Estimation Methods of Reference Evapotranspiration at Unsampled Locations

Transcription

1 New Mexico Institute of Mining and Technology Masters Independent Study Estimation Methods of Reference Evapotranspiration at Unsampled Locations Author: Kyle Smith Supervisor:Dr. Oleg Makhnin Dr. Oleg Makhnin An independent study submitted in fulfilment of the requirements for the degree of Master of Science in the June 2014

2 Declaration of Authorship I, Kyle Smith, declare that this independent study titled, Estimation Methods of Reference Evapotranspiration at Unsampled Locations and the work presented in it are my own. I confirm that: This work was done wholly or mainly while in candidature for a research degree at this University. Where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated. Where I have consulted the published work of others, this is always clearly attributed. Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work. I have acknowledged all main sources of help. Where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed myself. Signed: Date: i

3 You never know how strong you are until being strong is the only choice you have. Bob Marley

4 NEW MEXICO INSTITUTE OF MINING AND TECHNOLOGY Abstract Jan Hendrickx, Brian Borchers Department of Mathematics Master of Science Estimation Methods of Reference Evapotranspiration at Unsampled Locations by Kyle Smith Access to good interpolation products for reference evapotranspiration (ET) is imperative to many people like farmers, hydrologists and policy makers because reference ET at many locations is unknown. The Oklahoma Mesonet is a network of weather stations that take all measurements necessary to calculate reference ET. Interpolation methods based on the data from this network are explored; the best (smallest MSE) methods are identified. Regression kriging is used to identify subtle data errors. Kriging identified problematic data that were not representative on the area near the stations it since the data size was small. For regression kriging, various covariance models were explored and a non-stationary covariance model was found to best represent spatial variation. Inverse distance weighting (IDW) is compared to kriging and found to be a better interpolator during wintertime. Two forms of interpolate-then-calculate (ITC) are compared with each other and with calculate-then-interpolate (CTI). Generally, ITC had smaller estimation errors when solar radiation was known during summer times. It is recommended that ITC and IDW be used to interpolate reference ET for different parts of the year. Kriging is also used to help design a network of stations needed to make a good estimate of reference ET. 60 uniformly distributed stations serve as an adequate network.

5 Acknowledgements I would like to give special thanks to my project advisor Dr. Oleg Makhnin for always believing in me and being optimistic about this project. His instruction and guidance has allowed me to better understand the grand field of statistics. I would also like to thank Dr. Jan Hendrickx for all of the useful discussions we have had, especially when it comes to understanding more about hydrology and the importance of data quality. My thanks also goes to Dr. Brian Borchers who has helped me find out what I am capable of doing when I thought otherwise. Thanks to Michael Wine for being especially interested in this project and giving me a different perspective of modeling. Other thanks go to my family and friends who have supported me through this program and always gave words of encouragement. iv

6 Contents Declaration of Authorship i Abstract iii Acknowledgements iv Contents List of Figures List of Tables Abbreviations v vii ix x 1 Introduction Interpolation Evapotranspiration Data Description Data Set Data Set Utilizing the data Background Regression Kriging Trend Fitting Simple Kriging Maximum Likelihood Inverse Distance Weighting Cross-Validation Methods Quality Control Data Trend Estimation Kriging v

7 Contents vi 3.2 Covariance Model Model Comparison Subset Selection CTI and ITC Results Flagging Faulty Data April August Maximum Likelihood Estimator May IDW versus Kriging Subset Selection CTI and ITC Conclusion Discussion Bibliography 64

8 List of Figures 1.1 Measured Solar Radiation and Clear Sky Radiation at MIAM Station Observations for June QQ plot of Rn in April Empirical Covariance of Rn in April RMSE of Rn Estimates in April Map of RMSE of Rn in April QQ plot of Temperature in April Empirical Covariance of Temperature in April RMSE of Temperature Estimates in April Map of RMSE of Temperature in April QQ plot of the Psychrometric Constant in April Empirical Covariance of the Psychrometric Constant in April RMSE of the Psychrometric Constant in April Map of RMSE of the Psychrometric Constant in April QQ plot of Mean Actual Vapor Pressure in April Empirical Covariance of the Mean Actual Vapor Pressure in April RMSE of Mean Actual Vapor Pressure in April Map of RMSE of Mean Actual Vapor Pressure in April QQ plot of Wind Speed in April Empirical Covariance of Wind Speed in April RMSE of Wind Speed in April Map of RMSE of Wind Speed in April QQ plot of SVP in April Empirical Covariance of SVP in April RMSE of SVP in April Map of RMSE of SVP in April QQ plot of SVP Slope in April Empirical Covariance of SVP Slope in April RMSE of SVP Slope in April Map of RMSE of SVP Slope in April QQ plot of Wind Speed in August Empirical Covariance of Wind Speed in August RMSE of Wind Speed in August Map of RMSE of Wind Speed in August QQ plot of Mean Actual Vapor Pressure in August Empirical Covariance of Mean Actual Vapor Pressure in August vii

9 List of Figures viii 4.35 RMSE of Mean Actual Vapor Pressure in August Map of RMSE of Mean Actual Vapor Pressure in August QQ plot of SVP in August Empirical Covariance of SVP in August RMSE of SVP in August Map of RMSE of SVP in August QQ plot of SVP Slope in August Empirical Covariance of SVP Slope in August RMSE of SVP Slope in August Map of RMSE of SVP Slope in August QQ plot of the Psychrometric Constant in August Empirical Covariance of the Psychrometric Constant in August RMSE of the Psychrometric Constant in August Map of RMSE of the Psychrometric Constant in August Station Observations for May Empirical Covariance of Reference ET One Sill Parameter Comparison IDW and Kriging from D IDW and Kriging from D RMSE L curve for D1 of June Selected Stations for D1 of June RMSE L curve for D2 of June Selected Stations for D2 of June CTI and ITC

10 List of Tables 4.1 Data Transformations Data Flaggging IDW and Kriging from D IDW and Kriging from D CTI and ITC ix

11 Abbreviations MLE MSE RMSE QQ ET IDW CTI ITC ASCE PM SVP DOY Maximum Likelihood Estimator Mean Square Error Root Mean Square Error Quantile Quantile EvapoTranspiration Inverse Distance Weighting Calculate Then Interpolate Interpolate Then Calculate American Society of Civil Engineers Penman-Monteith Saturated Vapor Pressure Day Of Year x

