Lag (Days)
Figure 2: Autocorrelation in daily precipitation values at a sample rain gauge in Kentucky
When estimated data is included in a precipitation data set, the characteristic autocorrelations at different lag intervals should not change.
19
3.1.8 Probability Density Function
Preserving the probability density function involves matching the frequencies of each bin of a precipitation depth histogram based on data sets with observed data and data sets including infilled data. Quantitatively, this can be represented as a sum of absolute differences in frequencies between the two data sets for each rainfall depth interval.
3.2 Regional Statistics
3.2.1 Sum of absolute differences in Site Statistics
Most of the site statistics previously presented can be reformed into regional statistics. This may be accomplished by way of taking the absolute difference between a given site statistic at the base station and the identical site statistic at a neighboring station. This process should be completed with each of the neighboring stations; the summation of these absolute differences would serve as a regional statistic. The lower the sum of absolute differences, the more likely the regional statistic is preserved. The general formulation is as follows:
(3.12)
Where is the reported regional statistic related to site statistic, ; is a site statistic of the observed precipitation data at the base station; is the site statistic at neighboring station ; is the site statistic of the precipitation data set that includes estimated values; and refers to the total number of stations excepting the base station. This sum of absolute differences procedure can be applied to evaluate whether site-to-site means, standard deviations, correlations,
Gamma distribution parameters, ratio of dry to wet days, Markov Chain transition probabilities, etc. have been preserved when a precipitation data set includes estimated values.
20
The rain gauge-to-rain gauge (regional) statistics of focus in this study are formulated as follows:
3.2.1.1 Regional Mean
(3.13)
Where characterizes the preservation of station-to-station mean precipitation, is the observed mean precipitation at the base station, is the mean precipitation at the base station after infilling of missing values has been completed, is the observed mean precipitation at a neighboring station , and is the number of rain gauge stations excepting the base station.
3.2.1.2 Regional Standard Deviation
(3.14)
Where characterizes the differences between station-to-station standard deviation, is the observed standard deviation of precipitation values at the base station, is the standard deviation of precipitation values at the base station after infilling of missing values has been completed, and is the observed standard deviation of precipitation values at a neighboring station .
3.2.1.3 Regional Skewness
(3.15)
Where characterizes the differences between station-to-station skewness, is the observed skewness of precipitation values at the base station, is the skewness of precipitation
21
values at the base station after infilling of missing values has been completed, and is the observed skewness of precipitation values at a neighboring station .
3.2.1.4 Regional Correlation
(3.16)
Where characterizes the differences between station-to-station correlations, is the correlation between observed precipitation values at the base station and neighboring station ;
is the correlation between precipitation values at the base station after infilling has been completed and neighboring station .
3.2.1.5 Regional Gamma Distribution Shape Parameter
(3.17)
Where characterizes the differences between station-to-station gamma distribution shape parameters, is the shape parameter of observed precipitation values at the base station,
is the shape parameter of precipitation values at the base station after infilling has been completed, and is the gamma distribution shape parameter of observed precipitation values at neighboring station .
3.2.1.6 Regional Gamma Distribution Scale Parameter
(3.18)
Where characterizes the differences between station-to-station gamma distribution scale parameters, is the shape parameter of observed precipitation values at the base station,
22
is the scale parameter of precipitation values at the base station after infilling has been completed, and is the gamma distribution scale parameter of observed precipitation values at neighboring station .
3.2.1.7 Regional Dry Days to Wet Days Ratio
(3.19)
Where characterizes the differences between station-to-station dry days to wet days ratios, is the dry days to wet days ratio from observed precipitation values at the base station,
is the dry days to wet days ratio from precipitation values at the base station after infilling has been completed, and is the dry days to wet days ratio from observed precipitation values at neighboring station .
3.2.1.8 Regional Markov Dry/Wet Spell Transition Probabilities
(3.20)
where
(3.21)
(3.22)
Where characterizes the differences between station-to-station dry/wet spells transition probabilities, is the summation of the absolute differences between the transition probabilities between the observed values at the base station and observed values at station , and
23
is the summation of the absolute differences between the transition probabilities between the values at the base station after infilling has been completed and observed values at station .
The definitions of , , , and has preceded, and subscripts and refer to the base station and neighboring station respectively. indicates that the probability is calculated from a data set containing infilled values.
3.2.1.9 Regional Autocorrelation
(3.23)
Where characterizes the differences between station-to-station temporal autocorrelations, is the summation of temporal autocorrelation values from one to 14 day lag at the base station before infilling, is the same summation after infilling, and is the summation of temporal autocorrelation values from one day to 14 day lag at neighboring station using observed values.
3.2.1.10 Regional Histogram Bin Values
(3.24)
Where characterizes the differences between station-to-station histogram bins of precipitation thresholds, is the frequency of precipitation values within the histogram bin from observed values at the base station, is the frequency of precipitation values within the histogram bin from values at the base station after infilling, and is the frequency of precipitation values within the histogram bin from values at neighboring station .
24
3.3 Modifications to Interpolation Methods
In order to overcome some of the inherent limitations of interpolation methods for infilling missing precipitation data, the corrective strategies mentioned in this section may be adopted. The square-root transformation is a corrective technique applied before an interpolation procedure is commenced. The dry and wet days corrections are carried out after an interpolation procedure has been completed.
3.3.1 Square Root Transformation
A problem that may arise with interpolation is the introduction of heteroscedasticity, a bias
in which the variances of predictions determined by regression do not remain constant (Knaub Jr.
2006). The ideal interpolation method for missing precipitation should show no trend in residuals,
regardless of whether the observed precipitation being estimated is large or small in magnitude,
as such bias would adversely influence hydrologic or climatic trend analysis conducted on the
completed precipitation data set.
If interpolation results introduce heteroscedasticity, one method to overcome this issue is to
apply a square root transformation to the data involved in the interpolation and then to perform
interpolation with the transformed data. The results may then be back-transformed after
interpolation and re-evaluated for the absence heteroscedasticity.
3.3.2 Dry Days Correction
There are two common limitations of precipitation interpolation methods that estimate
missing precipitation at a base station using data from a global or local neighborhood of stations:
(1) underestimation of base station rainfall when no precipitation is recorded at the neighboring
stations, but precipitation actually occurred at the base station; and (2) overestimation of base
25
station rainfall when precipitation is measured at neighboring stations but, no precipitation actually occurred at the base station (Teegavarapu and Chandramouli 2005).
To resolve case 1 underestimation, data from external sources, such as radar-based precipitation estimates, may be employed (Teegavarapu and Chandramouli 2005); however, the reliability of radar-based measurements is a subject of controversy (Young et al. 1999; Adler et al. 2001). In the absense of external data, this limitation may be overcome by selecting a sufficient number of neighboring rain gauges to participate in interpolation such that the probability that all selected rain gauges would simulatenously report zero precipitation is minimal.
To resolve case 2 overestimation, interpolation results may be corrected by considering the magnitude of readings at neighboring stations and adjusting the estimated base station rainfall accordingly (Teegavarapu and Chandramouli 2005). For example, if the average reported rainfall at the neighboring stations is less than a selected threshold value, the estimated precipitation at the base station would be taken as zero. The formulation is given as follows:
(3.25)
Subject to:
(3.26)
Where refers to the threshold precipitation limit.
Another option to resolve case 2 overestimation is to consider the dry days of a Single Best
Estimator (SBE) station and to replace the estimated precipitation at the base station with zero 26
values accordingly (Teegavarapu and Chandramouli 2005). The selection of the SBE station may be open to further investigation; however, it has been selected as the neighboring station with the highest positive correlation with the base station (Eischeid et al. 2000). The formulation is given as follows:
(3.27)
Subject to:
(3.28)
A stochastic element may be introduced to the dry days correction technique by incorporating the joint probability of no rain observed at the base station given that no rain was observed at the SBE station. The joint probability is incorporated such that the dry days
- and that essential site specific statistics are more accurately preserved. The formulation is given as follows.
(3.29)
Subject to:
(3.30)
27
Where is a number between 0 and 1 inclusive selected at random for each time interval
. And is the joint probability of no rain observed at the base station given that no rain was observed at the SBE station as evaluated from available observed precipitation records.
The SBE remedial method may be further modified into a Multiple Best Estimator (MBE)
method such that if the observed precipitation readings at a local subset of neighboring stations
that are highly correlated with the base station are zero, then the estimated precipitation for the base station is taken as zero. The formulation is given as follows:
(3.31)
Subject to:
(3.32)
Where refers to the observed precipitation readings at the selected neighboring station . refers to the count of the selected neighboring gauges. For all remedial methods involving a Best Estimator(s), the positive correlations with the base station should be greater than 0.50.
Because dry days correction methods rely on a neighboring site(s) and are intended to reduce estimation errors at the base station, the influence of these aforementioned remedial strategies on the preservation of site and regional statistics will be of interest.
28
3.3.3 Wet Days Correction
A post-correction procedure to address underestimation of rainfall during high precipitation events at a rain gauge involves using an SBE station under the assumption that the
SBE station is most-likely to have captured extreme precipitation events at the base station while other neighboring stations may have not. To apply this correction, the historical maximum observed precipitation at the SBE station is recorded, and whenever the observed rainfall at the
SBE station is at least 50 percent of the maximum recorded precipitation, the estimated precipitation at the base station is replaced with the observed SBE precipitation for that time interval. The formulation is given as follows:
(3.33)
Subject to:
(3.34)
Where is the maximum historical preserved precipation at the SBE station and is a constant taken as 0.50 in this context.
3.4 Newly Proposed Interpolation Methods
In order to overcome the actual or perceived shortcomings of traditional non-naïve interpolation methods in minimizing estimation error and preserving site and regional statistics, newly proposed interpolation methods and their performance relative to traditional methods are to be investigated. 29
3.4.1 Multi-Objective Function Weighting (MOFW) Method
One of the perceived limitations of the previously mentioned Single Objective Function
Weighting (SOFW) Method is that the optimized objective function (i.e. RMSE, , MAE, MRE,
AE) is primarily focused on minimizing estimation errors without direct consideration of preserving site and regional statistics. Any preservation of such statistics in the SOFW Method is assumed to be accomplished by way of minimizing error in estimation of daily precipitation values; however, this assumption has yet to have been validated.
A newly proposed Multi-Objective Function Weighting (MOFW) Method is intended to directly address the minimization of estimation error and the minimization of differences between summary site and regional statistics between observed and estimated precipitation data sets. The proposed multi-objective function is a summation that includes at least one parameter focused on optimizing estimation error (e.g. RMSE, , MAE, MRE, AE) and five additional parameters directly related to preserving summary site and regional statistics. Each of these parameters is weighted by a constant such that appropriate significance is given to each parameter in the optimization procedure.
(3.35)
Subject to:
30
(3.36)
and
(3.37)
The terms of the multi-objective formulation are listed in Table 1.
31
Table 1: Listing of variables of multi-objective function
Term Definition , , , , Magnification constants applied to the terms of the multi-objective function such that the optimization procedure gives appropriate significance to each , of the terms. selected objective function associated with optimizing estimation errors
(e.g. RMSE, , MAE, MRE, AE) . weight assigned to the observed precipitation value at station . , lower and upper bounds of each assigned weight, , respectively. observed precipitation value at time at the base station. estimated precipitation value at time at the base station. observed precipitation value at time at station . number of time intervals number of rain gauging stations, excepting the base station. arithmetic mean of observed precipitation values at the base station. arithmetic mean of estimated precipitation values at the base station. arithmetic mean of observed precipitation values at station . standard deviation of observed precipitation values at the base station. standard deviation of estimated precipitation values at the base station. standard deviation of observed precipitation values at station . correlation between observed preciptiation values at station and observed precipitation values at the base station. correlation between observed preciptiation values at station and estimated precipitation values at the base station.
With the proper initialization of weights to the terms of the multi-objective function, it is hypothesized that the MOFW Method will outperform traditional interpolation methods with regards to preservation of site and regional statistics with only minimal loss of fidelity with regards to minimizing estimation errors.
