Discretization scale dependencies of the ensemble flow range versus catchment area relationship in distributed hydrologic modeling

Journal of Hydrology (2006) 328, 242– 257

available at www.sciencedirect.com

journal homepage: www.elsevier.com/locate/jhydrol

Discretization scale dependencies of the ensemble flow range versus catchment area relationship in distributed hydrologic modeling Theresa M. Carpenter *, Konstantine P. Georgakakos Hydrologic Research Center, 12780 High Bluff Drive, Suite 250, San Diego, CA 92130, United States Scripps Institution of Oceanography, UCSD, La Jolla, CA 92093-0224, United States Received 6 November 2004; received in revised form 30 November 2005; accepted 17 December 2005

KEYWORDS Distributed hydrologic modeling; Flow simulation; Spatial scaling; Simulation uncertainty

Summary Earlier research results have shown that the range of the ensemble flow simulations from a well-validated operational hydrologic model possesses a well-defined log-linear relationship to catchment area for which the ensembles are generated. The ensemble range is a function of both parametric and rainfall input uncertainty. In this paper the sensitivity of this scaling relationship for simulation uncertainty to the numerical discretization scale of the distributed model is quantified when the model is driven by WSR-88D (NEXRAD) radar data and for two catchments in the south-central United States. The distributed model used is subcatchment (rather than pixel) based, has components that are based on operational models employed in the US National Weather Service, and has been validated for the two application catchments of this work as part of the Distributed Model Intercomparison Project (DMIP). The approach taken develops a new parsimonious model for spatially correlated radar-rainfall errors with radar-pixel error variance that depends on the magnitude of the observed rainfall. Monte Carlo methods are used to translate parametric and radar-rainfall input uncertainty to ensemble flow simulations at a number of subcatchments of varying size. Selected events in the period from May 1993 to July 1999 were analyzed. The results confirm that the ensemble flow range is dependent on subcatchment scale with a well-defined log-linear relationship for both application catchments. This finding holds for all uncertainty scenarios examined and across all the scales of distributed model discretization. It is further found that a higher model discretization leads to a shorter range of simulation uncertainty for subcatchment spatial scales that are substantially larger than the average subcatchment size. Flow simulation uncertainty that is due to either parametric or input uncertainty alone scale in a way that is similar to the combined uncertainty scaling. ª 2006 Elsevier B.V. All rights reserved.

* Corresponding author. Tel.: +1 858 794 2726; fax: +1 858 792 2519. E-mail address: [email protected] (T.M. Carpenter). 0022-1694/$ - see front matter ª 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jhydrol.2005.12.008

Discretization scale dependencies of the ensemble flow range

Introduction Exploration and use of distributed hydrologic models for flow simulation and forecasting in operational and planning studies has been a research focus over several decades (e.g., Singh and Frevert, 2002; Beven, 2002). Expansion of effort has occurred in recent years as computing power has increased and availability of spatial data has proliferated. This includes the deployment of operational weather radar and satellite which provide precipitation estimates with high resolution in space and time (e.g., Smith et al., 1996; Owe et al., 2001), and the availability of high resolution digital spatial databases and geographic information systems with hydrologic analysis capability (e.g., Gurnell and Montgomery, 2000). However, extensive implementation of distributed hydrologic models in operational flow forecasting has been limited due to significant uncertainty in defining the necessary model parameters and input with high spatial resolution. In recent years there has been a focused effort to understand the sources of modeling and simulation uncertainty from distributed hydrologic models using operational quality data (e.g., Smith et al., 2004; Koren et al., 1999, 2004; Georgakakos and Krzysztofowicz, 2001; Refsgaard and Knudsen, 1996). The aim of this paper is to explore the relationship between uncertainty in precipitation input and model parameters and the model simulated streamflow for a well-validated distributed hydrologic model and for different scales of spatial discretization of the model. This paper expands recent work by the authors that examines the sensitivity of ensemble streamflow simulations to parametric and input uncertainties (Carpenter et al., 2001; Carpenter and Georgakakos, 2004b). An extensive Monte Carlo numerical simulation experiment has been undertaken to explore the sensitivity of ensemble flow simulations to input and parameter uncertainty at varying scales of distributed model resolution for two catchments in the south-central United States. The application catchments have drainage areas that are similar to areas for which operational forecasts are produced by the US National Weather Service (e.g., on the order of a few thousand km2). The distributed model employed, HRCDHM, is subcatchment based and uses adaptations of operational

243 hydrologic model components for runoff generation and flow routing. The study catchments are introduced in the following section, including a description of the model discretization scales used in this study. Third section discusses model calibration, estimation of parameters for each discretization scale, and the uncertainty model for the parametric errors. A new parsimonious model of the uncertainty in radar-based precipitation input that is based on pixel scale uncertainty and associated spatial correlation structures is described in the fourth section. Fifth section presents a validation of the ensemble flow simulations produced by the HRCDHM when used in conjunction with the models of uncertainty of the third and fourth sections. Discussion of study results is presented in the sixth section. Seventh section summarizes the conclusions and offers potential future research directions.

