Inter-comparison of spatial estimation schemes for precipitation and

Transcription

1 Inter-comparison of spatial estimation schemes for precipitation and temperature Yeonsang Hwang 1,2,Martyn Clark 2,Balaji Rajagopalan 1,2, Subhrendu Gangopadhyay 1,2, and Lauren E. Hay 3 1 Department of Civil, Environmental and Architectural Engg., University of Colorado, Boulder, CO 2 Co-operative Institute for research in Environmental Sciences (CIRES), University of Colorado, Boulder, CO 3 U.S. Geological Survey, Denver, CO Abstract Distributed hydrologic models typically require spatial estimates of precipitation and temperature from sparsely located observational points to the specific grid points. We compare and contrast the performance of several statistical methods for the spatial estimation in two climatologically and hydrologically different basins. The seven methods assessed are: (1) Simple Average; (2) Inverse Distance Weight Scheme (IDW); (3) Ordinary kriging; (4) Multiple Linear Regression (MLR); (5) PRISM (Parameter-elevation Regressions on Independent Slopes Model) based interpolation; (6) Climatological MLR (CMLR); and (7) Locally Weighted Polynomial Regression (LWP). Regression based methods that used elevation information showed better performance, in particular, the nonparametric method LWP. LWP is data driven with minimal

2 assumptions and provides an attractive alternative to MLR in situations with high degrees of nonlinearities. For daily time scale, we propose a two step process in which, the precipitation occurrence is first generated via a logistic regression model, and the amount is then estimated using the interpolation schemes. This process generated the precipitation occurrence effectively. The results shown in this paper will help guide the selection of appropriate spatial interpolation methods for use in watershed models for stream flow simulation, forecasts, and also downscaling of Global Climate Models (GCM) outputs. Submitted to Water Resources Research July 2004

3 1 Introduction Accurate simulations of streamflow from physical watershed models are often limited by the ability to capture the spatial variability of precipitation throughout a river basin [Syed et al., 2003]. Watershed models require estimates of precipitation and temperature among other variables on a regular grid or at sub-basins [Leavesley et al., 1996]. Even though the accuracy of precipitation measurements has been improved, problems still exist, in terms of both sparse spatial coverage of the observations, and difficulties in measuring precipitation during, for example, snow events. Furthermore, precipitation being an intermittent variable property makes it more difficult to estimate throughout the basin from sparse observational network. Spatial interpolation schemes are required that can provide estimates of the mean and uncertainty of precipitation and temperature at required locations (i.e. on a regular grid or at centers of sub-basins). Various methods have been developed for this purpose - these range from simple averaging methods (e.g. Arithmetic mean, Thiessen polygons), to physically based estimates such as lapse rates, to complex statistical methods (multiple linear regression, locally weighted polynomial, kriging, optimal interpolation, etc). Regional climate models are also used to estimate spatial variability in precipitation and other surface climate fields. The skill of these different methods depends on the space and time scales of the precipitation estimate. For example, precipitation estimates at daily (or sub-daily) time scales and small sub-basin areas (e.g. horizontal length scales <50km) are complicated by the intermittent properties of precipitation [Jothityangkoon et al., 2000; Gupta, 1993; Thornton et al., 1997]. Furthermore, the accuracy of different methods is regionally dependent. Methods that work well in regions dominated by large-scale frontal systems may not work well in regions dominated by sporadic thunderstorm activity. The growing needs of complex hydrologic models require estimations of precipitation

4 and temperature at small time (typically, daily or six hourly) and space scales. The purpose of this paper is to provide a comprehensive assessment of the advantages and limitations of different statistical techniques that are used to estimate spatial variability in precipitation and temperature. This assessment will be conducted on both daily and monthly time scales for two river basins in different hydro-climate regimes in the contiguous United States (U.S.). The different statistical methods are: Simple Average; Inverse Distance Weight Scheme (IDW); Ordinary Kriging; Multiple Linear Regression (MLR); PRISM (Parameterelevation Regressions on Independent Slopes Model) based interpolation; Climatological MLR (CMLR); and Locally Weighted Polynomial Regression (LWP). The two river basins presented here (shown in Figure 1) are Animas River at Durango, Colorado (Animas) and Alapaha River at Statenville, Georgia (Alapaha). The paper is organized as follows. Section 2 provides literature review of spatial interpolation methods and section 3 provides description of the spatial interpolation schemes used in this paper. Section 4 discusses the experimental design followed by descriptions of data in Section 5. Results from monthly and daily time scale analysis are presented in section 6. Summary and discussion of the results conclude the paper (section 7). 2. Literature review Several authors have discussed advantages and limitations of various interpolation schemes for precipitation and temperature [Dingman, 1994; Myers, 1994; Dirks et al., 1998; Lanza et al., 2001; Fassnacht et al., 2003] (see Table 1.). These methods range from simple average methods (e.g. Thiessen polygon [Thiessen, 1911]) to distance based methods such as inverse distance weighting scheme and ordinary kriging [Franke and Nielson, 1980; Creutin and Obled, 1982] to