12 To my mother and father xi

13 Chapter 1 Introduction 1.1 Interpolation Geostatistical interpolation is found in many applications of hydrology because it is helpful in estimating parameters of interest. Most times, measurements at particular places can be taken. The measurements consume time and money. Since taking measurements cannot be done everywhere all the time, estimating a specific value using interpolation methods with surrounding measurements is a logical step in finding out the behavior at the location. When finding out the right kind of interpolation to do, there is much to consider. First and foremost, the data quality must be checked. If the data does not have good data quality then any estimation made from the data set does not have good quality. Second, it s important to choose a good interpolation method. Kriging is one of the most popular methods since it comes with uncertainty estimates and is, in fact, the best (in the sense of minimizing MSE), under certain assumptions. Other methods like inverse distance weighting (IDW), and nearest neighbor do not take variation into account but they can still prove to be useful and in some cases better than kriging. When deciding on a method, there are many ways to follow through with it. For kriging, there is simple kriging, cokriging, regression kriging, universal kriging, etc. Before applying a kriging method the assumptions should be satisfied (or at least approximately) and the available data should be picked so that it makes a good prediction. ArcGIS is quite flexible to do kriging such as using kriging for only nearby data. The definition of nearby is subjective and usually involves all stations within a certain radius of the prediction location, or a specific number of nearest neighbors. ArcGIS is also flexible with IDW on these aspects too. Each method has its own internal intricacies. The most common way to test whether one method is superior to another is to use cross-validation. There are many ways to use cross-validation and so its application must be appropriate. 1

14 Chapter 1. Introduction 2 Another way to improve estimation techniques is to consider the specific application. When something does not sound right, such as having a negative height, then methods and formulas can be reevaluated. After applying the correct methods, and checking it, one can actually be more knowledgeable about the system. This information is useful because it can help develop or maintain a system of measurements stations. The use of extra information can also be helpful to make a design. In Section 1.2, reference evapotranspiration (ET) will be described and an equation will be given to calculate reference ET. This section is meant to get the reader familiar the application at hand. Section 1.3 describes where the data came from. There are two sets of data which differ in quality and calculated reference ET. Section 1.4 is an overview of the rest of the paper. Chapter 2 is about background information. This will talk about how least squares parameter estimation works. It also goes into calculating distances on a sphere, and IDW interpolation. The main result of regression kriging is introduced and the maximum likelihood method is discussed. Validation of models is also introduced by exploring the concept of cross-validation. Chapter 3 is about the methods used that include identifying odd data, comparing covariance models using likelihood methods, comparing kriging and IDW, using interpolation to find the number of stations needed to make a good estimate, and how calculate-then-interpolate compares with interpolate-then-calculate in varying situations. The fourth chapter gives results from applying the methods in chapter 3. What is most intriguing about this chapter is that sometimes simpler models are best for interpolation, extra information helps to some degree and the number of stations used to calculate reference ET at this scale is more than enough. Chapter 5 gives possible reasons for the results of chapter 4 and future work. 1.2 Evapotranspiration Most people are familiar with the concept of evaporation. Evapotranspiration is evaporation that comes from the ground and from plants, and water transpired from plants. Reference evapotranspiration (ET) is a quantity that describes the rate at which water vaporizes from a reference surface crop. The reference crop must be a well watered plant like tall grass alfalfa. This quantity is useful to people like farmers, planners, hydrologist and policy makers because it will allow them to make better decisions about water usage. The literature on calculating reference ET is quite ubiquitous. Some methods are better than others and each method depends on the kind of information available to people. Some sources to make the calculation are on [1 6]. The most accepted model is

15 Chapter 1. Introduction 3 the ASCE Penman-Monteith (PM) equation which has the form: ET r = (R n G) + γ Cn T +273 u 2(e s e a ) + γ(1 + C d u 2 ) Where ET r = standardized reference crop evapotranspiration for tall surfaces (mm/day) R n G = calculated net radiation at the crop surface (MJ/m 2 day) = soil heat flux density at the soil surface (MJ/m 2 day) T = mean daily air temperature at 1.5 to 2.5 m height ( C) u 2 e s e a = mean daily wind speed at 2 m = saturation vapor pressure at 1.5 to 2.5 m = mean actual vapor pressure at 1.5 to 2.5 m = slope of saturation vapor pressure-temperature curve (kp a C 1 ) γ = psychrometric constant (kp a C 1 ) C n = constant that changes with reference type and calculation time step (K mm s 3 /Mg day) C d = constant that changes with reference type and calculation time step (s m 1 ) Most of these values are calculated from measured data. The standard way of calculating each parameter is described in [1] and [2]. The same references offer guidelines on how to take measurements. Sometimes all the measurements are not taken but there are ways around this by using equations like the Makkink equation or the Priestely-Taylor equation or the Hargreaves equation [3]. There are many forms of evapotranspiration. The two most famous ones are the tall crop reference ET and the short crop reference ET. Tall crop is helpful for studying the water needs of alfalfa and short crop for studying smaller types of vegetation. Sometimes reference ET can be found at a rate that depends on hours instead of days and in some cases it is found at a rate depending on weeks and months. There are current areas of research related to estimating reference ET. Some of which use algorithms on satellite images called surface energy balance models in raster format as mentioned in [7]. Price has also used satellite thermal images to find out the spatial variability in reference ET [8]. Another way reference ET can be interpolated is by cokriging with elevation. It was found that in some cases cokriging with elevation helped in estimating reference ET [9] by lowering average error in the state of Oregon. Sometimes it is helpful to look at how other quantities, like precipitation, have been estimated. For weather stations in Spain, utilizing multivariate methods and the knowledge of GIS have helped to point out that elevation, distance to coast, and distance from the west were important factors to consider when looking at precipitation [10]. The idea of mountains in a certain vicinity gives information about spatial stationarity and the use