Study catchments The study catchments are located in the southern Central Plains of the United States. They are the 1232-km2 Blue River basin with outlet near Blue, Oklahoma (USGS gauge #7332500) and the 2482-km2 Illinois River basin with outlet near Tahlequah, Oklahoma (USGS gauge #7196500). The basins are characterized by mildly sloping terrain, significant agricultural (pasture) land use, and fairly uniform soil types. As shown in Fig. 1, the Blue River basin is located in southern Oklahoma and is characterized by a fairly narrow, elongated shape. The Illinois River basin is to the northeast of the Blue River basin and drains portions of Arkansas and Oklahoma. These basins were both included in the US National Weather Service (NWS) sponsored Distributed Model Intercomparison Project (DMIP) and are described by Smith et al. (2004). In DMIP, HRCDHM performed well in simulating the observed streamflow for the study catchments (Reed et al., 2004; Carpenter and Georgakakos, 2004a). In this study, each catchment was subdivided into a number of subcatchments for numerical modeling purposes. The subcatchment delineation process is GIS-based, using 90-m resolution digital elevation data (USGS GTOPO data), 200m resolution land use data (USGS LULC), and stream network data (EPA RF3 Reach Files). Three scales of spatial

Illinois River catchment, outlet near Tahlequah, OK A = 2524 km2 Blue River catchment, outlet near Blue, OK A = 1232 km2

Figure 1

Location of study catchments.

244

T.M. Carpenter, K.P. Georgakakos

discretization for the model were used for each study catchment, representing high-, medium-, and low- resolution distributed modeling scales. The model subcatchments for each catchment and for each discretization scale are shown in Fig. 2. For the Blue River, the high-, medium-, and lowresolution discretization scales yielded average subcatchment sizes of 30-, 59-, and 104-km2, respectively. For the Illinois River the average subcatchment sizes for the high-, medium-, and low-resolution discretization scales were 42-, 84-, and 152-km2, respectively. Geometric characteristics of the subcatchments at the various discretization scales (including the average area, and the median value and range of areas, stream lengths, and stream slopes) are given in Table 1.

Parameter estimation and parametric uncertainty model The hydrologic model, HRCDHM (Carpenter et al., 2001; Carpenter and Georgakakos, 2004a), includes components for mean areal precipitation estimation based on radar precipitation, runoff generation based on the Sacramento soil moisture accounting model (Burnash et al., 1973), temporal

(a) Low Resolution 2 Avg Subcatchment Area = 104km

(b) Medium Resolution 2 Avg Subcatchment Area = 59.4km

(c) High Resolution 2 Avg Subcatchment Area = 30.4km

distribution of runoff at the subcatchment outlet, and kinematic channel routing between subcatchments and to the catchment outlet. The model is forced by precipitation input and reference potential evapotranspiration, and produces estimates of soil moisture content, soil moisture fluxes between two soil layers, subcatchment runoff (surface and sub-surface components) and channel flow at the catchment outlets. The model uses a set of parameters describing moisture storage capacities, withdrawal rates, characteristics of percolation to the lower soil layer, within-catchment time delay between runoff production and appearance of flow at the catchment outlet, and cross-sectional geometry characteristics of main river stems. Values for the model parameters were estimated for the medium-resolution discretization scale for both study catchments during DMIP as reported in Carpenter and Georgakakos (2004a). The calibration was based on historical hydrometeorological data with hourly resolution for the period 5/1993–7/1999, including stream discharge at the catchment outlets, WSR-88D (NEXRAD) Stage III radar precipitation, and energy forcing data (consisting of air temperature, air pressure, solar radiation, and relative humidity used to estimate the potential evaporation rate). This calibration resulted in a set of ‘‘nominal’’ model

(a) Low Resolution 2 Avg Subcatchment Area = 152km

(b) Medium Resolution 2 Avg Subcatchment Area = 83.6km

(c) High Resolution 2 Avg Subcatchment Area = 41.8km

Figure 2 Illustration of the delineated subcatchments for the three discretization scales for the Blue River (left column) and for the Illinois River (right column) catchments. The subcatchment boundaries are shown in black, with the delineated stream network given by the gray lines.