3.4.2 Threshold-Based Localized Optimization
One of the assumed limitations of traditional interpolation methods is that a single linear combination of weights are assigned to each of the neighboring stations involved in interpolation 32
without any regard for the possibility of the site-to-site relationships between the base station and its neighboring stations to significantly change in time. These changes may be linked to seasonal relationships between stations or prevalent weather patterns at certain time periods, changes that may be represented by select preciptiation thresholds. It is hypothesized that if precipitation data sets are separated based on select thresholds and optimization is carried out separately on each data subset that estimation errors will be minimized and site and regional statistics will be preserved with an efficiacy superior to that of traditional interpolation methods mentioned in this study.
3.4.2.1 Global Mean Precipitation Threshold
One method of separating precipitation data sets into subsets for independent optimization is to use a threshold based on the the global mean of observed precipitation at the rain gauge stations involved in interpolation. An example of this is to select a median precipitation level and to separate precipitation entries into two subsets, one subset having entries with a global mean less than the threshold, and the second subset having entries greater than or equal to the selected threshold. Interpolation can then be carried out on each of these subsets.
3.4.2.2 Single Best Estimator Threshold
Another method to separate precipitation data is to select a precipitation threshold of a
Single Best Estimator (SBE) station. All precipitation entries in which the precipitation at the
SBE station is less than the selected threshold will be placed in one subset, and all entries greater than or equal to the threshold will be placed in the second subset. Further investigation may be made into the ideal selection of an SBE precipitation threshold such that estimation error is minimized and site and regional statistics are preserved.
33
3.4.3 Bootstrap Method
A newly proposed method for determining an ideal set of weights for neighboring stations to allow for an interpolation procedure to minimize estimation error and preserve site and regional statistics involves the use of a bootstrapping resampling procedure. Bootstrapping generally belongs in the class of estimators that use resampling with replacement from a base data set to create empirical distributions (Hilmer 2010). This procedure may be implemented by creating a model calibration data set with a series of randomly selected (with replacement) time intervals from the original precipitation data set. Single Objective Function Weighting (SOFW)
Method interpolation can be carried out on this data set using sum of least squares optimization to assign weights to neighboring rain gauges for interpolation. The model is then validated on a separate precipitation data set and performance measures are evaluated. This process may be repeated 1,000 3,000 times, and the set of weights that best preserves site and regional statistics while minimizing estimation errors may be adopted as an ideal interpolation technique to apply on the original data set. The strength of this procedure is assumed to be in its many iterations that allow for many possible solutions sets to be evaluated; in this regard, the bootstrap search is
- search method. However, bootstrapping may have considerable advantage over simplistic brute-force search in scenarios when the serial length of the original precipitation data set is not sufficient for the effective application of other interpolation techniques.
3.4.4 Fuzzy Logic Compromise Solution
It is possible that there exists a tradeoff between optimizing the performance of an estimation method with regards to error and preserving historical site and regional statistics. In a practical scenario in which estimation of missing precipitation data is necessary, a decision maker
34
may be more or less interested in minimizing errors than in preserving historical site and regional statistics. Examples of preference towards minimizing estimation error may involve scenarios in which a temporally limited historical precipitation data set is required for a short-term rainfall prediction. In such a scenario, site and regional statistics may not be of significant importance.
Examples of preference towards preserving historical site and regional statistics over minimizing estimation error may involve scenarios involving long term precipitation trend analysis and the development of hydrologic models for civil engineering design.
Fuzzy set theory has been used in the water resources field as tool to address uncertainty in hydrologic information (Ngo 2006) as well as to address the level of satisfaction of decision
tion of decision maker objectives are complicated by subjectivity or vagueness (Chaves et al. 2004; Fontane et al. 1997; Teegavarapu and Simonovic 2007; Ferreira 2009). In an unsymmetrical decision making environment, the incorporation of fuzzy mathematical programming formulations may find utility in providing a compromise estimation solution that strikes a balance between minimizing estimation error and preserving site and regional statistics according to the preferences of the decision maker.
Three formulations are used to obtain the solution of fuzzy mathematical programming
and Simonovic (1999). Original formulation refers to an objective function and constraints associated primarily with minimizing estimation error. For example, the selected objective function in this regard may be the multi-objective function including terms for minimizing estimation errors and preserving summary site and regional statistics (see equation (3.35) under the previously mentioned MOFW Method), but constrained to a minimum value of Root Mean
Squared Error (RMSE) as determined by the previously mentioned SOFW Method. Intermediate formulation refers to a modified original formulation in which fuzzy tolerances are added to the 35
constraints to provide for the preservation of site and regional statistics at the expense of reduced performance in minimizing estimation errors. Final formulation refers to a formulation with an objective of maximizing a membership function value using the Bellman and Zadeh (1970) criterion as explained by Zimmermann (1991). This final formulation balances the minimization of estimation errors with the preservation of site and regional statistics in accordance to the
preferences of the decision maker. It is important to note that membership functions are to be defined a priori for the objective function and fuzzy constraints involved in these formulations. In this study, in the absence of a clearly defined membership function, the function is assumed to be linear, with a value of 0 denoting preference towards the minimization of estimation errors and a value of 1 denoting preference towards preserving site and regional statistics.
The original formulation is given as follows:
maximize: (3.38)
Subject to:
(3.39)
(3.40)
(3.41)
36
Where the equations and inequalities defined above refer to classical, standard form linear mathematical programming mode formulation. , , , and are crisp numbers. The notation for this formulation is from Zimmermann (1991). The objective function in this formulation is referred to as , obtained when original mathematical formulation without any tolerances added to the constraints is solved.
The intermediate formulation is given as follows:
maximize: (3.42)
Subject to:
(3.43)
(3.44)
(3.45)
The objective function in this formulation is referred to as , obtained when tolerance, , is added to the limit in constraint (3.43).
The final formulation is given as follows:
37
maximize: (3.46)
Subject to:
(3.47)
(3.48)
(3.49)
(3.50)
(3.51)
The objective function values, and are used to develop a fuzzy membership function.
Membership function value, , is given as follows:
(3.52)
38
3.5 Collection and Validation of Daily Precipitation Data
The first step in analysis involves acquisition of a reliable and extensive data set of daily observed rainfall depths from a collection of rain gauges for a region of interest. Rain gauge measurements serve as a and despite their shortcomings, this tool and its variants are the most commonly used tools for the measurement of precipitation (Pedersen 2010). Other methods of recording rainfall magnitudes
(e.g. radar-based and satellite-based measurements) are only surrogate measures of precipitation and their results are commonly calibrated with data from rain gauges; surrogate measures of precipitation have also been a source of controversy (Adler et al. 2001; Young et al. 1999), and managing precipitation data sets containing surrogate measures of precipitation are beyond the scope of this study.
Hydrologic and climate models often require a relatively extensive spatio-temporal data set such that trend analysis may be conducted on the data. Interpolation methods for estimating missing precipitation data also require a data set that covers an area and time period that effectively captures the statistical characteristics of precipitation in the region, especially considering that a given region may be subject to climatic patterns, dry spells, wet spells, and extreme precipitation events with return periods in the order of decades. Considering these issues, it is understandable that national and international meteorological organizations have accepted that at least 30 years of data be used to determine climatic normals from climatic data sets
(Guttman 1989). In the case of point measured precipitation data, it is also important that measurement points are appropriately dispersed spatially through the region of interest.
Additionally, before observed precipitation data is used for interpolation, the observed precipitation data sets should be evaluated and validated for homogeneity. A homogeneous long- term climatic data series refers to a data set in which all variability and change in the data set is 39
due exclusively to climatic behavior (Aguilar et al. 2003). Relocation of monitoring stations, changes in instrumentation or methods of calculation, disruptive physical changes in the vincity of the gauge, and other non-climatic factors may obscure true climatic patterns in the data and bias results of climatological and hydrological studies (Costa and Soares 2009).
Wijngaard, et al. (2003) used four statistical tests to determine whether a daily precipitation and temperature data set for the European Climate was homogeneous; the tests may be generally applied to any climate scenario and have been adopted for this study. The tests are Standard
Normal Homogeneity Test (SNHT), Buishand Test, Pettitt Test, and Von Neumann Test, and their characteristics based on the explanation by Wijngaard, et al. (2003) are presented in Table 2.
Table 2: Listing of properties of homogeneity tests
Time of break in Assumes data Homogeneity Null Alternative Sensitivities homogeneity can is normally Test Hypothesis Hypothesis be located? distributed? Tails of time SNHT Values of the A step-wise Yes Yes series testing shift in the mean of Middle of Buishand variable are Yes Yes independent testing variable time series and values is Middle of Pettitt Yes No identically present. time series distributed (i.e. The series of Powerful at values is not all times Von Neumann homogeneou No No randomly (Addinsoft s). distributed. 2012)
The rejection or non-rejection of the null hypotheses for the aforementioned statistical tests
in which the null hypothesis of homogeneity is rejected in no more than one of the four tests at
Such a classification would be cause to inspect the data further before homogeneity can be 40
three or more tests and should subsequently not be considered for further analysis (Wijngaard et al. 2003).
3.6 Determining Historical Site and Regional Statistics of Observed Data
Site statistics are primarily defined by the parameters of statistical distributions that can be fitted to observed precipitation data at the missing station. Regional statistics are primarily defined by spatial correlations and other site-to-site statistics in a region in relation to the missing station. Ideally, when estimates of daily precipitation data are appended to an observed precipitation data set, the historical site and regional statistics should not change significantly. An explanation of the site specific statistics of interest in this study preceded in Chapter Three and are listed as follows:
Site Mean
Site Standard Deviation
Site Skewness
Non-rejection of Kolmogorov-Smirnov Test
Cumulative Density Function Visual Comparison
Gamma Distribution Shape and Scale Parameters
Dry to Wet Days Ratio
Markov Chain Transition Probabilities
Temporal Autocorrelation
Probability Density Function
An explanation of the regional statistics of interest in this study preceded in Chapter Three, and such statistics primarily consist of taking the sum of absolute differences amongst the 41
neighboring precipitation stations for each site statistic where applicable. The sum of the differences should be as minimal as possible. An explanation of the site statistics that can be incorporated into regional statistics of interest in this study preceded in Chapter Three and are listed as follows:
Site Mean
Site Standard Deviation
Site Skewness
Gamma Distribution Shape and Scale Parameters
Dry to Wet Days Ratio
Markov Chain Transition Probabilities
Temporal Autocorrelation
Probability Density Function
Additionally, the sum of differences in correlation between each neighboring station and the base station may serve as a regional statistic to be preserved.
3.7 Missing Data
Understanding and quantifying the spatial and temporal variability of precipitation in a watershed requires continuous precipitation data at different spatial and temporal scales (El Sharif and Teegavarapu 2011). However, data sets are populated with random errors or missing values which must be infilled before further analysis can be commenced. For relatively long serial precipitation data sets, it is not uncommon for about 20 percent of the data at a single station to be missing, and such an amount of missing precipitation data may significantly interfere with the extrapolation of site and regional statistics of a precipitation data set. 42
Figure 3 is a conceptual representation of missing data entries among a network of individual rain gauges ( through ) for given time intervals ( through ).
Figure 3: Representation of missing precipitation data entries among a network of rain gauges
In order to simplify the development of estimation models, this study assumes that at any given time interval, there is at most only one rain gauge station missing data as shown in Figure
3. To overcome this limitation in a practical scenario, separate models that only include stations with available observed precipitation data should be developed if more than one station is missing data for a given time interval.
43
3.8 Model Calibration Data and Validation Data
Figure 4: Development of calibration and validation data sets
The methods of interest for the estimation of missing precipitation data at a base station in this study involve a weighted summation of observed precipitation values at neighboring stations.
The selection of weights is achieved by the optimization of an objective function. This model is developed using a large randomly sampled subset of an entire time series of observed precipitation values at neighboring stations. In this study, the observed precipitation values at the base station are known for the entire time series, but the values the model is calibrated. Estimation results found using the model data are compared to observed precipitation values at the base station that were assumed to be missing.