5 techniques that make explicit use of topographic parameters such as elevation in the interpolation routine [e.g. Daly et al., 1994; Rajagopalan and Lall, 1998; Goovaerts 2000]. Simple methods such as spatial average, Thiessen polygon, and nearest neighbor are easy to implement and computationally very efficient. However, these methods provide poor estimates, especially in regions with sparse data network [Thiessen, 1911]. For example, if the grid cells in the hydrologic model use data from the nearest stations, discontinuity arises in the transition between the grid cells represented by two different observation stations (e.g., as in Thiessen polygon method). If all stations in a basin are averaged, then there will be no sub-basin variability at all [Dirks et al., 1998]. Methods that take into account the distances (i.e. distance between the estimation point and the observation stations) explicitly have the ability to address these limitations. Two of the early methods that use the distances are Inverse Distance Weighting (IDW) and Ordinary Kriging (OK). In IDW, weights are applied to the observational data based on the inverse of its distance from the estimation point - the distance is raised to some power [Bussieres and Hogg, 1989; Dirks et al., 1998]. Optimal power of the inverse distance weighting function could be calculated based on minimum error, but the power value of 2 is usually acceptable [Kruizinga and Yperlaan, 1978; Dirks et al., 1998]. OK [Tabios and Salas, 1985] develops weights for the surrounding stations based on both the co-variability between the stations and estimated co-variability between the surrounding stations and the estimation points. The spatial co-variability is often quite noisy (and is unknown between the surrounding stations and estimation points) so is modeled as a function of distance (e.g., using a variogram model). Fitting the variogram function is the key aspect of OK. Efforts

6 have been made to fit variograms objectively so as to improve the estimation [Todini et al., 2001]. In topographically complex regions, information on distance alone is insufficient to produce good spatial estimates. OK has been extended to include topographical information for better capturing the spatial variability of hydro-climate variables [Chua and Bras, 1982; Beek et al., 1992; Pardo-Iguzquiza, 1998; Prudhomme and Reed, 1999; Jeffrey et al., 2001; Kyriakidis et al., 2001; Erxleben et al., 2002]. For example, Goovarerts [2000] applied a regression scheme to capture spatial trends of precipitation field and used kriging on the residuals from the regression. As another extension of OK, indicator kriging can produce improved spatial variability estimation of intermittent rainfall field [Seo, 1996]. Indicator kriging uses a chosen threshold to transform the original values into indicator values (0 and 1). The indicator values are then analyzed to determine spatial variability based on experimental variograms. Kriging has also been applied in a two-step process of rainfall occurrence and amount estimation [Barancourt et al., 1992; Mackay et al., 2001]. An alternative to OK is Multiple Linear Regression (MLR). This popular theory is well developed [Helsel and Hirsch, 1992; Walpole et al., 1998, etc.]. Typically, topographical information such as latitude, longitude, and elevation is used to fit a linear relationship with the hydroclimate variable (precipitation or temperature). The fitted linear equation is then used to estimate the hydroclimate variables spatially [Ollinger et al., 1993; Kurtzman and Kadmon, 1999; Ninyerola et al., 2000; Marquinez et al., 2003]. For example, Daly et al. [1994] used regression techniques to estimate spatial variability of precipitation and other climate variables for the entire US. As part of this research, Daly developed the PRISM (Parameter-elevation Regressions on Independent Slopes Model) system, which has been used in many applications

7 [Church et al., 1995; Bishop et al., 1998]. Hay et al. [2002] used monthly climatological regression relationships to estimate the daily variability of climate variables (the stations selected for use in this method was optimized so as to maximize the fit between modeled and observed streamflow). Generally, methods that include elevation information tend to perform better, especially on low-density networks [Creutin and Obled, 1982]. Nonparametric methods that are data driven and do not require assumptions of the underlying function (e.g., linear) provide an attractive alternative. There are several nonparametric methods such as Splines, kernel-based [Owosina, 1992], and local polynomials [Loader, 1997; Rajagopalan and Lall, 1998]. Local polynomials are easy to implement and are skilful. The estimate at any point is based on a polynomial (of order p) fit to a small number (k) of its nearest neighbors. The polynomial order and the number of neighbors are obtained using objective criteria such as cross-validation measures, from the data. If p is equal to 1 (i.e. linear function) and the number of neighbors includes all the data points then this collapses to a traditional linear regression. Thus, the local polynomials can be thought of as a super set of linear regressions. The performance of nonparametric methods on a variety of synthetic and real data sets has been documented by Owosina [1992]. In many cases, simpler methods show moderate skill with less computational cost [Hevesi et al., 1992; Goodale et al., 1998; Hartkamp et al., 1999; Kurtzman and Kadmon, 1999; Parajka, 2000]. Application of the interpolation schemes on daily or sub-daily rainfall field tends to have higher variability compared to the monthly or annual analysis [Bussieres and Hogg, 1989; Jeffrey et al., 2001]. For example, skills of various kriging methods decrease with increasing spatial and temporal variability [Borga and Vizzaccaro, 1997; Hartkamp et al., 1999].

8 3 Interpolation methods We selected seven different methods for spatial interpolation with varying degrees of complexity for this inter-comparison effort. The methods are (1)Simple Average, (2)Inverse Distance Weight scheme, (3)Ordinary Kriging, (4)Multiple Linear Regression (MLR), (5) PRISM based interpolation, (6)Climatological MLR, (7)Locally Weighted Polynomial regression (LWP). For the daily time step analysis, logistic regression is also applied with the selected interpolation schemes. 3.1 Straight Average (SA) This (SA) method calculates the arithmetic mean of all the available observations in a basin. Thus the calculated mean is the estimate at any point in the basin. All of the observation points contribute equally to the estimate of the mean. Consequently, the resulting estimation doesn t show any spatial variability. However, this method is still valuable for the estimation of areal mean of hydroclimate variables on a small, dense, and well-distributed station network. The estimation of precipitation is given as, p 1 = n est p i n i= 1 (1) where p i is the observed precipitation at i th station, n is the total number of observation stations. 3.2 Inverse distance weight scheme (IDW)

9 The IDW method assigns weights based on the inverse of the distance to every data points located within a given search radius centered on the point of estimate The nearest observation station has the biggest weight and the most remote station has smallest weight. As a constraint, sum of the weights should be equal to 1. Traditional form of this method is, p est = n i= 1 w p i i (2) w = i 1 k di 1 d n j= 1 k j (3) where n is the number of stations in search radius, d i is the distance between estimation and i th observation points, d j is the distance between the estimation and each of the j observations, k is the power ( 1 k ). In this study, the revised weighting function introduce by Franke & Nielson [1980], which gave better result than the traditional function, was used: w i = R d i Rdi n R d j j= 1 Rd j 2 2 (4) where R is the maximum distance between the estimation and observation points inside the search radius. In equation (4) all weights are calculated based on the 3-D distance with the same units (meters). The estimates are less sensitive to the power of the distance function. Typically,