Discretization scale dependencies of the ensemble flow range Table 1

245

Geometric characteristics of subcatchments at various discretization scales Number of subcatchments

Average area (km2)

Median area (km2) (range)

Blue River High resolution Medium resolution Low resolution

41 21 12

30.4 59.4 104

30.7 (2.9–63.4) 38.0 (14.7–155.2) 33.2 (14.7–467.8)

6.31 (1.1–11.6) 8.61 (4.6–23.8) 8.39 (4.6–44.6)

0.0017 (0.0001–0.088) 0.0017 (0.0001–0.088) 0.0025 (0.0001–0.022)

Illinois River High resolution Medium resolution Low resolution

58 29 16

41.8 83.6 152

37.4 (3.2–87.9) 60.2 (8.6–206.2) 73.5 (12.3–413.2)

7.65 (1.3–15.5) 10.6 (3.9–28.4) 10.6 (3.9–43.3)

0.0021 (0.0001–0.042) 0.0022 (0.0001–0.038) 0.0027 (0.0001–0.042)

parameters, assumed uniform over the study catchment. Within each study catchment, distributed parameters were estimated by scaling these nominal uniform parameters based on the spatial variation of selected subcatchmentaveraged soil characteristics derived from the NRCS STATSGO database (NRCS, 1994). The selected soil characteristics were the spatially- and depth-averaged available water content, permeability and soil texture. These characteristics were used to scale the nominal parameters of the upper soil zone moisture capacities, interflow rate, and percolation parameters of the Sacramento model, respectively (see Carpenter and Georgakakos, 2004a for details). With the calibration and distribution of model parameters, HRCDHM reproduced the observed streamflow record well for both study catchments, with high correlation (>85%) and low bias (<2% of the mean flow for the period of record). Furthermore, the DMIP results also indicate skill in the reproduction of flows on scales smaller than those for which the model was calibrated (e.g., Reed et al., 2004; Carpenter and Georgakakos, 2004a). The focus of this research is on the sensitivity of simulated streamflow to input uncertainty at different scales of catchment discretization for selected events in the historical record. To obtain parameter values for the high and low resolution discretization scales of the present study, the Sacramento soil moisture capacity parameters were adjusted by a scaling factor of the form: pffiffiffiffi pffi d ¼ e0:01 AG ð1 eÞ ð1Þ rather than re-calibrating the model parameters for each discretization scale. This scaling factor, d, was developed in Carpenter and Georgakakos (2004a) and accounts for spatial rainfall variability effects on the soil moisture storage capacity estimates for scales differing from the scale of calibration. As applied in this research, AG in Eq. (1) denotes calibration area, and the parameter e is the ratio of application, A, to calibration, AG, area: e ¼ A=AG

ð2Þ

The scaling factor adjusts the ‘‘nominal’’ set of parameter values for the higher and lower resolution discretization scales based on the average subcatchment area of the selected and the calibration (intermediate) discretization scales. Table 2 presents the scaling factors and Sacramento model soil water capacities for the various discretization scales. Although the factor is close to unity, it was em-

Median stream length (km) (range)

Median slope (range)

Table 2 Scaling factors and sacramento model soil capacities for various discretization scales Medium resolution

High resolution

Low resolution

1.0 55 45 210 110 50

1.022 56 46 215 112 51

0.975 54 44 205 107 49

Illinois River d 1.0 UZTWC 100 UZFWC 45 LZTWC 90 LZFPC 64 LZFSC 45

1.027 103 46 92 67 46

Blue River d UZTWC UZFWC LZTWC LZFPC LZFSC

0.9688 97 43.6 87 63 43.6

ployed to be consistent with the fact that Sacramento model parameters are known to be scale dependent. The development of the scaling factors neglected other contributors to spatial dependence such as the vertical spatial variability of soil column and surface land cover. The application of the scaling factors was done to obtain uniform parameter estimates over the application catchment. Spatial distribution of these uniform parameter values for the high and low resolution discretization scales was again based on the spatial variation of the STATSGO soil properties for the subcatchments defined at these resolutions. Implementing the spatially distributed parameters for the high and low resolution discretization scales yielded similar simulation performance when compared to the observations over the calibration period (5/1993–7/1999). Again, low bias and high correlation were found for the high and low resolution discretization scales. For the Blue River, correlation was 86.2% and 86.7% for the low and high resolutions, respectively, with biases of <2%. On the Illinois River, the bias was 5% for the high resolution, and <1% for low resolution, with correlations of >86% in both cases. Parametric uncertainty was characterized for both the runoff-generation component and the flow routing component of the model. Multiple parameters of the Sacramento