Once the model is appropriately optimized, the same weighting scheme is applied to a second, smaller subset of data, for validation purposes. The performance of this validation data set is then evaluated. The estimates for precipitation at the base station in the validation data set are assumed to represent the missing values at the base station, for this reason, the length of the
44
calibration data set may be selected by random sampling up to two-thirds of the entire data set of observed precipitation entries, and the validation data set may be selected as the remaining one- third. For further analysis, the entire observed precipitation data set may be compared with the precipitation data set containing two-thirds of observed data and one-third of estimated data. A conceptual schematic of the development of calibration and validation data sets is shown in
Figure 4.
If the methodology of estimation has satisfactory performance in this optimization scenario in which all observed precipitation values are known, including that of the base station, then it is assumed that the estimation methodology may be generally applied in a practical scenario in
values at a base station are truly missing and in need of infilling.
3.9 Performance Measures
Various measures may be used to evaluate and rank the performance of estimation methodologies in relation to the minimization of estimation errors and the preservation of site and regional statistics. The following is an overview of such measures.
3.9.1 Validation of Results
Data sets completed by rainfall estimates for missing data should undergo tests to confirm data homogeneity and the absence of heteroscedasticity. Homogeneity may be confirmed by way of performing the battery of homogeneity tests mentioned previously, i.e. Standard Normal
Homogeneity Test (SNHT), Bushiand Test, Pettitt Test, and Von Neumann Test. A visual test may be used to confirm the absence of heteroscedasticity. A plot of squared residuals versus estimated precipitation values at the base station should exhibit no significant trend. An example of an estimated precipitation data set displaying negligible heteroscedasticity, as suggested by the 45
relatively low coefficient of determination for the best linear trendline fitted to the plot of residuals versus estimated precipitation, is shown in Figure 5.
Figure 5: Negligible heteroscedastiscity in a estimated precipitation data set for a rain gauge in
Kentucky, USA
3.9.2 Error Minimization Performance Measures
Error minimization performance measures involve comparing estimated precipitation entries at the base station with the observed precipitation entries at the base station for the same time intervals. Performance measures of consideration between these two data sets include Root-
Mean-Squared-Error (RMSE), Correlation Coefficient (CC), Mean Absolute Error (MAE), Mean
Relative Error (MRE), and Absolute Error (AE). The formulations are listed as follows:
(3.53)
46
Where is the observed precipitation value at the base station, at time, and is the corresponding estimated precipitation value. is the total number of observations within the validation data set, .
(3.54)
Where and refer to the observed and estimated precipitation data sets within the validation data set, , respectively.
(3.55)
(3.56)
It is noted that MRE only considers non-zero values of ; thusly, in this context refers to the total number of observations in the validation data set, , in which the observed base station precipitation value, is non-zero.
(3.57)
47
3.9.3 Site Statistics Preservation Performance Measures
Evaluating site specific performance measures involves comparing the full data set of observed precipitation values at the base station with the data set of precipitation values at the base station that includes the replacement of observed precipitation values with infill estimates.
An explanation of the site specific statistics of interest in this study preceded in Chapter Three.
3.9.4 Regional Statistics Preservation Performance Measures
Evaluating regional statistics performance measures involves comparing the full data set of observed precipitation values at the base station and neighboring stations with the data set of precipitation values at the base station that includes the replacement of observed precipitation values with infill estimates and the observed precipitation values at all neighboring stations. An explanation of the regional statistics of interest in this study preceded in Chapter Three, and primarily consist of taking the sum of absolute differences amongst the neighboring precipitation stations for each site statistic where applicable.
3.10 Ranking of Interpolation Methods
Once information is collected on the performance measures for each estimation method, a scoring system may be established such that interpolation methods may be compared to each other with regards to overall performance in minimizing estimation errors and preserving site and regional statistics. In this regard, performance measures evaluated for interpolation methods applied to a subset of all available rain gauges will be normalized in two stages.
In Stage I normalization, each performance measure is normalized to a value between
0.000 and 1.000 by way of dividing the performance measure by the maximum value of the given performance measure at the specified base station across multiple interpolation methods. In this 48
study, the ideal Stage I normalized performance measure has a magnitude of 0.000. The formulation for Stage I normalization is as follows:
(3.58)
Where refers to the normalized performance score between 0.000 and 1.000 for performance measure and interpolation method . is the result of performance measure for interpolation method , and is the maximum value of performance measure at the base station, regardless of the interpolation method.
Stage II normalization groups together Stage I normalized performance measures into three categories of performance: Estimation Errors, Site Statistics, and Regional Statistics. The grouping is completed as follows:
(3.59)
(3.60)
(3.61)
Where , , and refer to the summation of Stage I normalized performance measures related to minimization of estimation errors, preservation of site statistics, and preservation of regional statistics respectively for an interpolation method . Stage II
49
normalization is completed when , , and are normalized to a value between
0.000 (ideal) and 1.000 (non-ideal) as follows:
(3.62)
(3.63)
(3.64)
Where , , and are Stage II normalized performance measures characterizing minimization of estimation errors, preservation of site statistics, and preservation of regional statistics respectively for interpolation method applied at a given base station;
, , and refer to the maximum values of , , and respectively at a specified base station, regardless of interpolation method ; , , and refer to the minimum values of , , and respectively at a specified base station, regardless of interpolation method .
The final score assigned to an interpolation method at a given base station is determined by the summation of Stage II normalized performance measures. The final score can take a value between 0.000 (ideal) and 3.000 (non-ideal). The final score characterizing overall performance,
, is given as follows:
50
(3.65)
In this study, interpolation methods are evaluated for a subset (4 out of 15) base stations. A single best interpolation method for application at all base stations will be determined based on results from the specified subset of stations subject to the following conditions:
Condition One: A given interpolation method must appear in the top 10 performing methods with regards to overall performance at all base stations included in the subset.
Condition Two: Scores for , , and must each be below 0.500 to allow for at least median performance in all three categories of performance. This condition prevents a method with exemplary performance in preserving site and regional statistics, but low performance in minimizing estimation errors (or vice versa) from being recommended as an interpolation method for the infilling of missing daily precipitation data across multiple base stations.
Condition Three: If more than one method fulfills Conditions One and Two, then the best interpolation method will be selected as the method for which the summation of overall performance scores, , across all base stations included in the subset, is minimal.
51
4 Case Study
The objective of this study is to evaluate the efficacy of various traditional and newly proposed spatial interpolation techniques in preserving site and regional statistics in precipitation data while minimizing estimation errors for the estimation of missing daily precipitation data at a base station. What follows is an overview of the case study region (Kentucky, USA) in which the methodology presented in Chapter Three will be applied.
4.1 Background of Case-study Region
The state of Kentucky covers 104,623 square kilometers (40,395 square miles) of territory.
Its major axis is east to west from 80 34 west longitude and the width of the state spans from 36 the Eastern Coal Field, the Bluegrass, the Pennyroyal, the Western Coal Field, and the Jackson
Purchase. Most elevations are less than 107 meters (350 feet) above sea level, with its lowest elevation at approximately 76 meters (250 feet) above sea level (Kleber et al. 2000a). The climate is considered moderate in comparison with coastal and interior states with distinct seasons of equal length. According to literature, Kentucky annual mean precipitation varies according to latitude from 1,041 millimeters (41 inches) in the north to over 1,321 millimeters (52 inches) in the south (Kleber et al. 2000b).
In this study, 33 years (1971 2003) of daily observed precipitation readings from 15 rain gauge sites in the state of Kentucky, USA were available for analysis. Data was provided by the
52
Kentucky Agricultural Weather Center, University of Kentucky. The stations involved in the study are shown in Figure 6. To evaluate the performance of interpolation methods in minimizing estimation error, precipitation readings at a single station are assumed to be missing and are imputed using different interpolation methods.
Figure 6: Kentucky rain gauge stations
The normalized scoring method presented in Chapter Three was used to rank interpolation methods with regards to performance in minimizing estimation errors and preserving site specific
(single rain gauge) and regional (rain gauge-to-rain gauge) statistics.
53
5 Results
5.1 Computer Hardware and Software
Interpolation procedures in this study were primarily carried out on a notebook system with the following hardware and software specifications:
Processor: Quad-core 2GHz Intel Core i7-2630 CPU
8 GB RAM
Microsoft Windows 7 Home Premium 64-bit
Microsoft Office Excel 2007
Mathworks MATLAB 2010a
Microsoft Office Excel 2007 was the primary software used to perform solution searches, using its Generalized Reduced Gradient nonlinear Solver. Convergence was generally achieved in under 10 minutes for each interpolation method variant. The Bootstrapping Method as well as post-correction of interpolation methods were performed using Mathworks MATLAB 2010a.
5.2 Homogeneity of Original Historical Precipitation Data Sets
The original historical precipitation data set was tested for homogeneity using MeteoLab, an open-source Matlab toolbox for statistical analysis in meteorology (Santander Meteorology
Group 2012). Each of the four homogeneity tests were conducted at a significance level of 0.05
54
where applicable, and using the chronologically sorted precipitation data at each station. The results for the tests are shown in Table 3.
Table 3: Results of Homogeneity Tests on original precipitation data set
Homogeneity Test Number of Number of Station Von SNHT Pettitt Buishand Passes Failures Neumann 1 Fail Fail Pass Fail 1 3 2 Pass Fail Pass Fail 2 2 3 Pass Fail Pass Fail 2 2 4 Fail Fail Pass Fail 1 3 5 Fail Fail Pass Fail 1 3 6 Pass Fail Pass Fail 2 2 7 Fail Fail Pass Fail 1 3 8 Pass Fail Pass Fail 2 2 9 Pass Fail Pass Fail 2 2 10 Pass Fail Pass Fail 2 2 11 Pass Fail Pass Fail 2 2 12 Pass Fail Pass Fail 2 2 13 Pass Fail Pass Fail 2 2 14 Fail Fail Pass Fail 1 3 15 Pass Fail Pass Fail 2 2
Results show that the majority of stations in this study pass two homogeneity tests, namely, the SNHT and the Buishand test. All stations fail the Pettitt test and Von Neumann test. Results suggest that all stations should have their data carefully inspected for homogeneity. In this application, metadata gathered from the rain gauge stations involved in this study confirms that all data collected by the gauges were undisturbed by non-climatic events. In the absence of this metadata, it would have been recommended to exclude certain stations from this study, particularly Stations 1, 4, 5, 7 and 14 which failed three homogeneity tests. After data infilling, 55
the homogeneity results at any station should ideally not be any worse than the results from the testing of the original historical daily precipitation data set.
5.3 Normalized Performance Measures
To simplify the comparison of performance for each interpolation method, the performance measures evaluated in this study were normalized in two stages as mentioned in Chapter Three using the normalization constants presented in the following tables:
Table 4: Stage I normalization constants for Station 1
RMSE 7.116 mm 2.361 0.379 0.373 MAE 2.976 mm 0.212 MRE 1.645 mm 0.065 AE 8598.154 mm 0.155 SNHT 1 0.438 mm Pettitt Test 1 2.751 mm Buishand Test 1 0.472 Von Neumann Test 1 0.585 Heteroscedasticity 1 0.222 0.144 mm 17.757 0.512 mm 4.180 0.328 2.289 KS Test 1 0.677 0.023 1.452
56
Table 5: Stage I normalization constants for Station 2
RMSE 7.379 mm 2.844 0.379 0.409 MAE 2.960 mm 0.218 MRE 1.654 mm 0.026 AE 8,552.180 mm 0.157 SNHT 1 0.610 mm Pettitt Test 1 3.873 mm Buishand Test 1 0.351 Von Neumann Test 1 1.626 Heteroscedasticity 1 0.042 0.191 mm 10.645 0.635 mm 3.445 0.287 1.771 KS Test 1 0.103 0.021 1.276
57
Table 6: Stage I normalization constants for Station 3
RMSE 8.228 mm 3.006 0.586 0.390 MAE 4.044 mm 0.222 MRE 2.123 mm 0.111 AE 11,683.783 mm 0.163 SNHT 1 1.714 mm Pettitt Test 1 9.373 mm Buishand Test 1 1.087 Von Neumann Test 1 3.446 Heteroscedasticity 1 0.530 0.235 mm 42.085 0.709 mm 4.697 0.276 2.537 KS Test 1 0.187 0.038 0.969
58
Table 7: Stage I normalization constants for Station 15
RMSE 7.132 mm 2.675 0.414 0.384 MAE 3.037 mm 0.230 MRE 2.528 mm 0.068 AE 8,773.145 mm 0.198 SNHT 1 0.662 mm Pettitt Test 1 1.062 mm Buishand Test 1 0.508 Von Neumann Test 1 5.118 Heteroscedasticity 1 0.051 0.094 mm 3.158 0.488 mm 4.331 0.418 2.224 KS Test 1 0.135 0.025 1.851
Table 8: Stage II Normalization Constants
Station 1 3.803 5.000 4.201 10.280 1.781 8.422 2 3.758 5.000 3.483 10.243 1.023 8.561 3 3.087 5.000 4.519 11.595 1.918 7.675 15 3.451 4.961 3.453 12.053 2.065 6.882
5.4 Naïve Interpolation Methods
The summarized, normalized scores for the naïve Gauge Mean (GM) Method applied to a subset of Kentucky rain gauge stations is presented in Table 9. As expected, at all tested stations,
GM Method generally exhibited the worst performance of all interpolation methods evaluated in this study. As has been previously established, the GM Method is not recommended for the infilling of missing precipitation data; rather, this method serves as a benchmark such that any alternative or newly proposed interpolation method should at least outperform the GM Method
59
with regards to minimizing estimation errors and preserving site and regional statistics.