16 Chapter 1. Introduction 4 of certain models to reference ET was done by [9]. In many cases, different methods of interpolation are compared. Cokriging, residual kriging and ordinary kriging are compared using MSE, mean absolute error, and mean error percent for different months. Results have been found to vary as to which method is best [11]. Overall there are many approaches to interpolating reference ET but each place and station is unique and more knowledge of the various quantities will help to better understand how the system is behaving. 1.3 Data Description Weather stations from the Oklahoma Mesonet record data on humidity, temperature, wind speed, atmospheric pressure, solar radiation, and other environmental quantities. Specific quantities like air temperature, relative humidity, solar radiation, atmospheric pressure, and average windspeed are what is needed for the Penman-Monteith equation. The mesonet is composed of over 120 weather stations. The number of stations is not always constant. In the past there were fewer stations. Sometimes data is not recorded because of technical difficulties. Sometimes the data at a site is not good data because it might not give a good representative value at that site. This can happen by failing to account for sensor drift, dirty sensors that could be contaminated by bird feces or mud, or failure to calibrate a sensor. The great thing about the Oklahoma Mesonet is that the data collected is generally high quality data. Since station sites are obliged to give a good representative value for the area around it, a few recommendations were specified by the site standards committee which included reasonable accessibility, flat slopes, adequate plot of land, etc. The sensors used seem to be of high accuracy and sometimes the sensors are switched out for a different type of sensor. One of the goals of the Oklahoma Mesonet is to provide high quality data to be used for research purposes. Hence, quality assessment must be done to ensure that high quality data is obtained. The quality assessment of data is done primarily with a software program. Any odd data is flagged and given a grade. The data is analyzed by humans and determined to be bad or good data. If it is odd then the data is not entered into the system and the sensor is fixed by a technician. Overall much is done to make sure high quality data is obtained and improvements keep coming up with the data set [12 14]. Current ideas of quality assessment and assurance procedures can be found in [15]..

17 Chapter 1. Introduction Data Set 1 The data for this paper came from the Oklahoma Mesonet website [16]. Daily data was obtained from Since reference ET is the topic of interest, every station that has the proper measurements for a day was used and other data was not used. The proper measurements in this case refer to solar radiation, mean wind speed, air temperature, relative humidity, and atmospheric pressure. These measurements are taken at five-minute intervals; statistical data such as average, minimum and maximum values are computed daily. Dew point temperatures are not measured but computed from the measured air temperature and relative humidity. The data were used without any additional quality assessment except for removal of days with missing data. A total of about 549,000 daily observations were used for this study. Reference ET was calculated with the Penman-Monteith equation for daily tall grass reference ET [1] using as input the maximum and minimum air temperature, the Mesonet computed daily dew point temperature, the average daily wind speed, the total incoming daily solar radiation and the daily average atmospheric pressure. Some assumptions that should be noted are that the soil heat flux was assumed to be 0 since this is approximately true. C d = 0.38 s m 1 and C n = 1600 K mm s 3 /Mg day. Temperature was calculated by taking the mean of the maximum and minimum temperature; actual vapor pressure was not calculated from relative humidity measurments using equations 7 and 41 but from the Mesonet computed daily dew point temperature using equation 8; the saturated vapor pressure was calculated using equation 6 and 7; the psychrometric constant was calculated using equation 4; the slope of the saturation vapor pressure-temperature curve was calculated using equation 5; R n was calculated using equation and where in equation 16 the albedo coefficient is fixed to 0.23 but equation 20 has K ab which can be calculated using a different source [2] which has equations (2-4),(2-5) and (2-6). These equations have parameters in them but the parameters used, follow the most up to date values mentioned in the paper. It should also be noted that data from the Oklahoma panhandle was excluded since it is more difficult and more erroneous to make predictions in that area. It is more erroneous because the predictions made come from stations that do not surround the prediction locations. ArcGIS and R were used in this paper to make calculations. We call this data set D Data Set 2 The data for this paper came from the Oklahoma Mesonet website [16]. Daily data was obtained from Since reference ET is the topic of interest, every station

18 Chapter 1. Introduction 6 that has the proper measurements for a day was used and other data was not used. The proper measurements in this case refer to air temperature, relative humidity, solar radiation, wind speed and atmospheric pressure. These measurements are taken at five-minute intervals; statistical data such as average, minimum and maximum values are computed daily. Dew point temperatures are not measured but computed from the measured air temperature and relative humidity. Data set 2 underwent a rigorous quality assessment of its incoming solar radiation measurements following Allen et al. [2005]. Despite the advanced quality control of Mesonet, some of the radiation data contained errors well above 10 % (Figure 1.1) and as a consequence reference ETs will have errors of the same order of magnitude. In addition, all observations were removed that showed a difference of more than +/- 6% between the Mesonet solar radiation measurements and the quality assured Data Set 2. Many of the eliminated observations are probably reliable but since they may contain irregularities and compromise the overall quality of this study removal was the best option. A total of about 388,000 daily observations were used for this study in Data Set 2. Reference ET was calculated with the Penman-Monteith equation for daily tall crop reference ET [1] using as input the maximum and minimum air temperature, the Mesonet computed daily dewpoint temperature, the average daily wind speed, the total incoming daily solar radiation and the daily average atmospheric pressure. Some assumptions that should be noted are that the soil heat was assumed to be 0 since this is approximately true. C d = 0.38 and C n = Temperature was calculated by taking the mean of the daily maximum and minimum temperature; actual vapor pressure was calculated from relative humidity measurements using equations 7 and 41; the saturated vapor pressure was calculated using equations 6 and 7; the psychrometric constant was calculated using equation 4; the slope of the saturation vapor pressure-temerature curve using equation 5; R n was calculated using equation and where in equation 16 the albedo is set to 0.23 but equation 20 has K ab which was calculated using a different source [12] which has equations (2-4),(2-5) and (2-6). These equations have parameters in them but the parameters used, follow the most up to date values mentioned in the paper. It should also be noted that data from the Oklahoma panhandle was excluded since the single almost linear array of a few weather stations cannot be used for testing interpolation methods. ArcGIS and R were used in this paper to make calculations. Let s call this data set D1. An aridity correction was applied to the data following [17] and [18].