10 power value of 2 seems to work fine based on the error measures in applications of precipitation estimations [Kruizinga and Yperlaan, 1978; Dirks et al., 1998]. One can also incorporate elevation in calculating the distance thus bringing in the topographical information. 3.3 Ordinary Kriging (OK) Many spatial interpolation schemes are essentially based on the same form as equation (2) but use different weighting functions. The OK method calculates weights using a vector of covariances between the estimation point and surrounding observing stations (D) and also a matrix of covariances between all observing stations(c): 1 w = C D (5) where w is the vector of weight, C is the covariance matrix between observation locations, D is the vector of covariance between estimation point and observation locations. Because the covariance structure between the estimation and observations points is unknown, a model which is a function of distance is needed to determine the weights. This model is referred to as the variogram [Journel, and Huijbregts, 1978]. To derive this variogram model, covariance is estimated between all possible observation points within a specified distance. A smooth function is then fit to the observed covariances. This fitting process involves selecting a function (e.g., spherical, exponential etc) and its associated parameters (range, nugget, sill, power etc.). The variogram fitting is to an extent subjective. Given the limited choice of functions, the function selected often provides a poor fit to the covariance estimates from the observations.

11 Consequently, the estimates from equation (5) using the fitted variogram will have high variability i.e. poor performance. This is one of the key drawbacks of OK. To illustrate this point, the variogram for a simple application of well pumping surface estimation problem is shown in Figure 2. A variogram for January monthly mean precipitation from the Animas basin is also shown. It can be seen that the best fitted variogram does not capture the spatial covariability of the data very well (shown as points on the graphs). If the underlying surface is highly nonstationary (such as the case in this example) obtaining a best fit variogram will be difficult, thus, greatly limiting the applicability of Kriging. 3.4 Multiple Linear Regression (MLR) The MLR method assumes that a linear relationship between the predictor variables (typically topological variables) and a known response variable (precipitation or temperature) can be fitted and used to estimate the response variable at any desired locations. The model is of the form: p est = b0 + b1 x + b2 y + b3 z (6) where x,y,and z are dependent variables of latitude, longitude and elevation, respectively. b 0, b 1, b 2, b 3 are the regression coefficients. These regression coefficients are estimated by minimizing the squared errors. Because of the strong orographic effects on the temperature and precipitation [e.g., Henry, 1919; Sevruk et al., 1998], elevation is included as a predictor variable in most spatial interpolation researches on MLR. In this formulation, a separate regression equation may be

12 used for each time step (e.g., months, days). For this research, latitude, longitude, and elevation are used for the predictor variables, and the separate regression models are used for each time step. 3.5 Climatological MLR (CMLR) In MLR, the regression coefficients are used to fit a best model to estimate spatial variability on each time step. In CMLR, monthly trend (parameters for each climate variables) is assumed to be preserved through the finer time scale (say, daily time series) but the optimal intercept of the regression model can be changed for each time step. As the first step, fixed seasonal coefficients are calculated from the basin climatology. Using the monthly total (or monthly averaged for temperature) and the predictor variables x, y, z (latitude, longitude, and elevation, respectively), monthly MLR equations can be set up similar to that described in section 3.4. In traditional MLR method, coefficients in the model are calculated repeatedly on each time step. Secondly, the CMLR method changes intercept of the monthly model for shorter time scale estimation based on a group of anchor stations [Hay et al., 2002]. This intercept is calculated using mean values of climate variables and x, y, and z coordinates of the selected group of observation stations. With the fixed b 1, b 2, and b 3, intercept b 0 is determined with the 'optimal' anchor station sets by Exhaustive Search (ES) analysis [Wilby et al., 1999]. Several error measures could be used to find best anchor stations. Hay and McCabe [2002] also found that the accuracy of runoff estimation did not show significant improvement with more than three anchor stations. Different from the study of Hay et al. [2002], root mean square error between the dropped observation (cross validated) and estimation was tested as the objective function of the

13 optimal intercept calculation in this research. In addition, the nearest three station set was used as the anchor stations and all the results presented in this paper are based on this station choice option. After computing the optimal intercept for each given time step and for each estimation point, climate variables are calculated by the following linear regression equation: p est = b0, opt + b1, mth x + b2, mth y + b3, mth z (7) where b 1, mth, b, mth b, 2, and 3 mth are the regression coefficients based on climatology (monthly), b 0,opt is the intercept for each time step. 3.6 PRISM-based interpolation The PRISM method is a regression-based spatial interpolation method using climate data. It was developed to model the strong orographic effect on precipitation and give better estimation on mountainous terrain. PRISM develops weighted regression functions of elevation and precipitation to predict the precipitation on each cell s elevation [Daly et al., 1994]. Using this method, the Spatial Climate Analysis Service (Oregon State University) has constructed climatological maps of precipitation and other variables on a 2-km grid for the contiguous U.S. The PRISM climatology is used for interpolation. Extracting monthly and daily information from the PRISM climatology involves the following steps: (i) Anomalies of precipitation and temperature are calculated based on monthly mean values. The temperature anomalies are calculated as the difference between the observation and longterm mean monthly mean value. The precipitation anomalies are calculated as the ratio of the observation and long-term mean monthly total value.