Interestingly, the GM Method exhibited relatively similar performance to other interpolation methods with regards to minimizing estimation errors; however, GM Method exhibited relatively poor performance in preserving both site and regional statistics. This finding suggests that minimizing estimation errors in an interpolation scenario using traditional performance measures does not necessarily ensure the maintenance of historical site specific and regional statistics.
Rather, the accuracy of precipitation estimates using this naïve method may still be questionable, as evidenced in the figures below even if traditional performance measures associated with minimizing estimation error exhibit similar performance to other interpolation methods showing significantly superior overall performance.
Table 9: Summarized Performance Measures for Gauge Mean Method
Station Estimation Site Regional Overall Score Rank Errors Statistics Statistics (1-71)
1 0.475 1.000 1.000 2.475 71 2 0.334 1.000 1.000 2.334 71 3 0.807 1.000 1.000 2.807 62 15 0.543 1.000 1.000 2.543 71
60
Figure 7: Accuracy of daily precipitation estimates with GM Method at Station 1
Figure 8: Accuracy of daily precipitation estimates with GM Method at Station 2
61
Figure 9: Accuracy of daily precipitation estimates with GM Method at Station 3
Figure 10: Accuracy of daily precipitation estimates with GM Method at Station 15
With regards to preservation of data homogeneity, at Stations 1 and 2, the post-infilling precipitation data sets passed the SNHT and Buishand homogeneity tests. However, the other two stations, 3 and 15, failed the SNHT test for homogeneity in addition to failing the Pettitt and Von
Neumann tests, thereby performing worse than the original historical precipitation data set with regards to data homogeneity.
62
With regards to heteroscedasticity, there was no discernible trend in residuals when using the GM Method. There was also no significant deviation from the historical temporal autocorrelation of daily precipitation values using this method.
Modification of the GM Method to include dry/wet days post correction based on a Single
Best Estimator station did improve the performance of the GM Method in minimizing estimation errors and in preserving site and regional statistics. However, the post-corrected GM Method only achieved below median ranking compared to other post-corrected and non-post-corrected interpolation methods.
As shown in the following figures, the post-correction allowed for the cumulative density function of non-zero precipitation values at the base station to approach the historical cumulative density function. Despite this visual improvement, this remedial post-correction did not allow the
GM Method to pass the Kolmogorov-Smirnov two-sample test, except when the post-correction was applied to the GM Method for Station 3 out of the four evaluated stations. Post-correction also appeared to lower the mean daily precipitation at each base station to lower than historical levels, while not significantly affecting the standard deviation of precipitation values at a base station when compared to the un-corrected GM variant.
63
Original Method With Dry/Wet days Post -Correction
Figure 11: Site Summary Statistics for GM Method at Station 1
64
Original Method With Dry/Wet days Post -Correction
Figure 12: Site Summary Statistics for GM Method at Station 2
65
Original Method With Dry/Wet days Post -Correction
Figure 13: Site Summary Statistics for GM Method at Station 3
66
Original Method With Dry/Wet days Post -Correction
Figure 14: Site Summary Statistics for GM Method at Station 15
With regards to summary regional statistics, post-correction of the data had deleterious effects on the preservation of regional historical mean daily precipitation while not having a significant effect on preservation of regional standard deviations and improving station-to-station correlations. This is exemplified in the figures below based on results from applying GM with post-correction at the four stations tested in this experiment.
67
Original Method With Dry/Wet days Post -Correction
St ation (on figure) Actual Station Station (on figure) Actual Station 1 15 8 8 2 2 9 9 3 3 10 10 4 4 11 11 5 5 12 12 6 6 13 13 7 7 14 14 Figure 15: Regional Summary Statistics for GM Method at Station 1
68
Original Method With Dry/Wet days Post -Correction
Station (on figure) Actual Station Station (on figure) Actual Station 1 15 8 8 2 1 9 9 3 3 10 10 4 4 11 11 5 5 12 12 6 6 13 13 7 7 14 14 Figure 16: Regional Summary Statistics for GM Method at Station 2
69
Original Method With Dry/Wet days Post -Correction
Station (on figure) Actual Station Station (on figure) Actual Station 1 15 8 8 2 1 9 9 3 2 10 10 4 4 11 11 5 5 12 12 6 6 13 13 7 7 14 14 Figure 17: Regional Summary Statistics for GM Method at Station 3
70
Original Method With Dry/Wet days Post -Correction
Station (on figure) Actual Station Station (on figure) Actual Station 1 1 8 8 2 2 9 9 3 3 10 10 4 4 11 11 5 5 12 12 6 6 13 13 7 7 14 14 Figure 18: Regional Summary Statistics for GM Method at Station 15
71
Table 10: Summarized Performance Measures for SBE Method
Station Estimation Site Regional Overall Rank Errors Statistics Statistics Score (1-71)
1 1.000 0.000 0.000 1.000 28 2 1.000 0.000 0.000 1.000 5 3 0.423 0.000 0.000 0.423 1 15 0.751 0.044 0.186 0.981 25
The summarized, normalized scores for the naïve Single Best Estimator (SBE) Method applied to a subset of Kentucky rain gauge stations is presented in Table 10. Stations 8, 4, 9, and
12 were the SBE stations for Station 1, 2, 3, and 15 respectively, with SBE stations exhibiting a historical positive correlation with their respective base stations of approximately 0.60 or greater.
The SBE Method, despite the simplicity of its application, exhibited top to above median ranking
in preserving site and regional statistics. However, with regards to error minimization, SBE
Method is generally a poor performer and is thusly disqualified as a method to be considered for the infilling of missing precipitation data. The scatter plots in the figures below suggest the poor performance of the SBE Method in minimizing estimation errors.
72
Figure 19: Accuracy of daily precipitation estimates with SBE Method at Station 1
Figure 20: Accuracy of daily precipitation estimates with SBE Method at Station 2
73
Figure 21: Accuracy of daily precipitation estimates with SBE Method at Station 3
Figure 22: Accuracy of daily precipitation estimates with SBE Method at Station 15
With regards to preservation of data homogeneity, at Stations 1 and 15, the post-infilling precipitation data sets passed the same homogeneity tests as the original, non-imputed historical precipitation data sets. However, the other two stations, 2 and 3, failed the SNHT test for homogeneity in addition to failing the Pettitt and Von Neumann tests.
With regards to heteroscedasticity, the SBE Method applied at the four stations did not exhibit a discernible trend in residuals. There was also no significant deviation from the historical 74
temporal autocorrelation of daily precipitation values using this method. The SBE Method applied at the stations of interest also passed the Kolmogorov-Smirnov two sample test when only non-zero precipitation values were considered. With regards to summary site statistics, the SBE method did not appear to consistently overestimate or underestimate the site mean or site standard deviation. The occurrence of overestimation or underestimation in this regard is station-dependent and not method-dependent.
75
Original Method
Figure 23: Site Summary Statistics for SBE Method at Station 1
76
Original Method
Figure 24: Site Summary Statistics for GM Method at Station 2
77
Original Method
Figure 25: Site Summary Statistics for SBE Method at Station 3
78
Original Method
Figure 26: Site Summary Statistics for SBE Method at Station 15
With regards to summary regional statistics, site-to-site differences in mean daily precipitation, standard deviations, and correlations were preserved relatively well compared to other interpolation methods evaluated in this study. Interestingly, the SBE Method appeared to be one of the only methods evaluated in this study that could simultaneously maintain historical regional mean and regional standard deviations compared to other interpolation methods 79
evaluated in the study. This result was especially evident when the SBE Method was applied to
Station 2 as shown in Figure 28.
As mentioned previously, Station 2 was a station which failed an additional homogeneity test when SBE Method was applied to infill missing data; however, the station with imputed values using the SBE method exhibited remarkable performance in maintaining historical site and regional statistics, suggesting that poorer performance with regards to maintaining data homogeneity may be overlooked provided that the historical statistics of intended focus are sufficiently preserved.
80
Original Method
Station Actual Station Actual (on figure) Station (on figure) Station 1 15 8 8 2 2 9 9 3 3 10 10 4 4 11 11 5 5 12 12 6 6 13 13 7 7 14 14 Figure 27: Regional Summary Statistics for SBE Method at Station 1
81
Original Method
Station Actual Station Actual (on figure) Station (on figure) Station 1 15 8 8 2 1 9 9 3 3 10 10 4 4 11 11 5 5 12 12 6 6 13 13 7 7 14 14 Figure 28: Regional Summary Statistics for SBE Method at Station 2
82
Original Method
Station Actual Station Actual (on figure) Station (on figure) Station 1 15 8 8 2 1 9 9 3 2 10 10 4 4 11 11 5 5 12 12 6 6 13 13 7 7 14 14 Figure 29: Regional Summary Statistics for SBE Method at Station 3
83
Original Method
Station Actual Station Actual (on figure) Station (on figure) Station 1 1 8 8 2 2 9 9 3 3 10 10 4 4 11 11 5 5 12 12 6 6 13 13 7 7 14 14 Figure 30: Regional Summary Statistics for SBE Method at Station 15
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5.5 Inverse Distance Method
The summarized, normalized scores for the Inverse Distance Weighting (IDW) Method applied to a subset of Kentucky rain gauge stations is presented in Table 11. The IDW Method exhibited relatively poor overall performance when compared to all interpolation methods evaluated in this study at each of the tested stations, with below median performance in preserving site and regional statistics relative to other evaluated methods. With regards to error minimization, the IDW Method exhibited above median to median performance; however, the performance in error minimization did not transfer to similar (median to above median) performance in preserving site and regional statistics.
Table 11: Summarized Performance Measures for Inverse Distance Method
Station Estimation Site Regional Overall Rank Errors Statistics Statistics Score (1-71)
1 0.242 0.877 0.852 1.971 69 2 0.173 0.836 0.759 1.768 58 3 0.529 0.702 0.763 1.994 54 15 0.471 0.896 0.907 2.275 61
With regards to preservation of data homogeneity, at Stations 1, 2, and 3, the post-infilling precipitation data sets passed the same homogeneity tests as the original, non-imputed historical precipitation data sets. However, station 15 failed the SNHT test for homogeneity in addition to failing the Pettitt and Von Neumann tests.
With regards to heteroscedasticity, the IDW Method applied at the four stations did not exhibit a discernible trend in residuals. There was also no significant deviation from the historical temporal autocorrelation of daily precipitation values using this method. None of the stations
85
passed the Kolmogorov-Smirnov two sample test when only non-zero precipitation values were considered.