19 Chapter 1. Introduction 7 Figure 1.1: Plots of measured solar radiation (Rsm) before (top) and after (bottom) correction using the clear-sky radiation (Rso) calculated using the ASCE approach at station MIAM in It was determined that during DOY 1-60 the Rsm requires a correction multiplication by 0.836; DOY by 0.886; DOY by 0.937; DOY by 0.974; DOY by 0.952; DOY needs no correction. Personal communication with Hendrickx in May 2014.

20 Chapter 1. Introduction Utilizing the data In this paper, we are developing a kriging model for the interpolation of reference ET data, and comparing it to other models. Models in this case can mean something of the same form but with different parameter values, such as two spherical covariance models with different sill parameters, or totally differing forms such as a spherical model and an exponential model. Regression kriging is the main type of kriging done with the data set. To account for spatial variation we use a trend equation which accounts for spatial variation in terms of longitude and latitude for D1 and easting and northing for D2. Then a nonstationary covariance model is constructed. It is tested using crossvalidation. The development of the model and the different comparisons made to other models, gives further direction for the future. Some comparisons made include using a kriging method for which the covariance model is chosen by comparing a stationary model and a nonstationary model using the likelihood ratio test. Other possible models can be found by finding the maximum likelihood values. The kriging method is also compared to IDW interpolation in terms of 10 fold cross-validation. The kriging method was also used to identify problematic measurements by seeing how good the kriging method does at the station. Kriging was also used to help see how many stations need to be present to make a good interpolation by using a greedy algorithm approach. Calculate-then-interpolate reference ET means using the parameters in the PM equation to calculate reference ET and then estimate reference ET at an unsampled location from the calculated reference ET. Interpolate-then-calculate means use known parameter values to estimate parameters at an unsampled location and then calculate reference ET from this information. This paper also compares whether calculate-then-interpolate does better (in terms of 10 fold cross-validation) than interpolate-then-calculate. Other forms of the interpolate-then-calculate method are also incorporated.

21 Chapter 2 Background 2.1 Regression Kriging Regression kriging is an interpolation method that utilizes the tools of trend fitting and then simple kriging. The predictions are based on the multivariate distribution of the variables for each station. Trend fitting is typically done using the least squares method. After trend fitting one will find that the average of the residuals will have mean of zero and the residuals themselves will have normal behavior. This is enough to follow through with simple kriging. With simple kriging, available data is used to make predictions on areas unknown based on covariances from one station to another. Kitanidis book gives a more thorough understanding of geostatistics [19] Trend Fitting With trend fitting a model is proposed. Suppose the model is ET ri = β 1 + β 2 Elev i + β 3 T Max i + ε i, i = 1,..., n where n is the number of all of the measurements taken. This model can be represented in matrix notation as T = Xβ + E 9

22 Chapter 2. Background 10 where T, E R n, X R n 3, and β R 3. It should be apparent that the model matrix is 1 Elev 1 T Max 1 1 Elev 2 T Max 2 X =.. 1 Elev n T Max n Then since the goal is to minimize E 2 the solution β = X Y will reveal the best estimates for the parameter. After doing this, one can look at the histograms of the residuals, and make a QQ plot so as to make sure it follows a normal distribution. The statistical significance of each parameter can be found as well by looking at the p-value. If the p-value is small then that is evidence that the covariates (elevation, and maximum temperature) are critical to finding the predicting variable ( ET r). Another useful statistic is the R 2 value since it accounts for the proportion of variation explained Simple Kriging After calculating the best parameters one can use that and find the residuals of the data (E). The residuals for each station follow a multivariate normal distribution. f(e) = 1 ( exp 1 ) det Cov(2π) m/2 2 (E E)T Cov 1 (E E) Where Cov is the covariance matrix and m is the number of stations. Each residual will correspond to a station and a certain day. The residuals can then be denoted with different indices E i,j where i will correspond to the station and j a day. Not all of the stations will have evapotranspiration values for the same days. Thus when calculating the covariance between two stations, one must look at all days in common for which evapotranspiration values are available. The covariance calculated uses R s built in covariance function. The sample covariance between each station is found by the formula Cov(A, B) = 1 m 1 m (a i ā)(b i b) where a i and b i are random samples from the random variables A and B respectively. The covariance for each pair of stations is formatted in a covariance matrix Cov and a distance matrix is also calculated. The distance matrix consists of distances between all i=1

23 Chapter 2. Background 11 pairs of stations. For D1, this is found by using the great circle calculation dist(u 1, U 2 ) = 2 sin 1 ( ( ) φ sin 2 2 ( ) ) λ + cos φ 1 cos φ 2 sin 2 2 where φ i will correspond to latitude in radians and λ will correspond to longitude in radians. Distances for D2, are found using the UTM coordinates and the Pythagorean Theorem. Once all pairs of distances are found it is organized in a distance matrix where the entries will correspond to the covariance matrix. A plot of distances versus covariances is displayed and from there a covariance function is found. The covariance function has a range and a sill parameter. Sometimes it can have a nugget parameter that accounts for noise. In most cases the covariance function is nonlinear, but a transformation to the function can linearize it. After doing so, the best parameters can be found using least squares or maximum likelihood estimation. The best parameters are implemented in the covariance function and a new covariance matrix Σ 2,2, is found. Suppose a prediction needs to be made for some unsampled locations. The covariances of the unsampled location will be found using the covariance function. The covariance matrix of the unsampled locations will be called Σ 1,1 and the covariances between unsampled locations and predictor locations is stored in a vector covariance Σ T 1,2. A large covariance matrix will be constructed and assembled as Σ = ( ) Σ1,1 Σ 1,2 Σ 2,1 Σ 2,2 The best linear unbiased predictor is E 1 = Σ 1,2 Σ 1 2,2 E 2 where E 1 are the predictions of unsampled locations and E 2 is the available data. The variance of the estimate is [20] V ar(e 1 ) = Σ 1,1 Σ 1,2 Σ 1 2,2 Σ 2,1 2.2 Maximum Likelihood Estimating parameters can be done using the maximum likelihood approach. This approach works by first considering a probability distribution for the data. A likelihood function is constructed utilizing the data and the maximum of the function is found by varying parameters. The parameters will constitute the maximum likelihood estimators (MLE). Sometimes a logarithmic transformation is performed on the likelihood function