14 (ii) The anomalies are interpolated to each estimation point using IDW (other schemes may be used as well). (iii) PRISM climatology is added to the interpolated values to get the final estimates. The PRISM value for each estimation point is selected from the nearest PRISM grid point value and added to the interpolated value: t p est est = t = a p a + t prism p prism (8) where t, p are the estimated value of climate variables for each station, t a, p are the a est est interpolated climate variable anomalies, t, p are the monthly PRISM values from the prism prism nearest grid point. For the daily time scale precipitation estimation, monthly mean precipitation is divided by the number of days of each month to get anomalies. PRISM values are added in the same way. 3.7 Locally Weighted Polynomial method (LWP) The locally weighted polynomial method is similar to the MLR (section 3.4) but the regression equation is developed using nearest neighbors. A general form of local regression with one predictor variable is, p = µ ( ) + ε (9) i x i i where µ (x) is the appropriate polynomial function, ε i is the estimation error.

15 The polynomial function involved in this model can be linear or any higher order but linear function was used for this research [Loader, 1997]. This function is fitted by the minimizing a locally weighted least square in a given sliding window. However, order of the estimation model should be carefully selected in order not to give unwanted higher variance in estimation [Loader, 1999]: n i= 1 xi x 2 W ( pi ( a0 + a1( xi x))) (10) h where W( ) is the weighting function, h is the window width, a 0, a 1 are the coefficients. Optimal model fit is determined with proper neighbor size around each estimation point. Using GCV (General Cross Validation statistics), best neighbor size is determined based on the number of predictor variables and estimation error: GCV n e 2 i i= 1 n = m 1 n 2 (11) where e i is the error, n is the number of data points, m is the number of parameters. In this research, the statistical package LOCFIT [Loader, 1992] was used to fit locally weighted polynomial model and estimate the variability of climate variables. During the model fitting, the ratio of neighbor size and the total observation should be greater than certain value. It depends on the number of predictor variables and number of available data points [Loader, 1992]:

16 2 m + 1 α min (12) n where α is (neighbor size)/(number of observations). For the cross validation analysis in this research, lower bound of the alpha was set to (13/n). The theoretical lower bound could be 7 (= 2*3+1), but it s still very close to the edge of the model capacity. In this research, minimum number of observation of 13 was selected to get smoother but stable estimation. 3.8 Extensions to daily precipitation interpolation In practice, most watershed models require precipitation estimates at a daily time step. Jeffrey et al. [2001] suggest two different approaches to get daily interpolated climate variables; (1) direct interpolation from daily record and (2) generation of the daily values from monthly interpolated values. They adopted the second approach in their study. Either of the methods could be problematic if sufficient daily observations are not available. Spatial interpolation methods, such as those described in the previous section can be used on a daily time scale. However, due to the intermittent property of daily rainfall, interpolating several zero values tend to produce unrealistic rainfall fields at the daily time scale not to mention the potential for generating negative values. We use two methods for spatial interpolation at the daily time scale: (1) the interpolation methods described above are applied to the daily data and negative estimates are replaced with zero and (2) logistic regression is used to estimate rainfall occurrence and the interpolation methods described above are applied at locations where rainfall occurrence is generated. The

17 schematic of applying logistic regression to precipitation occurrence in a basin is shown in Figure 3. The station data sets are first transformed into a time series of occurrence (1 = wet days and 0 = dry days). Then, the precipitation amounts are interpolated only on estimated wet days. Similar to the standard least square regression model (6), the logistic model is consisted with regression constant β 0 and slopes β k for each predictor variables x k [Clark et al., 2004]: p = exp 1 ( β + β x + β x β ) k x k (13) where, as in ordinary least squares, the beta values are the regression constants, p is the probability of precipitation occurrence. The climatological rainfall probability for each dropped stations are also calculated using the rainfall probability of the surrounding stations from historical data through logistic regression in order not to give negative or probabilities exceeding unity. If the estimated probability of rainfall from the logistic regression is less (larger) than the climatological rainfall probability (threshold) then, the estimating station is set to dry (wet). The climatological probability of precipitation is calculated as, all rain day Climatolog ical rainfall probability = (14) # of available stations all rain day # of rainfall stations

18 The main purpose of two-step process is obtaining realistic rainfall occurrence using logistic regression before applying the interpolation methods to estimate the precipitation amounts. Due to data limitations we use three interpolation schemes (IDW, MLR, and LWP) in conjunction with the logistic regression. 4. Experimental Design The models described above were applied to monthly and daily precipitation and temperature observations from two basins. The flowchart for the estimation process for monthly time scale is shown in Figure 4. Three groups of analysis were arranged according to the available interpolation schemes shown in Table 2. The flowchart of the two-step method is also shown in Figures 5. The models are fitted on the observations and a suite of performance measures estimated in a cross-validated mode for comparison. The performance metrics are described below. 4.1 Measures for performance comparison Various measures of performance are used in this research for comparison. Depending on the measure, choice of basin and the interpolation method, the ranking between the schemes can change considerably [Bussieres and Hogg, 1989]. In addition to this arbitrariness, a method that is more elaborate than warranted by the quality of the data could give numerically superior results but the result could be misleading under the uncertainty of the record itself. In order to avoid possible unwanted overestimation of each interpolation schemes, only minimum effort was done on model tuning and each model was kept as simple as possible. The following measures were chosen for comparison:

19 a) Bias of the mean of climate variables on each station through time step: M i = N t = 1 p N ie ( t) N t = 1 p N io ( t) (15) where N is the number of time steps, p ie is the estimated climate variable at station i, p io is the observed climate variable at station i. b) Bias of the variance of the climate variables on each station through time step: V i N N 1 2 = [ pie ( t) pie ] [ p N 1 t= 1 t= 1 io ( t) p io ] 2 (16) where p is the mean of climate variable at station i. c) Spearman rank correlation coefficient between the observed values and cross-validated estimates at each station: r = n 6 i= n( n d 2 i 1) (17) where d i is the difference between the ranks assigned to the two variables, n is the number of pairs of data. For monthly (daily) time scale the monthly (daily) precipitation and temperatures are estimated (in a cross-validated mode) for each year at each station. The Spearman rank