Table 12: Summarized Performance Measures for post-corrected IDW Method
Station Estimation Site Regional Overall Rank Errors Statistics Statistics Score (1-71)
1 0.213 0.503 0.481 1.197 37 2 0.141 0.364 0.647 1.152 17 3 0.292 0.262 0.324 0.878 20 15 0.306 0.576 0.687 1.569 54
Modification of the IDW Method to include dry/wet days post correction based on a Single
Best Estimator station significantly improved the performance of the method in all three performance categories (Estimation Errors, Site Statistics, and Regional Statistics) and thusly improved the overall score of the IDW Method, generally raising the ranking of the method from below median - lowest ranking to above median - median performance relative to other interpolation methods. Without dry/wet days post correction, all IDW Method variants failed the
Kolmogorov-Smirnov two sample test when only non-zero precipitation values were considered.
With correction, Stations 2 and 3 passed the statistical test; however, Stations 1 and 15 did not.
In this study, the post-correction tended to result in a lower mean daily precipitation reading at a site. As shown in the figures below, without post correction, the IDW Method preserved both site and regional mean daily precipitation values. However, a slight reduction in the mean from historical levels, caused by dry/wet days post-correction, generally disturbed the ability of IDW Method to preserve the site and regional mean. Post-correction only slightly improved the performance of regional standard deviations and regional station-to-station correlations.
86
Post-correction did not allow the IDW Method to improve with regards to passing homogeneity tests. Rather, Station 3 failed an additional test, the SNHT, when post-correction was applied to the IDW Method results. The figures below present results related to the preservation of summary site and regional statistics for the IDW Method and also show the affects of post-correction on the results. Results suggest that careful consideration should be made with regards to applying dry/wet days post-correction to IDW Method or whether an alternative interpolation regime should be utilized instead.
87
Original Method With Dry/ Wet days Post -Correction
Figure 31: Site Summary Statistics for IDW Method at Station 1
88
Original Method With Dry/Wet days Post -Correction
Station (on figure) Actual Station Station (on figure) Actual Station 1 15 8 8 2 2 9 9 3 3 10 10 4 4 11 11 5 5 12 12 6 6 13 13 7 7 14 14 Figure 32: Regional Summary Statistics for IDW Method at Station 1
89
Original Method With Dry/Wet days Post -Correction
Figure 33: Site Summary Statistics for IDW Method at Station 2
90
Original Method With Dry/Wet days Post -Correction
Station (on figure) Actual Station Station (on figure) Actual Station 1 15 8 8 2 1 9 9 3 3 10 10 4 4 11 11 5 5 12 12 6 6 13 13 7 7 14 14 Figure 34: Regional Summary Statistics for IDW Method at Station 2
91
Original Method With Dry/Wet days Post -Correction
Figure 35: Site Summary Statistics for IDW Method at Station 3
92
Original Method With Dry/Wet days Post -Correction
Station (on figure) Actual Station Station (on figure) Actual Station 1 15 8 8 2 1 9 9 3 2 10 10 4 4 11 11 5 5 12 12 6 6 13 13 7 7 14 14 Figure 36: Regional Summary Statistics for IDW Method at Station 3
93
Original Method With Dry/Wet days Post -Correction
Figure 37: Site Summary Statistics for IDW Method at Station 15
94
Original Method With Dry/Wet days Post -Correction
Station (on figure) Actual Station Station (on figure) Actual Station 1 1 8 8 2 2 9 9 3 3 10 10 4 4 11 11 5 5 12 12 6 6 13 13 7 7 14 14 Figure 38: Regional Summary Statistics for IDW Method at Station 15
95
5.6 SOFW Method
The summarized, normalized scores for the Single Objective Weighting (SOFW) Method applied to a subset of Kentucky rain gauge stations is presented in Table 13. The best objective functions from the ones evaluated with SOFW Method variants are AE for Station 1,2, and 3 and
MRE for Station 15. The AE objective function was the best performing in three of the four evaluated stations, possibly suggesting that minimizing AE may have universal superiority with regards to overall performance amongst the SOFW non-corrected variants. Results suggest that the strength of SOFW variants lie in minimizing estimation errors at the expense of preserving site and regional statistics. Relative to the other interpolation methods explored in this study,
SOFW Method achieved below median rankings.
Table 13: Summarized Performance Measures for SOFW Variants
Station Objective Function Estimation Site Regional Overall Rank Errors Statistics Statistics Score (1-71)
AE 0.078 0.769 0.722 1.569 47 MAE 0.080 0.771 0.724 1.575 52 1 RMSE 0.083 0.834 0.791 1.709 59 0.083 0.834 0.791 1.709 60 MRE 0.279 0.784 0.734 1.797 66 AE 0.002 0.896 0.610 1.508 43 MAE 0.001 0.897 0.610 1.509 45 2 RMSE 0.028 0.896 0.612 1.536 47 0.041 0.974 0.723 1.739 57 MRE 0.314 0.783 0.692 1.789 59 AE 0.122 0.672 0.593 1.387 26 MAE 0.122 0.674 0.592 1.388 28 3 MRE 0.184 0.660 0.609 1.453 36 0.100 0.775 0.722 1.597 49 RMSE 0.103 0.785 0.728 1.616 51 MRE 0.312 0.402 0.539 1.254 37 MAE 0.109 0.575 0.610 1.294 43 15 AE 0.109 0.578 0.619 1.307 44 RMSE 0.091 0.627 0.678 1.397 46 0.091 0.628 0.679 1.399 47
96
With regards to preservation of data homogeneity, at Stations 1 and 15, the post-infilling precipitation data sets passed the same homogeneity tests as the original, non-imputed historical precipitation data sets. However, the other two stations, 2 and 3, failed the SNHT test for homogeneity in addition to failing the Pettitt and Von Neumann tests.
With regards to heteroscedasticity, none of the SOFW variants applied at the four stations exhibited a discernible trend in residuals. There was also no significant deviation from the historical temporal autocorrelation of daily precipitation values using this method.
Table 14: Summarized Performance Measures for SOFW Variants with post correction
Station Objective Estimation Site Regional Overall Rank Function Errors Statistics Statistics Score (1-71)
AE 0.092 0.238 0.400 0.730 10 MAE 0.103 0.241 0.403 0.747 11 1 MRE 0.239 0.284 0.429 0.952 24 RMSE 0.070 0.454 0.444 0.968 25 0.077 0.460 0.445 0.982 26 AE 0.032 0.355 0.606 0.993 4 MAE 0.042 0.353 0.606 1.002 6 2 RMSE 0.058 0.350 0.606 1.014 9 0.018 0.385 0.653 1.056 12 MRE 0.249 0.375 0.651 1.276 22 AE 0.147 0.118 0.241 0.506 6 MAE 0.161 0.117 0.246 0.524 10 3 0.115 0.173 0.271 0.560 11 RMSE 0.113 0.178 0.291 0.581 14 MRE 0.220 0.098 0.300 0.619 16 MRE 0.344 0.013 0.176 0.533 3 MAE 0.125 0.147 0.326 0.598 6 15 AE 0.134 0.148 0.332 0.614 9 0.086 0.202 0.392 0.680 13 RMSE 0.104 0.202 0.396 0.702 17
Modification of the SOFW Method to include dry/wet days post correction based on a
Single Best Estimator station significantly improved the overall score of the SOFW Method
rformance 97
relative to other interpolation methods as shown in Table 14. Some variants were able to achieve top rankings. Post correction of SOFW variants also indicates a trade-off between the minimization of estimation errors and preservation of site and regional statistics; for a relatively small loss in performance in minimization of estimation errors, significant improvements in preservation of site and regional statistics were achieved. Without dry/wet days post correction, all SOFW Method variants failed the Kolmogorov-Smirnov two sample test when only non-zero precipitation values were considered. Post-correction allowed for the passing of this test and its visual counterpart as shown in the cumulative density plots in the figures below.
In this study, the post-correction tended to result in a lower mean daily precipitation reading at a site. As shown in the figures below, a slight reduction in the mean from historical levels generally disturbed the ability of SOFW Method variants to preserve the regional mean.
This was the case at all four stations except for Station 15, in which post correction on the SOFW
Method variant with an MRE objective function reduced the site mean daily precipitation to closer to historical levels, and as a result, more accurately preserved the regional mean daily precipitation statistic as shown in Figure 46. Post-correction appears to slightly improve the performance of preservation of regional standard deviations and station-to-station correlations.
Interestingly, post-correction allowed the variants that originally failed three homogeneity tests at Station 2 to fail only two of the four homogeneity tests. At Station 3, post-correction did not allow for the passing of two homogeneity tests when three had failed without post-correction.
The figures below present results related to the preservation of summary site and regional statistics for the SOFW Method variants with the best performance (before post-correction) at each of the selected stations. The figures also show the affects of post- results.
98
Original Method With Dry/Wet days Post -Correction
Figure 39: Site Summary Statistics for SOFW Method (AE Objective Function) at Station 1
99
Original Method With Dry/Wet days Post-Correction
Station (on figure) Actual Station Station (on figure) Actual Station 1 15 8 8 2 2 9 9 3 3 10 10 4 4 11 11 5 5 12 12 6 6 13 13 7 7 14 14 Figure 40: Regional Summary Statistics for SOFW Method (AE Objective Function) at Station 1
100
Original Method With Dry/Wet days Post -Correction
Figure 41: Site Summary Statistics for SOFW Method (AE Objective Function) at Station 2
101
Original Method With Dry/Wet days Post-Correction
Station (on figure) Actual Station Station (on figure) Actual Station 1 15 8 8 2 1 9 9 3 3 10 10 4 4 11 11 5 5 12 12 6 6 13 13 7 7 14 14 Figure 42: Regional Summary Statistics for SOFW Method (AE Objective Function) at Station 2
102
Original Method With Dry/Wet days Post -Correction
Figure 43: Site Summary Statistics for SOFW Method (AE Objective Function) at Station 3
103
Original Method With Dry/Wet days Post-Correction
Station (on figure) Actual Station Station (on figure) Actual Station 1 15 8 8 2 1 9 9 3 2 10 10 4 4 11 11 5 5 12 12 6 6 13 13 7 7 14 14 Figure 44: Regional Summary Statistics for SOFW Method (AE Objective Function) at Station 3
104
Original Method With Dry/Wet days Post-Correction
Figure 45: Site Summary Statistics for SOFW Method (MRE Objective Function) at Station 15
105
Original Method With Dry/Wet days Post-Correction
Station (on figure) Actual Station Station (on figure) Actual Station 1 1 8 8 2 2 9 9 3 3 10 10 4 4 11 11 5 5 12 12 6 6 13 13 7 7 14 14 Figure 46: Regional Summary Statistics for SOFW Method (MRE Objective Function) at Station 15
106
Post-correction of the SOFW Method variants also allowed for the more accurate preservation of Markov chain transition probabilities as shown in Figure 47.
107
Station Original Method With Dry/Wet days Post -Correction
1
2
3
15
Figure 47: Markov chain transition probabilities for best SOFW Method Variant at base station 108
5.7 MOFW Method
The summarized, normalized scores for the top ranking, non-corrected Multi-Objective
Weighting (MOFW) Method variants applied at four Kentucky rain gauge stations are presented in Table 15. Non-corrected MOFW Method variants were able to achieve above median ranking in all tested stations; however, other non-corrected MOFW Method variants were from the lowest ranking methods.
Table 15: Summarized Performance Measures for top ranking, non-corrected MOFW Variants
Estimation Site Regional Overall Rank Station Objective Function Errors Statistics Statistics Score (1-71)
1 RMSE + Site Stats 0.175 0.392 0.223 0.790 12
2 RMSE + Site Stats 0.288 0.576 0.332 1.196 18
3 AE+100(Site+Regional) 0.160 0.592 0.499 1.250 22
15 RMSE + Site Stats 0.378 0.336 0.257 0.971 21
Results suggest that the following multi-objective formulation from those evaluated in this study may have universal superiority over other evaluated MOFW variants across multiple base stations:
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Minimize :
(5.1)
Subject to:
(5.2)
(5.3)
(5.4)
(5.5)
(5.6)
With regards to preservation of data homogeneity, at Stations 1 and 15, the post-infilling precipitation data sets passed the same homogeneity tests as the original, non-imputed historical precipitation data sets for variants listed in Table 15. However, Stations, 2 and 3, failed the SNHT test for homogeneity in addition to failing the Pettitt and Von Neumann tests.