24 Chapter 2. Background 12 and the resulting function is called the log-likelihood function. Maximizing that function can be done by finding the MLE. Suppose the probability distribution is the multivariate normal distribution and there are n data vectors. Then the likelihood function with the covariance matrix as a parameter is l(σ) = n i=1 1 ( exp 1 ) det Σ(2π) m/2 2 (E i E) T Σ 1 (E i E) Since this function has an exponent, a logarithmic transformation can be used to take advantage of this property. The log likelihood function is L(Σ) = n ( 1 ) ( ) 2 (E i E) T Σ 1 1 (E i E) log det Σ(2π) m/2 i=1 and the MLE is ˆΣ = arg max Σ L(Σ). Sometimes testing one model over another is done to see how one model performs in comparison to another. Consider the likelihood statistic ( ) L( Σ) λ = 2 log L(Σ) Here the model changes with respect to the covariance parameter. The statistic lies on a χ 2 distribution and the p-value it corresponds to indicates how much more significant a model is compared to another [21, 22]. 2.3 Inverse Distance Weighting Inverse distance weighting (IDW) is another interpolation method commonly employed by hydrologists. The idea of IDW is to make predictions of unsampled locations by considering how far away sampled locations are from the unsampled locations. The formula would be m Y j = λ i Y i i=1 where λ i = 1 d p i,j m i=1 1 d p i,j and p > 1 and in most cases p is optimized in terms of MSE [23].

25 Chapter 2. Background Cross-Validation Cross-validation is one method that is used to verify how good interpolation estimates are in terms of MSE. Many forms of cross-validation exist. Leave one out cross-validation and k-fold cross-validation are conventional forms of validation. The way leave one out cross-validation works is first, identify all available stations that have wind speed data for one day. Pick one station for which data will be omitted. Predict wind speed for that station using data from other stations using some interpolation method. Find the difference between the predicted value and the true value. Repeat this method for all other stations. If looking at more than one day, then do this for all other days. Add up the squared differences and divide by the total number of observations and this will give the MSE. k-fold cross-validation is different because the number of stations is randomly divided up into k subgroups. Then one subgroup has data omitted and predictions are made on this subgroup. Differences are found between the true data and the predicted data. The same differences are found by predicting other subgroups and an MSE is found.

26 Chapter 3 Methods 3.1 Quality Control In many cases the data may be unsatisfactory in terms of being a good representative of the area around it. Although there are certain measures that are taken by people who administer the data for stations in Oklahoma some of it may still be of bad quality. Since accurate interpolation methods need good data, it is imperative to identify problematic data. This can be done using kriging. Kriging will provide a prediction and the deviation from the observed value will indicate problematic data. Since data set 1 has not been filtered, it is more likely to have odd measurements and as a result will be the only data set used for this section Data The data set consists of climate stations in the state of Oklahoma. It is found on the Oklahoma Mesonet Website [16]. The stations take measurements of various weather conditions such as temperature, solar radiation, humidity, atmospheric pressure, etc. These measurements are enough for it to be used to calculate the amount of reference ET at each weather station location. The climate stations have been making measurements since before 1997 and continue to do so to this day. Data will only be considered from 1997 to There have been a total of 129 climate stations during this period. Consider Figure 3.1 where time is listed in days. The first 31 points correspond to 1997, points correspond to 1998 and so on. When the stations first began making measurements there were fewer stations that made measurements than there are now. Gradually as time progressed more and more weather stations were added. From there were roughly 60 stations, from 14

27 Chapter 3. Methods 15 Figure 3.1: Number of stations taking measurements versus time for all days in June there were roughly on average 86 stations per day and from 2007 to 2013 there were roughly on average 104 stations per day. Data for 2014 is not being considered for this data set. All of the weather stations are well spread out over the state of Oklahoma. Mesonet data sets are presented on a day to day basis. When calculating reference ET, the well-accepted ASCE Penman-Monteith method is used to make the calculations. This method uses input data of net radiation of crop surface, soil heat flux density at the soil surface, mean daily temperature at height 1.5 to 2.5 meters, saturation vapor pressure at 1.5 to 2.5 meters, mean actual vapor pressure at height 1.5 to 2.5 meters, slope of the saturation vapor pressure-temperature curve, psychrometric constant, and constants that change with reference type. It is also the case that an aridity adjustment was made to better estimate reference ET. Aridity adjustments account for the dryness in the air and was also used by Allen to better estimate reference ET. Sometimes reference ET is calculated to be less than 0 and this cannot occur so it is set to It would have been set to 0 but sometimes certain mathematical functions (like log) require the value to be bigger than 0. Luckily reference ET is less than 0 in very rare cases. The literature for the derivation of reference

28 Chapter 3. Methods 16 ET is quite ubiquitous and will not be discussed here. An interesting discussion of its derivation is discussed in [24]. To better estimate reference ET it is imperative to have consistency in the set up of the weather stations and their sensors. This consistency means that the weather stations should have some characteristics that are quite common. For instance, it is best if the weather stations are on a flat area, away from forests, residential areas, and rivers. Unfortunately it is not possible for all these ideal conditions to be met and so sometimes exceptions must be made. It is also the case, that the data is frequently monitored to make sure realistic measurements are being made and that nothing is wrong with the measurements. It turns out that if a sensor is making odd measurements then it is checked and the data thrown out if it is found that the instrument is defective. It is also the case that the sensors are calibrated every so often to ensure that high quality data is obtained. Without good data quality the information that can be obtained from these stations would not be good quality either. The overall goal of these weather stations is meant to serve as a good representative of the area around it Trend Estimation One of the things that can be done to ensure good data quality is to compare it to an interpolation method. Interpolation methods can help identify which stations are acting odd and which may not be a good representative of the area around it. If this turns out to be the case, then an odd station is flagged and looked at more closely. This includes looking at possible reasons for why odd data is taking place. In this case regression kriging is the interpolation to be used. Other methods can also be used but regression kriging takes into account the spatial variation along with accounting for important factors affecting reference ET. With this interpolation method, it is going to be difficult to make predictions at the panhandle and so the data for these station will be excluded and interpolations will not be made for this location. There is also lots of variability in the data when comparing winter data to summer data. A solution to this is to consider all data for one month. This means have all data from all stations for all days and for all years of one month. First consider wind speed measurements. A trend line will be fitted to wind speed measurements. The goal of the trend fit is such that it is explains the spatial variation, has normal residual behavior and is as simple as possible. Meeting all of the requirements of the trend equation may not be possible so the best compromise must be taken. This trend line will take the form of a linear first order polynomial with variables that depend on longitude, and latitude. Fitting a trend line is done by using a least squares method.