20 correlation is computed between the two. Thus, obtaining as many correlation coefficients as there are number of stations. Some advantages of the rank correlation coefficients are reported over the traditional correlation coefficient. First, the rank correlation coefficient doesn t assume that the relation between two variables is linearly related. Second, the rank correlation coefficient does not need normality [Walpole et al., 1998]. Because systematic bias may lead erroneous correlation, this measure was used with RMSE for better performance assessment. d) Root mean square error of the estimated values: RMSE N t= 1 = ( p ie ( t) p N io ( t)) 2 (18) e) Inter-station correlation between the observed values and estimation at each time step. In the case of the monthly time step, for each year the cross-validated estimates and the observed values at all the stations are correlated, thus, obtaining as may correlation coefficients as the number of years. 5 Data Two basins with different climatic characteristics chosen for this study (Figure 1) as mentioned earlier are Animas and Alapaha. Animas basin is snowmelt driven with occasional rain-on-snow events during winter and, have large relief with elevation ranging from 680m to 3700m. The Alapaha basin is at a lower elevation dominated by rainfall events. Among the

21 various characteristics of watersheds topography has significant impact on spatial variability estimation of climate variables. Daily minimum and maximum temperature and precipitation recorded at all the observational locations were compiled from National Weather Service (NWS) and snow telemetry (SNOTEL) databases [Hay et al., 2002] shown in Table 2. To be consistent across the basins, the data from 1979 onwards were used in the study. 5.1 Data quality control Measurement and recording errors are common in hydroclimate data sets and can impact the interpolation schemes [Dingman, 1994]. Reek et al. [1992] noticed that there is significant number of erroneous daily values in climate variables records. The errors include data entry, recording, and reformatting errors. Some of the selected data quality control methods examined in their research (1) extreme outlier detection, (2) diurnal change limit, (3) inconsistency, (4) spike check, (5) Z-score test, (6) same value repeat check were applied on the data sets chosen for this study. These methods are described in the work of Reek et al. [1992]. In order to obtain stable estimates the minimum number of observations required by each method is different (Table 3). For cross validation purpose, IDW required at least two available stations because the weight can be given as one if there is only one station available except for the estimation point. The MLR method needs at least five available points because there are four unknown coefficients to be determined (coefficients for three coordinates and a intercept) except for the estimation point that is being hold through the evaluation process. Data variability varies quite a bit between months and basins, as can be seen in Figure 6. Figure 6 shows the graph of available total days versus available stations. Each four rows show

22 the graphs for the basins used in this research. First two columns are for January and the rest are for July. For example, the first graph (row 1, column 2) shows that most of the stations (15~17) maintain the entire time steps in January data series of Animas basin. Then, available numbers of days drop rapidly beyond 17 stations. However, entire time steps are available on less then half the total stations on the January series of Alapaha basin (2 nd row, 1 st column). As a result, a combination of interpolation methods will have to be used to address the data availability issue. This will be mentioned in the following section. 6 Results A comparison of the performance on monthly and daily time scales following the flowcharts in Figure 4 and 5 was performed incorporating the data requirements for the different methods (Table 3). The methods are applied for all months each basin. Results for January and July are shown as representative of the wet and dry seasons in the basins. But first, a short comparison on synthetic data is presented. Data is generated to mimic a ground water surface due to four pumping wells (see Figure 2). It can be seen that the surface is highly nonlinear with sharp gradients and curvature at the well points. As mentioned earlier, the best fitted variogram (Figure 2) does not capture the spatial co-variability of the data very well (shown as points on the graphs). Consequently, the estimates are also poor. We applied MLR and LWP to this data set. As expected, the MLR fits a global linear function, which clearly is not suitable for this data set, thus, completely smoothing out the nonlinearity, while the LWP, being a local estimation method, captures the variability much better. This is demonstrated by the fact that the median biases of the estimates of the data points are and -0.02, and the RMSE values are 0.28 and 0.5, respectively, for LWP and MLR. Significant improvements in the LWP estimates are apparent.

23 Extensive comparisons on a variety of synthetic data sets (Owosina, 1992) also bear out this conclusion. 6.1 Monthly precipitation and temperature Cross-validated estimates are used to calculate the performance measures. Figures 7 and 8 shows the boxplots performance measures for January and July monthly total precipitation and maximum temperature, respectively, in the two basins Animas and Alapaha (all of the graphs use millimeter for precipitation and degree Celsius for temperature). The length of the boxes indicates the interquartile range of the measures from cross validated estimates at each location in the basin and the whiskers show the 5 th and 95 th percentile range; the horizontal line in the box is the median of the estimates. Larger box length indicates increased variability in the estimates. In each of the plots, median value of the best performer is shown as a gray solid line. While there is surprising similarity in the skill among different techniques, several observations are apparent: (i) The SA method performs poorly on almost all the measures (ii) Methods using elevation perform well. In particular, the LWP method seems to show relatively good performance (smaller box widths and median value close to zero). (iii) For precipitation, all methods displayed a negative bias in the variance this is expected as the noisy character of individual station time series is smoothed. To assess the performance of each method in reproducing observed spatial variability, Spearman rank correlations are computed between the cross-validated estimates and the observed values on the spatial map. The resulting set of spatial correlations are shown as boxplots in Figure 9. It can be seen that the correlations are higher for January precipitation