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With regards to heteroscedasticity, none of these MOFW variants applied at the four stations exhibited a discernible trend in residuals. There was also no significant deviation from the historical temporal autocorrelation of daily precipitation values using this method.
Table 16: Summarized Performance Measures for top ranking, post-corrected MOFW Variants
Station Objective Function Estimation Site Regional Overall Rank Errors Statistics Statistics Score (1-71)
1 RMSE + Site Stats 0.221 0.012 0.153 0.387 1
2 RMSE + Site Stats 0.308 0.235 0.339 0.882 2
3 AE+Regional 0.132 0.116 0.243 0.491 2
15 RMSE + Site Stats 0.391 0.033 0.000 0.424 1
Modification of the MOFW Method variants in this section to include dry/wet days post correction based on a Single Best Estimator station significantly improved the overall score of the
MOFW Method variants as shown in Table 16. After post-correction, some MOFW variants were able to achieve top ranking with above median performance in all three categories of performance measures. Results further confirm the existence of a trade-off between minimization of estimation errors and the preservation of site and regional statistics as a slight reduction of performance in minimization of estimation errors was coupled with a significant increase in performance in preserving site and regional statistics. The top ranked post-corrected MOFW variants exhibited above median performance in preserving regional statistics, even when terms involving summary regional statistics preservation were excluded from the multi-objective formulation.
Like when applied to the SOFW variants, the dry/wet days post correction allowed for the passing of the Kolmogorov-Smirnov two sample test, improvement in preservation of the historical cumulative density function for non-zero precipitation values, and improvement in
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preservation of historical Markov chain transition properties. There was no improvement with regards to allowing stations that had failed three homogeneity tests after infilling to pass at least two homogeneity tests.
5.8 Threshold Method: Global Mean Daily Precipitation
The summarized, normalized scores for the top ranking variants Mean Daily Precipitation
Threshold Optimization Method without post-correction applied to a subset of Kentucky rain gauge stations is presented in Table 17. In this study, the mean daily precipitation magnitude selected to separate time intervals for optimization was 25mm, as results from interpolation methods (e.g. SOFW Method variants) evaluated previously involving un-separated precipitation data sets appeared to adopt an underestimation bias between 25 and 30 mm as exemplified in
Figure 48.
Table 17: Summarized Performance Measures for 25mm Global Mean Daily Precipitation Threshold Method (non-corrected)
Station Objective Function Estimation Site Regional Overall Rank Errors Statistics Statistics Score (1-71)
1 AE 0.076 0.773 0.732 1.581 53 2 MAE 0.036 0.925 0.635 1.596 50 3 MAE 0.124 0.687 0.609 1.420 31 15 MAE 0.098 0.651 0.679 1.428 48
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Figure 48: Estimation accuracy at Station 3 for non-corrected SOFW Method with MAE objective function
The best objective functions for the stations evaluated with Threshold Mean Precipitation
Method variants are AE for Station 1 and MAE for Stations 2, 3, and 15. The top ranking, non- corrected Mean Threshold Method variants generally achieved below median rankings, with the weakness of the method primarily in the preservation of site and regional statistics. At all four evaluated stations the non-corrected Mean Threshold Method was outperformed by the non- corrected SOFW Method for the same objective function. This finding suggests that the additional computational effort of the Mean Threshold Method, when compared to the SOFW
Method, is not worthwhile.
With regards to preservation of data homogeneity, at Stations 2, 3 and 15, the post-infilling precipitation data failed the SNHT test for homogeneity in addition to failing the Pettitt and Von
Neumann tests.
With regards to heteroscedasticity, none of the Mean Threshold Method variants applied at
the four stations exhibited a discernible trend in residuals. There was also no significant deviation
from the historical temporal autocorrelation of daily precipitation values using this method.
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Table 18: Summarized Performance Measures for 25mm Global Mean Daily Precipitation Threshold Method (non-corrected)
Station Objective Function Estimation Site Regional Overall Rank Errors Statistics Statistics Score (1-71)
1 AE 0.041 0.251 0.411 0.704 6 2 MAE 0.039 0.383 0.632 1.055 11 3 AE 0.138 0.123 0.245 0.506 5 15 AE 0.111 0.108 0.395 0.614 10
Modification of the Mean Threshold Method variants to include dry/wet days post correction based on a Single Best Estimator station significantly improved the overall score of the
Mean Threshold Method variants, generally raising the overall ranking of the best performing of
study. Without dry/wet days post correction, all Mean Threshold Method variants failed the
Kolmogorov-Smirnov two sample test when only non-zero precipitation values were considered.
The post-correction allowed for the passing of this test in the majority of cases and its visual counterpart as shown in the cumulative density plots below.
In this study, the post-correction tended to result in a lower mean daily precipitation reading at a site, as found with post-correction of other interpolation methods. As shown in the figures below, a slight reduction in the mean from historical levels generally disturbed the ability of Mean Threshold Method variants to preserve the regional mean when the original interpolation had already underestimated the historical mean. Post-correction did not generally appear to significantly improve or worsen the performance of regional standard deviations or regional station-to-station correlations.
Post-correction allowed the Mean Threshold Method variants that originally failed three homogeneity tests at Stations 2 and 15 to fail only two of the four homogeneity tests. At Station
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3, post-correction did not allow for the passing of two homogeneity tests when three had failed without post-correction.
Original Method With Dry/Wet days Post -Correction
Figure 49: Site Summary Statistics for Mean Threshold Method (AE Objective Function) at Station 1
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Original Method With Dry/Wet days Post -Correction
Station (on figure) Actual Station Station (on figure) Actual Station 1 15 8 8 2 2 9 9 3 3 10 10 4 4 11 11 5 5 12 12 6 6 13 13 7 7 14 14 Figure 50: Regional Summary Statistics for Mean Threshold Method (AE Objective Function) at Station 1
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Original Method With Dry/Wet days Post -Correction
Figure 51: Site Summary Statistics for Mean Threshold Method (MRE Objective Function) at Station 2
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Original Method With Dry/Wet days Post -Correction
Station (on figure) Actual Station Station (on figure) Actual Station 1 15 8 8 2 1 9 9 3 3 10 10 4 4 11 11 5 5 12 12 6 6 13 13 7 7 14 14 Figure 52: Regional Summary Statistics for Mean Threshold Method (MRE Objective Function) at Station 2
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Original Method With Dry/Wet days Post -Correction
Figure 53: Site Summary Statistics for Mean Threshold Method (MAE Objective Function) at Station 3
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Original Method With Dry/Wet days Post -Correction
Station (on figure) Actual Station Station (on figure) Actual Station 1 15 8 8 2 1 9 9 3 2 10 10 4 4 11 11 5 5 12 12 6 6 13 13 7 7 14 14 Figure 54: Regional Summary Statistics for Mean Threshold Method (MAE Objective Function) at Station 3
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Original Method With Dry/Wet days Post -Correction
Figure 55: Site Summary Statistics for Mean Threshold Method (MRE Objective Function) at Station 15
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Original Method With Dry/Wet days Post -Correction
Station (on figure) Actual Station Station (on figure) Actual Station 1 1 8 8 2 2 9 9 3 3 10 10 4 4 11 11 5 5 12 12 6 6 13 13 7 7 14 14 Figure 56: Regional Summary Statistics for Mean Threshold Method (MRE Objective Function) at Station 15
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Table 19: Performance comparison between Mean Threshold Method and SOFW method for identical objective functions
Rank (1 -71) 25 mm Global Objective Station Mean SOFW Function Threshold no threshold Method AE 6 10 1 MAE 23 11 CORREL 29 26 MAE 11 6 2 RMSE 13 9 AE 14 4 AE 5 6 3 MAE 8 10 CORREL 13 11 AE 10 9 15 MAE 11 6 CORREL 14 13
While post-correction did improve overall performance of the Mean Threshold Method variants, rankings of the variants were generally lower than rankings of post-corrected SOFW methods for the same objective function at the four evaluated stations as shown in the shaded entries of Table 19, further suggesting that the additional computational effort of the Mean
Threshold Method may not be worthwhile when other, simpler methods are able to provide similar or superior results.
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5.9 Threshold Method: Single Best Estimator (SBE)
The summarized, normalized scores for the uncorrected SBE Precipitation Threshold
Optimization Method applied to a subset of Kentucky rain gauge stations is presented in Table
20. In this study, the SBE station for each base station was selected as the station with the highest positive correlation with the base station. Stations 8, 4, 9, and 12 were the SBE stations for
Station 1, 2, 3, and 15 respectively, with SBE stations exhibiting a historical positive correlation with their respective base stations of approximately 0.60 or greater.
Table 20: Summarized Performance Measures for SBE Precipitation Threshold Method
Station Estimation Site Regional Overall Rank Errors Statistics Statistics Score (1-71)
1 0.018 0.844 0.740 1.602 56 2 0.010 0.697 0.628 1.335 27 3 0.000 0.780 0.708 1.488 39 15 0.000 0.590 0.730 1.320 45
To determine the SBE precipitation threshold to separate data at each base station, an iterative analysis was conducted for the SBE station for each base station in which the sum of least squares interpolation method was applied using different SBE thresholds. The SBE threshold that resulted in the lowest RMSE was selected as the ideal SBE threshold. Figure 57 presents the RMSE performance measure on validation data set results at Station 15. For Station
15, the ideal SBE threshold was selected as 13 mm. Ideal SBE thresholds were selected as 28 mm, 10.8 mm, and 13.3 mm for Stations 1, 2, and 3 respectively.
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Figure 57: Selection of an SBE Threshold for Station 15
Relative to other interpolation methods evaluated in this study, the strength of the SBE
Threshold method is in the minimization of estimation errors; however, the method exhibits below median performance in preserving site and regional statistics. As such, the SBE Threshold
Method is generally classified with a below median ranking in overall performance.
With regards to preservation of data homogeneity, at Stations 1, 2, and 15, the post- infilling precipitation data sets passed the same homogeneity tests as the original, non-imputed historical precipitation data sets. However, Station 3 failed the SNHT test for homogeneity in addition to failing the Pettitt and Von Neumann tests.
With regards to heteroscedasticity, none of the SBE Threshold Method experiments applied at the four stations exhibited a discernible trend in residuals. There was also no significant deviation from the historical temporal autocorrelation of daily precipitation values using this method.
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Table 21: Summarized Performance Measures for post-corrected SBE Precipitation Threshold Method
Station Estimation Site Regional Overall Rank Errors Statistics Statistics Score (1-71)
1 0.000 0.474 0.445 0.919 20 2 0.059 0.228 0.529 0.816 1 3 0.064 0.270 0.388 0.722 18 15 0.030 0.207 0.417 0.653 12
Modification of the SBE Threshold Method to include dry/wet days post correction based on a
Single Best Estimator station significantly improved the overall score of the SBE Threshold
median ranking relative to other interpolation methods. Without dry/wet days post correction, all SBE Threshold Method applications failed the Kolmogorov-Smirnov two sample test when only non-zero precipitation values were considered. The post-correction allowed for the passing of this test and its visual counterpart at Stations 2, 3, and 15 as shown in the cumulative density plots in the figures below.
In this study, the post-correction tended to result in a lower mean daily precipitation reading at a site. As shown in the figures below, a slight reduction in the mean from historical levels generally disturbed the ability of SBE Threshold Method variants to preserve the regional mean. Post-correction slightly improved the performance of regional standard deviations or regional station-to-station correlations. Post-correction did not allow stations to pass or fail additional homogeneity tests compared to the SBE Threshold Method without post-correction.