29 Chapter 3. Methods 17 Once a trend is fitted, it is subtracted from the measured wind speed data and the residuals are obtained. The residuals should have mean 0 and the distribution must be normal. Normality is checked by looking at the histogram of the residuals and making a normal Q-Q plot. The R 2 value can also be obtained and the statistical significance of the parameters indicate which covariates serve a greater purpose. Since the data is real world data, it may be difficult to get normal behavior from residuals so transformations may be warranted. The transformations will be performed on the prediction variable which in this particular case is the wind speed data. After transformations are done, a trend equation is fitted, the statistical significance of the covariates is examined and the R 2 value is looked at too. Once all models are found for each prediction model for one month, the same will be done for the rest of the months of the year Kriging Suppose that the appropriate trend model is chosen for a prediction variable, then the model is subtracted and residual data is constructed. Simple kriging can now take place. The residual data is now used to find the empirical covariances between each station. The empirical covariances are organized in a covariance matrix. A matrix of distances between each station is also constructed and the values for covariance and distance are plotted against each other using a scatter plot. Distances are calculated using the great circle formula. The scatter plot will suggest a covariance model. If there are any covariances that behave out of the norm, then a relation between these covariances and a station will be identified and a scatter plot of correlation versus distance is constructed. This scatter plot will be used to see if the station is behaving out of the norm and it if is then it will be flagged. For the predictor variable, the covariance model will be station dependent. The covariance model parameters will be found using least squares. Since the model is nonlinear, it is linearized by taking a logarithmic transformation. After finding the parameters, the estimated covariance matrix is found and predictions can be made. Identifying odd stations will depend on looking at the relative RMSE. This process works by looking at a day where the data is available for one station, then prediction of the station is made from other stations with data that day. This is done for all days of the month and for all years and the RMSE is calculated for that station. The RMSE is found for each station. If any stations have a relatively high RMSE, such as more than 3 standard deviations away from the mean, then the location of the odd stations is identified. It will be flagged if the odd stations lie in the geographic interior but will be given extra consideration to not be flagged if not near the interior.

30 Chapter 3. Methods Covariance Model Suppose that the predictor variable is reference ET and that a trend model can be constructed. Then the empirical covariance matrix is created from the residuals. It appears that due to the large relative differences in magnitude, a nonstationary covariance model is warranted. Consider the nonstationary covariance model Cov(Z i, Z j ) = σ i σ j exp ( γ x i x j 2 ) where Z i is the residuals between station i and station j, σ i is the sill parameter for station i, γ is the range parameter and x i is the location of station i. Covariances between each station are known through this model but when the sill parameter for one location is not known then the sill parameter is found by finding the geometric mean between all stations. Testing whether a nonstationary and stationary model is superior will be done using the likelihood ratio test for data set D1. Data set D2 will not be explored here. This being the case, the best parameters will be found using the maximum likelihood test. The stationary model will be Cov(Z i, Z j ) = σ 2 exp ( γ x i x j 2 ) and the likelihood function will be the products of the multivariate distribution where the parameters will be the parameters of the covariance function. One problem associated with calculating the maximum likelihood estimate is that it assumes that there is no missing data. In practice, this is not true. To make up for missing data, missing data is filled in by data used by the nearest station. After this occurs, the likelihood value can be found. Filling in missing data is no problem when a few stations are missing measurements, but to fill in more than that might be misleading and thus lead to very incorrect results. Since the number of stations changes from year to year one should take this into account. It is the case that from 1997 to 1999 there are around 60 stations, and then from 2000 to 2006 there are around 100 stations and then after that there are about 115 stations. Thus, when filling in data, it is best to pick one of the time periods. It is also the case that the rough average number of stations that fail a day is about 5 and so filling in data using this method should not be too bad. It is of interest that the MLE be found for each model case. This is done using R s built in Nelder-Mead algorithm. After MLEs are found for each model it is compared to the other model using the likelihood ratio test.

31 Chapter 3. Methods Model Comparison There are many interpolation methods used to predict evapotranspiration. These generally include many methods like spline fitting, inverse distance weighting, nearest neighbor, or kriging. One of the most popular is IDW. A comparison of IDW and regression kriging can be made to see which one is superior. This can be answered using 10 fold cross-validation. 10 fold cross-validation is more realistic in this case since there seems to be no more than 10% of stations failing at one time. Since there are three main times from 1997 to 2013 that have almost consistent number of stations it is best to note the different comparisons and see which one performs better in terms of MSE. The way this comparison is made is by using the interpolation method of calculated values of reference ET. Kriging was done in the same manner as in and and IDW made predictions purely on measurements made for the day. This will be done for D1 and D Subset Selection Predicting reference ET has been explored using different methodologies. These predicting techniques do a fair job with the number of stations available, but if less stations were available then how much would the accuracy decrease? One option that can be considered, is looking at what stations can be omitted when using an interpolation method. This option can be used to find out which stations need to be omitted that will not result in a bad interpolation. There are several methods that can be used to find out this best subset selection where best is meant as the smallest MSE. For multiple locations this means minimize MSE = tr(c j,j C k,j C 1 k,k C j,k)/n Where C j,j is a block matrix containing correlation values of unpredicted stations and C k,k is a block matrix corresponding to a selected subset, and N is the number of columns of C j,j. The block matrices will come from the following correlation matrix C = ( ) Cj,j C j,k C k,j The greedy algorithm approach will be used to solve this optimization problem. This approach will use the correlation matrix associated with the empirical correlation matrix that came from residuals of the fitted trend equation for ET. This method will first C k,k randomly select a station to be used for interpolation. Then the second station will