246

T.M. Carpenter, K.P. Georgakakos

model runoff generation component were considered, specifically the upper layer soil moisture capacities, the interflow rate, and the percolation parameters. Uncertainty in the lower soil layer parameters was not considered in this analysis due to the focus on significant flow events rather than long-term behavior. The soil model parameter uncertainty was defined again using the STATSGO soils property database. From the range of soil property values specified in the database, the upper and lower bounds of the available water content and permeability soil properties for each subcatchment were determined and used to derive the parameter uncertainty bounds for the upper layer soil capacities and interflow rate by scaling the nominal parameter values. For the parameters of the percolation function of the Sacramento model, the parameter error ranges were based on the uncertainty in the soil texture property yielding interdependent uncertainty ranges for the percolation parameters (for details, see Carpenter and Georgakakos, 2004b). With no additional, a priori information on the distribution of the soil properties within the specified ranges, a uniform distribution of parameter values within the derived uncertainty bounds was employed. Thus, the generic formulation for soil model parameter uncertainty is a ¼ la þ ea

ð3Þ

where a is a soil model parameter, la is the mean value of that parameter, and ea is random error selected from a uniform distribution in the range [aL, +aU], with aL and aU defined for each subcatchment. Thus the uncertainty in soil model parameters reflects the uncertainty in soil properties within and across the subcatchments of the study watersheds. Uncertainty in the routing component involved the estimation of channel cross-sectional characteristics including the top width and hydraulic depth. The generic formulation of Eq. (3) is employed, with a being the model cross-sectional parameter, la the mean value of that parameter, and ea being the uniformly-distributed random error. For the routing model parameters, the random error was selected from a 50% uncertainty range about the mean value, i.e., [0.5la, +0.5la]. This formulation follows the routing model parameter uncertainty of Carpenter and Georgakakos (2004b), wherein they showed that the model was more sensitive to the soil parameters than to the routing parameters.

method (e.g., Mantoglou and Wilson, 1982), was abandoned in favor of a simpler but parsimonious formulation based on spatial moving averages. This analysis builds on previous work by the authors that employed uniformly- and exponentially-distributed MAP errors (Carpenter et al., 2001; Carpenter and Georgakakos, 2004b), and spatially-uncorrelated pixel-scale errors (Georgakakos and Carpenter, 2003). The radar-rainfall data cover the study catchments with a resolution (pixel size) of approximately 3.5 km · 3.5 km. The development of the radar-rainfall uncertainty model begins with the assumption that known biases have been removed through quality control procedures prior to uncertainty analysis. Thus, radar-rainfall errors on this scale are considered Gaussian with mean zero and given standard deviation r. Should any biases be known and present in the data, the model for pixel error uncertainty would include a non-zero mean, which will likely be a function of range from the radar (e.g., Seo et al., 2000). It is noted that the study catchments occupy a relatively small fraction of the radar umbrella and therefore differential range bias within each catchment would be small. Pixel radar-rainfall errors are typically decomposed into a component that measures uncertainty with respect to ‘‘ground truth’’ at the scale of the radar pixel, and a component that accounts for the scale difference between the radar measurement and the raingauge measurements that typically serve as ground truth (e.g., Ciach and Krajewski, 1999; Kitchen and Blackall, 1992). In this analysis we focus on the first component that consists of the errors at the scale of measurement of radar rainfall. This error is assigned a standard deviation r, which is represented as a non-increasing nonlinear function of pixel radar rainfall, R. The analysis of Seo (1998) indicates that bias corrected radar rainfall becomes substantially more accurate for measured radar rainfall greater than 25.4 mm/h, with an increase of accuracy toward this value from low R values. This unknown relationship between r and R is simplified in this work as indicated in Fig. 3, or mathematically:

Radar-rainfall and mean areal precipitation uncertainty models Precipitation input to the hydrologic models is derived from NEXRAD multisensor (quality-controlled) radar-rainfall estimates generated at the NWS Arkansas-Red Basin River Forecast Center. However, the model input is averaged at the scale of model resolution, i.e., as mean areal precipitation (MAP) estimates for each subcatchment. The formulation of uncertainty for radar rainfall first establishes a pixel scale uncertainty model and then upscales the uncertainty in radar-rainfall estimates to the spatial resolution of the hydrologic models, producing models of uncertainty for mean areal precipitation. Because of the intended use with ensemble simulation procedures for several historical time steps, use of CPU intensive procedures to preserve spatial correlation in the generated field, such as the turning bands

Figure 3 Standard deviation of radar-rainfall pixel error as a function of pixel rainfall value.