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Original Method With Dry/Wet days Post -Correction
Figure 58: Site Summary Statistics for SBE Threshold Method at Station 1
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Original Method With Dry/Wet days Post -Correction
Station (on figure) Actual Station Station (on figure) Actual Station 1 15 8 8 2 2 9 9 3 3 10 10 4 4 11 11 5 5 12 12 6 6 13 13 7 7 14 14 Figure 59: Regional Summary Statistics for SBE Threshold Method at Station 1
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Original Method With Dry/Wet days Post -Correction
Figure 60: Site Summary Statistics for SBE Threshold Method at Station 2
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Original Method With Dry/Wet days Post -Correction
Station (on figure) Actual Station Station (on figure) Actual Station 1 15 8 8 2 1 9 9 3 3 10 10 4 4 11 11 5 5 12 12 6 6 13 13 7 7 14 14 Figure 61: Regional Summary Statistics for SBE Threshold Method at Station 2
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Original Method With Dry/Wet days Post -Correction
Figure 62: Site Summary Statistics for SBE Threshold Method at Station 3
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Original Method With Dry/Wet days Post -Correction
Station (on figure) Actual Station Station (on figure) Actual Station 1 15 8 8 2 1 9 9 3 2 10 10 4 4 11 11 5 5 12 12 6 6 13 13 7 7 14 14 Figure 63: Regional Summary Statistics for SBE Threshold Method at Station 3
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Original Method With Dry/Wet days Post-Correction
Figure 64: Site Summary Statistics for SBE Threshold Method at Station 15
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Original Method With Dry/Wet days Post -Correction
Station (on figure) Actual Station Station (on figure) Actual Station 1 1 8 8 2 2 9 9 3 3 10 10 4 4 11 11 5 5 12 12 6 6 13 13 7 7 14 14 Figure 65: Regional Summary Statistics for SBE Threshold Method at Station 15
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5.10 Bootstrap Method
The summarized, normalized scores for the non-corrected Bootstrap Method applied to a subset of Kentucky rain gauge stations is presented in Table 22. In the current context, the best station weights were selected based on overall performance of the weights on the validation data set in the bootstrapping process. The bootstrapping process involved 3,000 iterations. In the validation data set used in the bootstrapping process, the sets of weights resulted in interpolation method variants exhibiting a widely varying range of overall performances. Case in point, at
Station 1, the validation data set to which 3,000 bootstrapping iterations were applied featured normalized overall performance scores ranging from 0.228 to 2.675 on a scale of 0 (top performance) to 3 (worst overall performance).
Table 22: Summarized Performance Measures for Bootstrap Method
Station Estimation Site Regional Overall Rank Errors Statistics Statistics Score (1-71)
1 0.088 0.831 0.798 1.717 61 2 0.286 0.971 0.743 1.999 70 3 0.150 0.840 0.802 1.793 53 15 0.167 0.629 0.689 1.485 50
The Bootstrap Method, despite 3,000 iterations used to select the best performing set of neighboring stations weights for missing data interpolation, exhibited relatively poor overall performance when compared to all interpolation methods evaluated in this study at each of the tested stations. With regards to error minimization, the Bootstrap Method, performed relatively well; however, the performance in error minimization did not transfer to superior performance in preserving site and regional statistics.
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With regards to preservation of data homogeneity, at Stations 1 and 15, the post-infilling precipitation data sets passed two homogeneity tests. However, Stations 2 and 3 failed the SNHT test for homogeneity in addition to failing the Pettitt and Von Neumann tests.
With regards to heteroscedasticity, the Bootstrap Method applied at the four stations did not exhibit a discernible trend in residuals. There was also no significant deviation from the historical temporal autocorrelation of daily precipitation values using this method. None of the stations passed the Kolmogorov-Smirnov two sample test when only non-zero precipitation values were considered.
Table 23: Summarized Performance Measures for post-corrected Bootstrap Method
Station Estimation Site Regional Overall Rank Errors Statistics Statistics Score (1-71)
1 0.094 0.445 0.453 0.992 27 2 0.337 0.510 0.635 1.481 38 3 0.151 0.211 0.302 0.665 17 15 0.132 0.233 0.432 0.797 19
Modification of the Bootstrap Method to include dry/wet days post correction based on a
Single Best Estimator station significantly improved the overall performance of the Bootstrap
Method, generally raising the ranking of the method from lowest ranking to above median- median performance relative to other interpolation methods. Without dry/wet days post correction, all Bootstrap Method applications failed the Kolmogorov-Smirnov two sample test when only non-zero precipitation values were considered. The post-correction allowed for the passing of this test and its visual counterpart as shown in the cumulative density plots in the figures below.
In this study, the post-correction tended to result in a lower mean daily precipitation reading at a site. As shown in the figures below, without post correction, the Bootstrap Method 136
preserved both site and regional mean daily precipitation values relatively well. However, a slight reduction in the mean from historical levels, caused by dry/wet days post-correction, generally disturbed the ability of the Bootstrap Method to preserve the site and regional mean. Post- correction slightly improved the performance of regional standard deviations or regional station- to-station correlations.
Post-correction did not allow the Bootstrap Method to improve with regards to passing homogeneity tests. The figures below present results related to the preservation of summary site and regional statistics for the Bootstrap Method and also show the affects of post-correction on the results.
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Original Method With Dry/Wet days Post -Correction
Figure 66: Site Summary Statistics for Bootstrap Method at Station 1
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Original Method With Dry/Wet days Post -Correction
Station (on figure) Actual Station Station (on figure) Actual Station 1 15 8 8 2 2 9 9 3 3 10 10 4 4 11 11 5 5 12 12 6 6 13 13 7 7 14 14 Figure 67: Regional Summary Statistics for Bootstrap Method at Station 1
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Original Method With Dry/Wet days Post -Correction
Figure 68: Site Summary Statistics for Bootstrap Method at Station 2
140
Original Method With Dry/Wet days Post -Correction
Station (on figure) Actual Station Station (on figure) Actual Station 1 15 8 8 2 1 9 9 3 3 10 10 4 4 11 11 5 5 12 12 6 6 13 13 7 7 14 14 Figure 69: Regional Summary Statistics for Bootstrap Method at Station 2
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Original Method With Dry/Wet days Post -Correction
Figure 70: Site Summary Statistics for Bootstrap Method at Station 3
142
Original Method With Dry/Wet days Post -Correction
Station (on figure) Actual Station Station (on figure) Actual Station 1 15 8 8 2 1 9 9 3 2 10 10 4 4 11 11 5 5 12 12 6 6 13 13 7 7 14 14 Figure 71: Regional Summary Statistics for Bootstrap Method at Station 3
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Original Method With Dry/Wet days Post -Correction
Figure 72: Site Summary Statistics for Bootstrap Method at Station 15
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Original Method With Dry/Wet days Post -Correction
Station (on figure) Actual Station Station (on figure) Actual Station 1 1 8 8 2 2 9 9 3 3 10 10 4 4 11 11 5 5 12 12 6 6 13 13 7 7 14 14 Figure 73: Regional Summary Statistics for Bootstrap Method at Station 15
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5.11 Fuzzy Logic Method
The summarized, normalized scores for the non-corrected Fuzzy Logic Method applied to a subset of Kentucky rain gauge stations is presented in Table 22. The Fuzzy Logic Method, achieved median below median ranking when compared to all interpolation methods evaluated in this study at each of the tested stations, with median below median performance in preserving site and regional statistics relative to other evaluated methods. With regards to error minimization, the Fuzzy Logic Method exhibited top above median performance.
Table 24: Summarized Performance Measures for Fuzzy Logic Method
Station Estimation Site Regional Overall Rank Errors Statistics Statistics Score (1-71)
1 0.085 0.626 0.512 1.223 39 2 0.092 0.768 0.492 1.352 28 3 0.172 0.698 0.559 1.429 34 15 0.207 0.528 0.529 1.264 38
The original and intermediate formulations of the Fuzzy Logic procedure relied on a multi- objective functions composed of the summation of RMSE and summary site and regional statistics as follows:
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Minimize :
(5.7)
Subject to:
(5.8)
(5.9)
(5.10)
In the original formulation, objective function (5.7) was constrained to the minimum value of the RMSE performance measure from the SOFW method. In the intermediate formulation, objective function (5.7) is constrained to the same value, except that a tolerance of 0.5 is added to the constraint to allow for the preservation of site and regional statistics at the expense of losing performance in minimizing estimation errors. The final formulation involved maximizing a membership function , a value between 0 and 1, as mentioned in Chapter Three. When equals
0, the final formulation is considered to prefer minimization of estimation errors over the preservation of site and regional statistics. When equals 1, the final formulation is considered to prefer the preservation of site and regional statistics over the minimization of estimation errors.
The gradient between these two extremes is assumed to be linear in this case study. The values for the stations evaluated in this study are presented in Table 25
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achieved by the Fuzzy Logic method in minimizing estimation errors while preserving site and regional statistics.
Table 25: Membership function values for Fuzzy Logic Method
Station Membership Function Value
1 0.679 2 0. 673 3 0.553 15 0. 559
With regards to preservation of data homogeneity, at Stations 1, and 15, the post-infilling precipitation data sets passed two homogeneity tests. Stations 2 and 3 failed the SNHT test for homogeneity in addition to failing the Pettitt and Von Neumann tests.
With regards to heteroscedasticity, the Fuzzy Logic Method applied at the four stations did not exhibit a discernible trend in residuals. There was also no significant deviation from the historical temporal autocorrelation of daily precipitation values using this method. None of the stations passed the Kolmogorov-Smirnov two sample test when only non-zero precipitation values were considered.
Table 26: Summarized Performance Measures for post-corrected Fuzzy Logic Method
Station Estimation Site Regional Overall Rank Errors Statistics Statistics Score (1-71)
1 0.109 0.144 0.297 0.551 4 2 0.134 0.375 0.499 1.007 7 3 0.160 0.146 0.196 0.501 3 15 0.183 0.127 0.272 0.582 4
Modification of the Fuzzy Logic Method to include dry/wet days post correction based on a Single Best Estimator station significantly improved the performance of the method in all three 148
performance categories (Estimation Errors, Site Statistics, and Regional Statistics) and thusly improved the overall score of the Fuzzy Logic Method, generally raising the ranking of the method to top ranking amongst all evaluated base stations. Without dry/wet days post correction, all Fuzzy Logic Method variants failed the Kolmogorov-Smirnov two sample test when only non- zero precipitation values were considered. With correction, all four evaluated stations passed the statistical test.
In this study, the post-correction tended to result in a lower mean daily precipitation reading at a site. As shown in the figures below, without post correction, the Fuzzy Logic Method preserved both site and regional mean statistics relatively well. However, a slight reduction in the mean from historical levels, caused by dry/wet days post-correction, generally disturbed the ability of Fuzzy Logic Method to preserve the site and regional mean. Post-correction appeared to slightly improve the performance of regional standard deviations and regional station-to-station correlations.
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Original Method With Dry/Wet days Post -Correction
Figure 74: Site Summary Statistics for Fuzzy Logic Method at Station 1
150
Original Method With Dry/Wet days Post -Correction
Station (on figure) Actual Station Station (on figure) Actual Station 1 15 8 8 2 2 9 9 3 3 10 10 4 4 11 11 5 5 12 12 6 6 13 13 7 7 14 14 Figure 75: Regional Summary Statistics for Bootstrap Method at Station 1
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Original Method With Dry/Wet days Post -Correction
Figure 76: Site Summary Statistics for Fuzzy Logic Method at Station 2
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Original Method With Dry/Wet days Post -Correction
Station (on figure) Actual Station Station (on figure) Actual Station 1 15 8 8 2 1 9 9 3 3 10 10 4 4 11 11 5 5 12 12 6 6 13 13 7 7 14 14 Figure 77: Regional Summary Statistics for Fuzzy Logic Method at Station 2
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Original Method With Dry/Wet days Post -Correction
Figure 78: Site Summary Statistics for Fuzzy Logic Method at Station 3
154
Original Method With Dry/Wet days Post -Correction
Station (on figure) Actual Station Station (on figure) Actual Station 1 15 8 8 2 1 9 9 3 2 10 10 4 4 11 11 5 5 12 12 6 6 13 13 7 7 14 14 Figure 79: Regional Summary Statistics for Fuzzy Logic Method at Station 3
155
Original Method With Dry/Wet days Post -Correction
Figure 80: Site Summary Statistics for Fuzzy Logic Method at Station 15
156
Original Method With Dry/Wet days Post -Correction
Station (on figure) Actual Station Station (on figure) Actual Station 1 1 8 8 2 2 9 9 3 3 10 10 4 4 11 11 5 5 12 12 6 6 13 13 7 7 14 14 Figure 81: Regional Summary Statistics for Fuzzy Logic Method at Station 15
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5.12 Best Interpolation Method
The top ten ranked interpolation methods from the 71 methods evaluated in this study for four base stations are listed in Table 27. Post-corrected variants of the MOFW, SOFW, and Fuzzy
Logic Method featured top rankings, fulfilling Condition One of the procedure defined for the selection of a best interpolation method to apply to multiple base stations. Condition Two requires that any top ranked method must have a performance score of less than 0.500 in
Estimation Errors, Site Statistics, and Regional Statistics. After these two conditions are fulfilled, only one method remains available for the best method selection, and thusly the requirement to fulfill Condition Three is moot. From the evaluated methods in this case study, the best interpolation method for application at all base stations in the case study is taken as the post- corrected variant of the Fuzzy Logic Method.