32 Chapter 3. Methods 20 be selected based on what other station is least correlated with the first station. The third station will be picked based on which station has the highest modified variance prediction. e.g. argmax j (C jj C 2,j C 1 2,2 C j,2) Where C is a correlation matrix which can be thought of as C = ( ) Cj,j C j,2 C 2,j C 2,2 and C j,j is a scalar corresponding to station j. The fourth station will be selected based on three stations from argmax j (C j,j C 3,j C 1 3,3 C j,3) Where C is the same correlation matrix which can be thought of as C = ( ) Cj,j C j,3 C 3,j C 3,3 and so on. The number of stations to pick will be different in each case. Since a random station is picked in each case it is more informative to take a look at the results after doing this 10 times to average them and make a plot of RMSE vs. number of stations. The plot will also be overlaid with predicting reference ET at 10 random locations each time and averaging them. The random selection is based on a uniform distribution where longitude ranges from 100 to 94.5 and latitude will range from 34 to 37. Then a set number of stations will be selected and the locations for the selection will be shown. This algorithm will be used for data set D1 and it can be applied to data set D2 as well. The difference is that for D2, UTM coordinates will be used. The differences for each data could indicate if quantity is better than quality. 3.5 CTI and ITC Another thing that can be considered is comparing the prediction methods of interpolatethen-calculate (ITC) or calculate-then-interpolate (CTI) for D1. D2 will not be considered for this section. The comparison (CTI and ITC) is important since sometimes partial data for a station may be available and it would be interesting to see how predicting the rest of the data to calculate reference ET would compare to just interpolating the

33 Chapter 3. Methods 21 reference ET from other stations. The interpolate-then-calculate (ITC) method works by finding all of the parameters used in the Penman-Monteith equation which interpolates by and and then it calculates reference ET. Ten fold cross-validation makes the comparison. This is by randomly leaving out all data from a tenth of the stations and then interpolating the values of the parameters for those stations and then calculating reference ET; the other thing that is done is calculating the reference ET values and then interpolating over the same stations left out. This process will resume until cross-validation is done. Another thing to consider and which might be more realistic is to look at how the parameters (from PM equation) for each station will be unavailable and some will be available. This means interpolation from other stations will be done to fill in the missing data and reference ET can be calculated. This can be compared with the usual method of calculating reference ET and then interpolating. However, the number of reference ET to use with this method will be based on which stations have missing parameter values for each day since if a parameter value is missing, reference ET cannot be calculated using (PM). The use of satellite data can also be used to improve interpolation by filling in missing data for net radiation and can be explored with a future project.

34 Chapter 4 Results 4.1 Flagging Faulty Data Since there are two data sets, D1 is more likely to have faulty data. It can be used to help find flawed data in D1 which can simultaneously be used to find flawed data in D April Net Radiation The net radiation for April had a trend fitted to it that was Rn 3 i = β 1 + β 2 Lat i + β 3 Lon i + ResRn i where ResRn i is the residual. Lat and Lon correspond to the latitude and longitude of the location of a measurement. Figure 4.1 displays the QQ plot of the residuals and Figure 4.2 displays the empirical covariance plot. Then a covariance function was created and it takes the nonstationary form Cov(ResRn i, ResRn j ) = σ i σ j exp ( γ x i x j 2 ) Estimating the parameters meant linearizing the function and then solving for it using least squares. An algorithm to find MSE from kriging for each station is: 1. Leave out data for a station 2. Predict ResRn for that station from all ResRn available on that day 22

35 Chapter 4. Results 23 Figure 4.1: Normal QQ plot for residual ResRn data Figure 4.2: Empirical Covariance for ResRn data

36 Chapter 4. Results Repeat steps 1 and 2 for all stations on all days Then use all predicted ResRn and true ResRn to compute the MSE for each station. The RMSE for the stations are in Figure 4.3. A map of the locations for the red dots relative to other stations is shown on Figure 4.4. Notice that since the odd stations are not that high and lie on the boudaries, no stations will be flagged here for Rn. Figure 4.3: RMSE of stations where red dots indicate are more than three standard deviations above the mean Temperature The temperature here came from averaging the maximum and minimum daily temperatures. The temperature for April had a trend fitted to it that was T i = β 1 + β 2 Lat i + β 3 Lon i + ResT i where ResT i is the residual. Figure 4.5 displays the QQ plot of the residuals and Figure 4.6 displays the empirical covariance plot. Then a covariance function was created and it takes the nonstationary form Cov(ResT i, ResT j ) = σ i σ j exp (γ x i x j 2 )

37 Chapter 4. Results 25 Figure 4.4: RMSE of stations where red dots indicate more than three standard deviations above the mean Figure 4.5: Normal QQ plot for residual ResT data

38 Chapter 4. Results 26 Figure 4.6: Empirical Covariance for ResT data Estimating the parameters meant linearizing the function and then solving for it using least squares. The same algorithm from net radiation was applied to find the odd stations. The results are on Figures 4.7 and a map on Figure 4.8. Due to the location of the odd stations, no stations will be flagged here. Psychrometric Constant The model for the psychrometric constant is γ i = β 1 + β 2 Lat i + β 3 Lon i + Resγ i Figure 4.9 displays the QQ plot of the residuals and Figure 4.10 displays the empirical covariance plot. RMSE from kriging is shown on Figure 4.11 and the location (on the edge of the state) on Figure Since the location of the station makes up for the odd behavior no stations will be flagged. Mean Actual Vapor Pressure A model for mean actual vapor pressure is eai = β 1 + β 2 Lat i + β 3 Lon i + Resea i where Resea i is the residual. Figure 4.13 displays the QQ plot of the residuals and Figure 4.14 displays the empirical covariance plot. Notice that there is obviously an odd station

39 Chapter 4. Results 27 Figure 4.7: RMSE of stations where red dots indicate are more than three standard deviations above the mean Figure 4.8: RMSE of stations where red dots indicate more than three standard deviations above the mean

40 Chapter 4. Results 28 Figure 4.9: Normal QQ plot for residual Resγ data Figure 4.10: Empirical Covariance for Resγ data