24 compared to July. The correlations from the straight average method are -1. This is because, in cross-validated the observation of a given year is dropped and is predicted using the rest of the data, as a result, if a large value is dropped the average from the rest of the data is going to be small and vice versa, thus, producing a correlation of -1. Note also the correlations on spatial map for CMLR, which occur because the CMLR method is using the coefficients obtained from climatology. CMLR method, which is based on linear regression theory like MLR, showed good performance especially on mountainous basin. However, CMLR s competitive but lower performance in Alapaha basin can be seen from the higher bias of mean compare to that of MLR (Figures 7 and 8). The main idea of the CMLR method is to get stable estimation using the longterm climatological trend through the basin for each time step. Keeping the climatological trend in the coefficients for predictor variables, response variables could be estimated with optimal intercept on each time step. However, if a basin has large year-to-year (day-to-day) variation on the climate variables, climatologically estimated model couldn t get a good estimation for monthly (daily) time step. In other words, if the slope of the linear fit is already out of phase, the intercept is not sufficient to catch the variation even though it s the optimal value. To illustrate this point, figure 10 shows the boxplots of the regression coefficients and the correlation between the observed and estimation of the MLR fits at each station for January and July precipitation in the Alapaha and Animas basin. For each station for each year an MLR fit is developed in a cross-validated mode, which is then used to estimate the dropped value. Thus, at each station we obtain an MLR fit for each year. The correlation and the regression coefficients from each fit are shown as boxplots for each station. Figure 10 allows us to assess (a) if the regression coefficients change substantially from year to year; and (b) if these changes are

25 predictable (e.g., as assessed through the correlation plot). We see that both conditions (a) and (b) are satisfied, meaning that the use of climatological regression relations, as in CMLR, will be unable to capture the year-to-year variability, and likely result in lower skill. The methods were applied to interpolate January precipitation on fine DEM grid (roughly 0.9km x 0.9km) in the Animas basin. The resulting surface maps are shown in Figure 11. Strong heterogeneity in the rainfall consistent with topography can be seen with more precipitation in the higher elevations and less at lower elevations. The straight average washes out the strong spatial heterogeneity and consequently, the interpolated values are close to 0. IDW and OK tend to over smooth, while MLR and LWP maps capture the spatial features very well. Comparing the performances between the distance-based and regression-based schemes (Figures 7, 8, 9) the importance of incorporating elevation information in the interpolation methods is apparent. 6.2 Model performances at daily time scale The first approach for interpolation of precipitation at daily time scales is to apply the methods on the daily data and replace any estimated negative values with zero. For temperature this is not an issue as it can take negative and positive values. The boxplots of the performance measures from this approach are shown in Figures 12 and 13. The observations made from Figures 7 and 8 are generally valid here as well: the regression-based methods using elevation seem to perform better. Performance of the interpolation methods on daily time scales was similar to those seen at the monthly time scales. Spatial map of daily precipitation on for Jan 25, 1979 from a selection of methods is shown in Figure 14. Notice the reduction in the magnitude of the interpolated values relative to the observed and also the over-smoothing by the

26 interpolation methods, thus, missing the spatial features seen in the observed. This is largely due to the fact that the rainfall occurrence is not well generated, which we hope to correct using the two-step approach. Logistic regression is first applied to estimate the precipitation occurrence and the interpolation schemes are then applied to estimate the precipitation at locations with precipitation occurrence generated by the logistic regression. For the two-step process the options available based on the data limitations are shown in Table 4 - Case 1 (Case 2) is where IDW (MLR) is used to interpolate precipitation amounts. As mentioned before, if the sample size on any day is small, then the IDW method is used for interpolation. Otherwise the regression based methods are used. Figure 15 shows the bar chart of the ratio of correct hits of the precipitation occurrence for the Animas and Alapaha River basins. We sum all the days on which the logistic regression estimated the precipitation occurrence correctly. Notice that using logistic regression greatly increases the hit rate (from 50 ~ 75% to 80%~91%) relative to using the MLR and setting the negative values to zero, for both the basins. The hit rate for Case 2 is slightly lower than for Case 1 because MLR s estimate of precipitation amounts can be less than zero. Rainfall occurrence estimation bias was also analyzed by contingency table and Kuiper s skill score (KSS) [Wilks, 1995]. The contingency table is consisted of four elements showing number of days included in each category as shown in Table 5. KSS is calculated as, KSS ( ad bc) = (19) ( a + c)( b + d) A value of 1.0 represents perfect estimation skill for this skill measure. Figure 16 shows the great improvement in rainfall occurrence estimation skill on both basins and seasons. However,

27 summer season s skill for the rainfall-dominated Alapaha basin was lower than winter season while no difference was observed on in the snow-melt dominated Animas basin. In the Animas basin, strong elevation effect is believed to provide similar estimation skill for both seasons. Figure 17 shows the boxplots of performance measures of interpolation of daily precipitation. Notice that the magnitudes of the biases are significantly smaller (and closer to zero) compared to that from the single step (i.e. not using the logistic regression for precipitation occurrence estimation) method in Figure 12. However, a slight negative bias is apparent. This is largely due to the threshold established in the logistic regression for generating the number of rainy days. 7 Summary and Discussion We compared a suite of spatial interpolation schemes for precipitation and temperature at the monthly and daily time scales. For interpolation of daily precipitation a two-step approach was used. In this, the spatial occurrence of precipitation is first generated using logistic regression and the precipitation amounts are estimated using the interpolation schemes. Based on the performance measures, regression based methods that used elevation information performed quite well in estimating the observations in a cross-validated mode. However, there are some issues around proper application of various spatial interpolation schemes. The difficulty with fitting a good variogram to the data rendered poor performance of Ordinary Kriging. There are variations of ordinary kriging method (indicator kriging, cokriging, universal kriging, etc) to improve the estimation skill with the consideration of topography and intermittency of rainfall events [Prudehomme and Reed, 1999; Barancourt et al., 1992]. However, even with this added complexity, one is not assured to find a variogram that has a good fit to the data.