Discretization scale dependencies of the ensemble flow range 2r ¼ 1  0:02R R 6 25 mm/h R 2r ¼ 0:5 R > 25 mm/h R

ð4Þ ð5Þ

Given this uncertainty model for pixel-scale radar-rainfall errors it is desired to arrive at a model of radar-rainfall error uncertainty on aggregate scales (i.e., pertaining to mean areal precipitation errors). Georgakakos and Carpenter (2003) derive an uncertainty model for mean areal precipitation on various scales from the pixel error model discussed, assuming that there is no spatial dependence of the errors of neighboring pixels. In this work we generalize their approach by allowing for significant spatial correlation of the pixel radar-rainfall errors. Due to the intended use of the model for uncertainty within a Monte Carlo sampling framework, a computationally parsimonious model is developed in the following. Consider a portion of the domain of radar pixels under the radar umbrella, and define a two-dimensional spatial window centered on pixel (i, j) as shown in Fig. 4. The centroids of the pixels are shown, and the window dimensions are characterized by the number of pixels that are contained within the window in each of the x and y dimensions, counting from the central pixel. There are L pixels to the right and L pixels to the left of pixel (i, j), and there are K pixels above and K pixels below pixel (i, j), all within the specified window of Fig. 4. For each pixel within the window centered at (i, j), define an independent and uniformly distributed random variable uni,nj in the range [0, 1], with mean 0.5 and variance 1/12. We may then define a spatially correlated random variable, ni,j, at (i, j) as follows: ni;j ¼

1 ð2K þ 1Þð2L þ 1Þ

ni¼þK X nj¼þL X

ðauni;nj  bÞ;

uni;nj  Uð0; 1Þ

ni¼K nj¼L

ð6Þ

247 where a and b are constants and U(0, 1) denotes the uniform probability distribution in the interval [0, 1]. Because ni,j is the sum of independent and uniformly distributed random variables (at least 9), it has a Gaussian probability distribution. The constants a and b are determined from the requirements of the second moment statistics imposed on ni,j. The requirement for zero mean ni,j (E{ni,j} = 0) leads to the condition: a ¼ 2b

ð7Þ

In addition, the requirement for a pre-specified variance equal to r2 ðEfn2i;j g ¼ r2 Þ leads to 8" #2 9 < ni¼þK = X X nj¼þL b2 E ð2uni;nj  1Þ ð8Þ ¼ r2 2 2 : ; ð2K þ 1Þ ð2L þ 1Þ ni¼K nj¼L This last relationship for independent uni,nj yields b2 ¼ r2 3ð2K þ 1Þð2L þ 1Þ

ð9Þ

or pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b ¼ r 3ð2K þ 1Þð2L þ 1Þ

ð10Þ

Spatial correlation for neighboring pixel variables (e.g., ni,j and ni,j+s) is due to the overlap of the windows used to define these variables. For example, along the x-axis, the lag-s correlation, qx(s), may be determined to be (following the arrangement of Fig. 4) qx ðsÞ ¼

Efni;j ni;jþs g Efn2i;j g

;

s < 2L þ 1

ð11Þ

which by using Eqs. (8) and (6), and the independence property of the uniform variable uni,nj, yields s ; s < 2L þ 1 ð12Þ qx ðsÞ ¼ 1  2L þ 1 The spatial correlation function along the y-axis in an analogous way may be found to be qy ðsÞ ¼

Figure 4 Radar pixel centroids in a two-dimensional spatial window centered on pixel (i, j).