41 Chapter 4. Results 29 Figure 4.11: RMSE of stations where red dots indicate are more than three standard deviations above the mean Figure 4.12: RMSE of stations where red dots indicate more than three standard deviations above the mean

42 Chapter 4. Results 30 in red. This station is ALVA and it will be flagged and omitted from further calculations. Afterwards the covariances between the station Resea data will be modeled using the same covariance function but with different parameters. The RMSE and RMSE outlier locations can be seen on Figures 4.15 and Since the outliers do not seem far off and the associated station locations are on the edge, no other stations will be flagged. Figure 4.13: Normal QQ plot for residual Resea data Mean Wind Speed The trend model for mean wind speed is U2i = β 1 + β 2 Lat i + β 3 Lon i + ResU2 i where ResU2 i are the residuals. The QQ plot and empirical covariance plot are on Figure 4.17 and 4.18 respectively. The same covariance model was used and the parameters were estimated by linearization and using least squares. The RMSE is on Figure 4.19 and a map of the locations on Figure Since the location makes up for the error, no stations will be flagged. Saturation Vapor Pressure The trend model is es i = β i + β 2 Lat + β 3 Lon i + Reses i

43 Chapter 4. Results 31 Figure 4.14: Empirical Covariance for Resea data Figure 4.15: RMSE of stations where red dots indicate are more than three standard deviations above the mean

44 Chapter 4. Results 32 Figure 4.16: RMSE of stations where red dots indicate more than three standard deviations above the mean Figure 4.17: Normal QQ plot for residual ResU2 data

45 Chapter 4. Results 33 Figure 4.18: Empirical Covariance for ResU2 data Figure 4.19: RMSE of stations where red dots indicate are more than three standard deviations above the mean

46 Chapter 4. Results 34 Figure 4.20: RMSE of stations where red dots indicate more than three standard deviations above the mean where Reses i are the residuals. The QQ plot and empirical covariance are on Figure 4.21 and 4.22 respectively. Using the same covariance form, the RMSE from kriging is shown on Figure 4.23 and its locations on Figure There are no serious outliers so nothing will be flagged SVP Slope The SVP slope trend model is i = β 1 + β 2 Lat i + β 3 Lon i + Res i where Res i are the residuals. The QQ plot of Res i is shown in Figure 4.25 and the empirical covariance is Figures 4.27 and 4.28 show no indication stations need to be flagged August Wind Speed

47 Chapter 4. Results 35 Figure 4.21: Normal QQ plot for residual Reses data Figure 4.22: Empirical Covariance for Reses data

48 Chapter 4. Results 36 Figure 4.23: RMSE of stations where red dots indicate are more than three standard deviations above the mean Figure 4.24: RMSE of stations where red dots indicate more than three standard deviations above the mean

49 Chapter 4. Results 37 Figure 4.25: Normal QQ plot for residual Res data Figure 4.26: Empirical Covariance for Res data

50 Chapter 4. Results 38 Figure 4.27: RMSE of stations where red dots indicate are more than three standard deviations above the mean Figure 4.28: RMSE of stations where red dots indicate more than three standard deviations above the mean

51 Chapter 4. Results 39 The model for the wind speed is U2i = β 1 + β 2 Lat i + β 3 Lon i + ResU2 i where the residuals ResU2 have a QQ plot on Figure 4.29 and an empirical covariance on Figure The RMSE of kriging and their locations shown on Figure 4.31 and 4.32, suggest that no stations need to be flagged. Figure 4.29: Normal QQ plot for residual ResU2 data Mean Actual Vapor Pressure The trend model for the mean actual vapor pressure is eai = β 1 + β 2 Lat i + β 3 Lon i + Resea i In this case the QQ plot for the residuals is shown in Figure 4.33 and the empirical covariance plot is on Figure RMSE of kriging predictions and the map is on Figure 4.35 and Figure WEBB will be flagged Saturated Vapor Pressure The trend model for saturated vapor pressure is es i = β 1 + β 2 Lat i + β 3 Lon i + Reses i

52 Chapter 4. Results 40 Figure 4.30: Empirical Covariance for ResU 2 data. Figure 4.31: RMSE of stations where red dots indicate are more than three standard deviations above the mean

53 Chapter 4. Results 41 Figure 4.32: RMSE of stations where red dots indicate more than three standard deviations above the mean Figure 4.33: Normal QQ plot for residual Resea data

54 Chapter 4. Results 42 Figure 4.34: Empirical Covariance for Resea data. Figure 4.35: RMSE of stations where red dots indicate are more than three standard deviations above the mean

55 Chapter 4. Results 43 Figure 4.36: RMSE of stations where red dots indicate more than three standard deviations above the mean where a corresponding QQ plot of Reses is shown on Figure 4.37 and the empirical covariance plot is shown on Figure Notice that there seems to be stations that need to be flagged but it was not, because the correlation seemed to be okay. RMSE for kriging on the residuals is shown on Figure 4.39 and the map on Figure Since the prediction for station WEBB has a prediction error of more than three standard deviations and it is more on the interior of the map it will be flagged. Slope of Saturated Vapor Pressure The model is i = β 1 + β 2 Lat i + β 3 Lon i + Res i The residuals (Res ) has a QQ plot on Figure 4.41 and an empirical covariance plot on Figure There appears to be one station that stands out on the covariance plot along with having odd correlations. This station is BBOW and it will be flagged and ommitted from further calculations. The errors associated with kriging are on Figure 4.43 and the location on Figure No other stations will be flagged for this case. Psychrometric Constant The model for this constant will take the form γ i = β 1 + β 2 Lat i + β 2 Lon i + Resγ i

56 Chapter 4. Results 44 Figure 4.37: Normal QQ plot for residual Reses data Figure 4.38: Empirical Covariance for Reses data.

57 Chapter 4. Results 45 Figure 4.39: RMSE of stations where red dots indicate are more than three standard deviations above the mean Figure 4.40: RMSE of stations where red dots indicate more than three standard deviations above the mean

58 Chapter 4. Results 46 Figure 4.41: Normal QQ plot for residual Res data Figure 4.42: Empirical Covariance for Res data.