28 Among the regression based methods, the main drawback of PRISM based method is the accuracy in the PRISM climatology, which has its own uncertainty. Also, the use of ratio corrections for precipitation does not preserve the observed probability density function of daily precipitation amounts (i.e., it alters the skewness). The CMLR method, which gives more stable regression model using climatology, was applied for the point estimation in this research and showed good performance on mountainous basin. In addition to this, there are efforts to improve the performance of regression based methods by using additional predictor variables such as slope, wind, and the distance from shoreline [e.g., Ninyerola et al., 2000; Marquinez et al., 2003]. The logistic regression significantly improved the estimation of rainfall occurrence and this coupled with the regression-based methods seems to be able to capture the high degree of spatial variability of precipitation on a daily time scale. Due to hydrologic variability in the watersheds it is often difficult to prescribe a single method for spatial interpolation [Dingman, 1994; Kruizinga and Yperlaan, 1978]. However, overall, our comparisons seem to suggest MLR and LWP as competitive methods for spatial interpolation of precipitation and temperature. In the application presented in this study topography is a dominant factor and the spatial variations of precipitation and temperature by and large follow linearly with topography. Hence, MLR and LWP performed comparably. Given the flexibility and data-driven aspect of LWP, it has the ability to capture any nonlinearity thus, making it more attractive. The IDW method is also a nonparametric method similar to LWP and hence, the results from both of them are comparable. However, LWP is a significant improvement to IDW theoretically. All of the methods used in this research are also applied on the other two basins (East fork of the Carson River near Gardnerville, Nevada (Carson) and Cle Elum River near Roslyn,Washington (Cle Elum)). The results were consistent through these two basins.

29 A proper rainfall regeneration method should be considered with rainfall intermittency estimation. In this research, the logistic regression showed no overall bias on the wet day regeneration above the threshold but the resulting rainfall estimation showed under estimation of the rainfall amount. This rainfall amount under estimation might be corrected by proper threshold and probability cutoff value, etc. Further research should be followed. There are issues beyond the analysis done in this study for the practical use of climate variable estimation. These include the method of data handling and sensitivity analysis of interpolation methods to runoff model. Effect of available data point size in a basin and the observation density should be checked because the density and quality of available data set is a critical issue on most interpolation methods. This may also include the issue around the search radius (related to the number of available data). To test this, density of the observation could be adjusted by randomly selecting the observation station in certain ratio to the total number of observation with varying search radius. The resulting relation between the network density and the search radius (number of observation) could be used as the guidance for interpolation scheme selection according to the network density. In addition, the impact of the possible outliers also needs to be checked by performing a series of interpolation analysis with a station dropped one by one. Because the climate variables are one of the basic input data for runoff models, the sensitivity of spatially interpolated product should be investigated. However, through the complicated parameter calibration and modeling procedures, subtle differences between the spatial interpolation methods (as shown in this study) may be unrecognizable in practical runoff analysis. Addition to this, there is no guarantee that the best point estimation method gives the best runoff estimation though runoff models. Among the interpolation methods used in this paper,

30 the main idea of CMLR method was developed to produce better rainfall amount because the accuracy of some hydrologic models like PRMS are more depends on the accurate estimation of rainfall quantity than spatial variability. This is one of the reasons that performance of CMLR method was not good in this research. Hay et al. [2002] also showed that the prediction of temperature is important in snow-melt dominated basins. The results shown in this paper will help selecting proper spatial interpolation schemes not only for watershed modeling but also for agricultural, climate impact studies using GCM, etc. However, the differences of performances between the interpolation schemes were often not clear. It is possible that the different estimations of the spatial variability will result in larger differences of simulated runoff through a hydrologic model so, the small differences between the spatial estimations turned into a critical issue, or vice versa. The next step of our research is to understand the sensitivity of the different spatial estimation on the streamflow simulation. This work is currently underway. Acknowledgments Partial support of this work by NOAA GAPP program (Award NA16GP2806) and the NOAA RISA Program (Award NA17RJ1229) is thankfully acknowledged.

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38 Draft Figure captions Figure 1: Map showing the study regions Figure 2: (a) Spatial map of the well draw down surface due to 4 pumping wells (b) Fitted Variogram (line) and observed estimates (dots) (c) Fitted variogram and observed estimates of January mean precipitation in the Animas basin. Figure 3: Schematic of application of logistic regression to the precipitation occurrence process Figure 4: Flowchart for spatial interpolation of monthly precipitation and temperature Figure 5: Flowchart for spatial interpolation of daily precipitation using the two-step process Figure 6: Number of observations available in each basin Figure 7: Boxplots of estimation bias, error, and correlation for January and July monthly total precipitation (over 12 available stations, Alphabets shown on x-axis [S/I/K/M/O/P/L] stands for [Straight average/idw/kriging/mlr/cmlr/prism/lwp]) Figure 8: Same as Figure 7 but for January and July monthly maximum temperature. Figure 9: Boxplots of correlations on spatial map for January and July monthly total precipitation Figure 10: Boxplot of MLR coefficients and correlation for estimation of January and July monthly total precipitation in Animas and Alapaha basin Figure 11: Spatial map of estimates from selected methods for January mean monthly total precipitation in Animas basin (Observed rainfall values are shown as white numbers on topography plot located at the upper left pane) Figure 12: Same as Figure 7 but for interpolation of daily precipitation 38

39 Draft Figure 13: Same as Figure 8 but for interpolation of daily temperatures Figure 14: Same as Figure 11 but for daily precipitation on for Jan 25, 1979 Figure 15: Bar charts of rainfall occurrence hit ratio (which is the ratio of correct estimation of rainy and dry days) simulated from different methods Figure 16: Skill of rainfall occurrence realization Figure 17: Same as Figure 7 but from daily interpolation of precipitation using the twostep process. 39