Table 27: Top Ten Ranked Interpolation Methods
Rank Station 1 Station 2 Station 3 Station 15 1 RMSE+Site SBE Threshold Naïve SBE RMSE+Site 2 AE+100(Site+Reg.) RMSE+Site AE +Reg. AE+100(Site+Reg.) 3 MAE+Site AE +Reg. Fuzzy Logic SOFW, MRE 4 Fuzzy Logic SOFW, AE AE+Site Fuzzy Logic 5 RMSE+Reg. Naïve SBE SOFW, AE (25mm) AE+Site 6 SOFW, AE (25 mm) SOFW, MAE SOFW, AE SOFW, MAE 7 AE +Reg. Fuzzy Logic AE +Site+Reg AE +Reg 8 AE+Site AE+Site SOFW, MAE (25mm) AE +Site+Reg. 9 AE +Site+Reg. SOFW, RMSE AE+100(Site+Reg.) SOFW, AE 10 SOFW, AE AE+100(Site+Reg) SOFW, MAE SOFW, AE (25mm) Note: all interpolation methods represent post-corrected variants
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5.13 Best Interpolation Method: Further Analysis
The post-corrected Fuzzy Logic Method was applied at all 15 base stations involved in this study. The estimated versus observed daily precipitation plots for all 15 base stations using this method are presented below. Results sugg minimizing estimation errors, overestimation errors for relatively low precipitation events and underestimation errors for relatively high precipitation events (above 25 mm) remain.
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Station 1 Station 2
Station 3 Station 4
Station 5 Station 6
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Station 7 Station 8
Station 9 Station 10
Station 11 Station 12
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Station 13 Station 14
Station 15 Figure 82: Accuracy of estimated precipitation values at base stations for Best Interpolation Method
Regarding homogeneity and hetereoscedasticity, results presented in Table 28 show that the majority of stations in this study pass two homogeneity tests, namely, the SNHT and the
Buishand test. All stations fail the Pettitt test and Von Neumann test. Stations 2, 3, 5, 7, and 8 fail the SNHT test in additon to the Pettitt and Von Neumann homogeneity tests. All base stations in this study evaluated under the post-corrected Fuzzy Logic Method exhibited no hetereoscedasticity.
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Table 28: Homogeneity and Hetereoscedasticity analysis for Best Interpolation Method
Homogeneity Tests Base Vonn Hetereoscedasticity Station SNHT Pettitt Buishand Neumann 1 Pass Fail Pass Fail Pass 2 Fail Fail Pass Fail Pass 3 Fail Fail Pass Fail Pass 4 Pass Fail Pass Fail Pass 5 Fail Fail Pass Fail Pass 6 Pass Fail Pass Fail Pass 7 Fail Fail Pass Fail Pass 8 Fail Fail Pass Fail Pass 9 Pass Fail Pass Fail Pass 10 Pass Fail Pass Fail Pass 11 Pass Fail Pass Fail Pass 12 Pass Fail Pass Fail Pass 13 Pass Fail Pass Fail Pass 14 Pass Fail Pass Fail Pass 15 Pass Fail Pass Fail Pass
The Fuzzy Logic Method with post-correction allows all base station estimates to pass the
Kolmogorov-Smirnov two sample test when only non-zero values are considered. With regards to the site mean, then this Fuzzy Logic Method variant sometimes overestimates the historical mean and sometimes underestimates it depending upon which station the method is applied to as shown in the following figures.
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Station 1 Station 2
Station 3 Station 4
Station 5 Station 6
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Station 7 Station 8
Station 9 Station 10
Station 11 Station 12
165
Station 13 Station 14
Station 15 Figure 83: Preservation of Site Mean for Best Interpolation Method
166
With regards to site standard deviations, the best method consistently underestimates the standard standard deviation of daily precipitation at a base station as shown in the following figures.
Station 1 Station 2
Station 3 Station 4
167
Station 5 Station 6
Station 7 Station 8
Station 9 Station 10
168
Station 11 Station 12
Station 13 Station 14
Station 15 Figure 84: Preservation of Site Standard Deviations for Best Interpolation Method
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With regards to the preservation of the regional mean, the best method performed reasonably well; however, select base stations had their respective regional daily mean precipitation more accurately preserved than others, as shown in the following figures.
Station 1 Station 2
Station 3 Station 4
Station 5 Station 6
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Station 7 Station 8
Station 9 Station 10
Station 11 Station 12
171
Station 13 Station 14
Station 15 Figure 85: Preservation of Regional Mean for Best Interpolation Method
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With regards to the preservation of the regional standard deviations, the best method performed reasonably well; however, select base stations had their respective regional standard deviations more accurately preserved than others, as shown in the following figures.
Station 1 Station 2
Station 3 Station 4
Station 5 Station 6
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Station 7 Station 8
Station 9 Station 10
Station 11 Station 12
174
Station 13 Station 14
Station 15 Figure 86: Preservation of Regional Standard Deviations for Best Interpolation Method
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With regards to the preservation of the regional correlations, the best method performed reasonably well; however, regional correlations were consistently overestimated, especially for neighboring stations with a historically high correlation with the base station, as shown in the following figures.
Station 1 St ation 2
Station 3 Station 4
Station 5 Station 6
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Station 7 Station 8
Station 9 Station 10
Station 11 Station 12
177
Station 13 Station 14
Station 15 Figure 87: Preservation of Regional Correlations for Best Interpolation Method
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With regards to the preservation of the Markov chain transition probabilities, they were well generally well preserved at all base stations as exemplified in one base station shown in
Figure 88.
Figure 88: Preservation of Markov chain transition probabilities at Station 7
With regards to the preservation of the temporal autocorrelation at a site, they were well generally well preserved at all base stations as exemplified in one base station shown in Figure
89.
Figure 89: Preservation of temporal autocorrelation at Station 12
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With regards to the preservation of the probability density function at a site, they were well generally well preserved at all base stations as exemplified in one base station shown in Figure
90.
Historical Precipitation Histogram Station 15 Estimated Precipitation Histogram Station 15 Figure 90: Preservation of Probability Density Function at Station 15
A regional variogram plots the variance in data between any two base stations as a function of the distance between the two stations. Comparison of variograms from the benchmark Gauge
Mean Method, a traditional Single Objective Function Method variant, and the Fuzzy Logic Best
Method variant illustrates an improvement in the historical variogram and the variogram for infilled precipitation data sets as the transition is made from GM to SOFW to Fuzzy Logic Best
Method as shown in the following figures.
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Figure 91: Regional variogram for GM Method
Figure 92: Regional variogram for SOFW Method (RMSE objective function)
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Figure 93: Regional variogram for Fuzzy Logic Method Variant (Best Method)
In this study, the post-corrected Fuzzy Logic Method is presented as a versatile spatial interpolation technique that is able to balance trade-offs between minimizing estimation errors and preserving site and regional statistics. Despite the efficacy of this method, this Fuzzy Logic variant does not have top ranking performance in every performance measure identified in this study at every base station. The issue of maintenance of data homogeneity at select stations is a concern. Results from the Fuzzy Logic Method are based on a linear preference assumption between minimization of estimation errors and preserving site and regional statistics, an assumption that may not represent all decision-maker scenarios for the infilling of missing precipitation data. With this limitations mentioned, this Fuzzy Logic Method variant is recommended for the infilling of missing precipitation data for the rain gauge network investigated in this study.
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6 Conclusions
6.1 Contributions of the Study
The analysis of the historical spatial and temporal characteristics of precipitation are of vital importance to civil engineers, hydrologists, water resource managers, climate scientists, and related stakeholders. A common data source for such analyses includes networks of rain gauges dispersed throughout a region of interest. As rain gauges are subject to random and systematic errors, robust spatial interpolation measures may be adopted to infill missing precipitation entries at a gauge, and with such infilling, estimation errors are to be minimized and essential site and regional statistics are to be preserved. Traditional interpolation procedures have been mainly focused on the minimization of estimation errors and have not been evaluated with regards to the maintenance of site statistics and gauge-to-gauge regional statistics.
In this study, essential site and regional statistics of interest for preservation have been identified and defined primarily as spatial correlations and other site-to-site statistics in a region.
Traditional and newly proposed interpolation methods have been evaluated for the preservation of site and regional statistics. With the application of these methods in a Kentucky case study region, newly proposed spatial interpolation techniques and variants of traditional infilling methods exhibited superior performance in preservation of site and regional statistics compared to traditional interpolation methods; however, it was possible for newly proposed methods to exhibit lesser performance than traditional methods with regards to the minimization of estimation errors.
infilling of missing precipitation data based on balancing these trade-offs between minimization 183
of estimation errors and maintenance of site and regional statistics. For the case study region, the best method was selected as the Fuzzy Logic Method with dry/wet days post-correction applied.
6.2 Limitations of the Study and Recommendations for Future Research
Results from this study are specifically applicable to the case study region; however, the methodology of adopting and evaluating interpolation techniques proposed in this study has general application. However, application of the methodology presented in this study to other case study regions should consider the following limitations and opportunities for further research:
Multi-station imputation: Data imputation in this study was conducted at only a single station for any given time interval and assumes that observed precipitation data is available at all rain gauge sites excepting the base station. Multi-station imputation for a given time interval may be of great utility in interpolation scenarios.
Solution Space Search Method: In this study, a Generalized Reduced Gradient (GRG) solver was used for the majority of interpolation variants to assign weights to neighboring stations. Such solvers analyze the solution space for a given objective function by analyzing the gradient of the objective function. The search of the solution space is initiated at a user-defined starting point and variables are modified within specified constraints until the solver converges to a local or global optimum solution. GRG Solver is considered robust in optimizing relatively simple non-linear
To overcome this limitation, the use of genetic algorithm solvers may be investigated as such algorithms, if properly initialized, are known to converge at global optimum solutions. Also, in this study, global interpolation techniques were utilized, that is, all neighboring rain gauges were
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involved in estimation of a missing precipitation entry at a base station. A recommendation for future research is to explore the efficacy of local interpolation methods by way of using mixed- integer non-linear programming with binary variables such that objective function optimization will only consider the estimation of missing data at a site.
Time-scale experiments: Proposed interpolation methods in this study have only been evaluated for infilling at the daily time scale. Suitability of interpolation techniques in this study for infilling at coarser and finer time scales is open to investigation.
Climate diversity: Methods proposed in this study have only been evaluated for a temperate climate region, namely, the state of Kentucky, USA. Suitability of interpolation techniques in this study for infilling at other climate regions may be a subject of interest.
Decision-maker scenarios: The importance of minimization of estimation errors and the preservation of site and regional statistics make take on different forms and preferences based on infilling scenarios a decision-maker is faced with. For example, managers of relatively small watersheds may be more concerned with the minimization of estimation errors than with the preservation of site and regional statistics. Hydrologists concerned with long-term spatial analysis of precipitation data may be equally concerned with estimation errors and site and regional statistics or perhaps more inclined towards preservation of site and regional statistics. The
MOFW variants and the Fuzzy Logic Method proposed in this study are uniquely formulated to address such decision-maker preferences and may be explored further.
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