40 Draft Tables Table 1. References of spatial interpolation on rainfall data Category Methods References Simple methods Distance based methods Topography involved methods Nearest neighbor, Arithetic mean, Thiessen polygon Inverse distance weighting, Thiessen (1911) Kruizinga and Yperlaan (1978), Franke and Nielson (1980), Bussieres and Hogg (1989), Dirks et al. (1998), Hartkamp et al. (1999) Ordinary kriging Creutin and Obled (1982), Tabios and Salas (1985), Barancourt et al. (1992), Borga et al. (1994), Borga and Vizzaccaro (1997), Sun et al. (2002), Syed et al. (2003) Cokriging, detrend kriging, indicator kriging, two-step process Multi-linear regression, PRISM, CMLR, locally weighted polynomial Chua and Bras (1982), Barancourt et al. (1992), Beek et al. (1992), Hevesi et al. (1992), Young (1992), Seo (1996), Pardo-Iguzquiza (1998), Kurtzman and Kadmon (1999), Prudhomme and Reed (1999), Goovaerts (2000), Kyriakidis et al. (2001), Mackay et al (2001), Todini et al. (2001), Jeffrey et al. (2001), Goovaerts (2000), Erxleben et al. (2002), Kastelec and Kosmelj(2002), Ollinger et al. (1993), Daly et al. (1994), Loader (1997), Goodale et al. (1998), Rajagopalan and Lall (1998), Loader (1999), Ninyerola et al. (2000), Hay et al. (2002), Fassnacht et al. (2003), Marquinez et at. (2003) Notice that papers present multiple methods are listed in the group by the method that showed best performance in the paper. Basin # of stations Table 2 Data availability in each basin Data Period Animas ~ Alapaha ~ Methods used Number of available observations >12 >4 >2 1. Straight average 2. IDW 3. PRISM 4. MLR 5. CMLR 6. Kriging 7. LWP Table 3 Data requirements for each interpolation scheme Schemes Min. # of sample* 1. Straight average 2. IDW 3. PRISM 4. MLR 5. CMLR 6. Kriging Explanation Straight average 2 Need min. of 1 excluding the estimation point itself IDW 2 same as above Kriging 4 Depends on the data structure MLR 5 # of parameters+1(intercept)+1(cross validation) CMLR 5 same as above PRISM 2 Same as IDW (PRISM data set is needed for application) LWP 10 (2*p+1)+3 * All for cross validation analysis 1. Straight average 2. IDW 3. PRISM 40

41 Draft Deg. of sophistification Table 4 Options for daily interpolation of precipitation in the two-step process [for all time steps] If (P log >P clim ) then, Case 1; Use IDW method for all days Case 2; Use IDW, for # of rainfall station <5 Use MLR, for all other days Case 3; Use IDW, for # of rainfall station <5 Use MLR, for 5 < # of rainfall station < 12 Use locally weighted polynomial, for # of rainfall station >12 P log ; Rainfall probability of the estimating station calculated by logistic regression P clim ; Rainfall probability from climatology of the basin Table 5. 2X2 contingency table Observed Wet Dry Estimated Wet a b Dry c d 41

42 Draft Figures Figure 1. Map showing the study regions 42

43 Draft Figure 2 (a) Spatial map of the well draw down surface due to 4 pumping wells (b) Fitted Variogram (line) and observed estimates (dots) (c) Fitted variogram and observed estimates of July mean precipitation in the Animas basin. 43

44 Draft Rainfall field Rainfall occurrence x y z r Interpolation by logit regression x y z r Basin Get interpolation on rainy stations x y z r Apply interpolation schemes Figure 3 Schematic of application of logistic regression to the precipitation occurrence process 44

45 Draft Figure 4 Flowchart for spatial interpolation of monthly precipitation and temperature Data Correction For all climate variables #>2 # of stations. #>12 2<#<4 over 80% 30~100% over 30% Straight avg. IDW PRISM Straight avg. IDW Kriging MLR CMLR PRISM Straight avg. IDW Kriging MLR CMLR PRISM LWP Performance comparison 45

46 Draft Figure 5 Flowchart for spatial interpolation of daily precipitation using the two-step process Daily Precipitation Data No # of station. >4 Yes Yes All stations dry No Use IDW, Logistic regression Pclim < Plogit rain Pclim > Plogit no rain Set, P=0 3-cases based on # of rain station on the given time step Case 1 Use IDW Case 2 #<4; IDW #>4; MLR Case 3 #<4; IDW 4<#<12; MLR #>12; LWP Performance comparison 46

47 Draft Figure 6 Number of observations available in each basin 47

48 Draft Figure 7 Boxplots of estimation bias, error, and correlation for January and July monthly total precipitation (over 12 available stations, Alphabets shown on x-axis [S/I/K/M/O/P/L] stands for [Straight average/idw/kriging/mlr/cmlr/prism/lwp]) 48

49 Draft Figure 8 Same as Figure 7 but for January and July monthly maximum temperature 49

50 Draft Figure 9 Boxplots of correlations on spatial map for January and July monthly total precipitation 50

51 Draft Animas Basin Alapaha Basin Figure 10 Boxplot of MLR coefficients and correlation for estimation of January and July monthly total precipitation in Animas and Alapaha basin 51

52 Draft Figure 11 Spatial map of estimates from selected methods for January mean monthly total precipitation in Animas basin (Observed rainfall values are shown as white numbers on topography plot located at the upper left pane) 52

53 Draft Figure 12 Same as Figure 7 but for interpolation of daily precipitation 53

54 Draft Figure 13 Same as Figure 8 but for interpolation of daily temperatures 54

55 Draft Figure 14 Same as Figure 11 but for daily precipitation on for Jan 25,

56 Draft Figure 15 Bar charts of rainfall occurrence hit ratio (which is the ratio of correct estimation of rainy and dry days) simulated from different methods Animas Basin Alapaha Basin KSS KSS January July January July 0 IDW MLR CASE1 CASE2 0 IDW MLR CASE1 CASE2 Figure 16. Skill of rainfall occurrence realization 56