Efni;j niþs;j g Efn2i;j g

¼1

s ; 2K þ 1

s < 2K þ 1

ð13Þ

The correlation function depends linearly on the spatial lag and it is inversely proportional to the linear size of the window (number of pixels included along a given direction). It is noted that the lag as specified here is an integer, and the correlation function at lag one along a given direction has a minimum non-zero value of 2/3, determined for the smallest possible size of the window (K = 1 or L = 1). In both cases of x and y spatial correlations, for s > 2L + 1 and s > 2K + 1, the value of the spatial correlation function is equal to zero. To allow for lower values of the spatial correlation function at lag one toward the value of 0.5, one may introduce a subdivision of the pixel space so that there are m  1 points between pixels along the x-direction and n  1 points between pixels along the y-direction, where m and n are integers. The methodology of constructing the Gaussian random variable ni,j remains the same, except that in this case there are a total of (2mK + 1)(2nL + 1) points on which an independent uniform random variable uni,nj is defined. In that case,

248 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b ¼ r 3ð2mK þ 1Þð2nL þ 1Þ

T.M. Carpenter, K.P. Georgakakos ð14Þ

and the pixel-to-pixel spatial correlation along x and y is ns ; ns < 2nL þ 1 2nL þ 1 ms ; ms < 2mK þ 1 qy ðmsÞ ¼ 1  2mK þ 1 qx ðnsÞ ¼ 1 

ð15Þ ð16Þ

For K and L equal to 1 and as m andn increase, the minimum correlation for adjacent pixels (s = 1) remains higher than 0.5, and it tends to the value of 0.5 for n and m becoming infinitely large. An example of a generated Gaussian n-field is shown in Fig. 5 for a mean zero, standard deviation 0.5, and a linear spatial correlation function that becomes zero at 5 pixel lengths (i.e., K = L = 2). The field covers a 100-pixel · 100pixel domain. The Gaussian character of the generated field may be seen in Fig. 6 showing the sample histogram constructed from the field of Fig. 5 together with a comparable sample histogram of generated Gaussian variables, and with the theoretical histogram of the Gaussian distribution at the same discretization levels of the n-axis of the histogram. Good agreement is shown. The Figure also shows the sample mean and standard deviation of the generated values that are in good agreement with the theoretical ones. Fig. 7 shows the sample linear correlation function (assumed the same for both x- and y-directions) of the generated field. For reference Fig. 7 also shows an exponential function with a spatial scale of 3 pixel lengths. It is apparent that for short (long) spatial lags the generated field shows higher (lower) correlations than an exponentially correlated field. The correlation is practically zero at a lag of 5 pixel lengths or longer. For substantial correlation of radar pixel errors (down to a lag-one spatial correlation of 0.5), the previous formulation may be used to form a parsimonious computational model for generating spatially correlated Gaussian random

variables with mean zero, with a given variance and with a linear spatial correlation function. Using this model and the pixel standard deviation from Eq. (5), one may develop a stochastic model for generating radar-rainfall pixel values, R0i;j , that contain errors with the prescribed characteristics: R0i;j ðtÞ ¼ Ri;j ðtÞð1 þ ni;j ðtÞÞ

ð17Þ

where Ri,j denotes observed pixel rainfall, and the standard deviation r embedded in the definition of the random error ni,j is defined from Eq. (5) for the (i, j)th pixel. To establish the relationships for uncertainty at the subcatchment or catchment scales, Monte Carlo sampling from the above spatially-correlated radar pixel error model was performed over each hour with precipitation in the historical record. At each hour, the spatially-correlated pixel error was sampled, applied to the observed radar rainfall, and then averaged to obtain mean areal precipitation (MAP) estimates for each subcatchment of the study basins. A total of 30 MAP estimates were produced at each time step for each subcatchment. From these samples, the standard deviation of MAP estimates were computed and used to develop a relationship between the uncertainty in the MAP estimates (as described by their standard deviation) and the magnitude of MAP for various subcatchment sizes in each catchment. Fig. 8 illustrates these relationships at the intermediate discretization scale for the two study catchments. For legibility, the plot presents a few selected subcatchment sizes rather than the set of all possible subcatchment sizes. Linear relationships are implied on the subplots for various subcatchment sizes (represented by the number of pixels, Np, contained within the subcatchment) and for subcatchment average precipitation less than 25 mm/h. For subcatchment average precipitation greater than 25 mm/h, a constant rainfall variability measure (2r/ Ravg) appears appropriate. This structure for subcatchment precipitation uncertainty follows that assumed at the radar pixel scale as given in Eqs. (4) and (5). Note that in the Figure panels, the lowest set of points shows the relationship for the entire catchment.

Validation of ensemble flow simulations

Figure 5 Sample error field ni,j for a domain of 100 by 100 radar pixels. The gray scale bar indicates the values of the errors generated. The intended statistics of the generated field are mean zero, standard deviation 0.5 and linear spatial correlation that becomes zero at a length of 5 pixel lengths.

With the uncertainty models defined, ensemble streamflow simulations were generated within a Monte Carlo framework for selected historical events and for each discretization scale. Three uncertainty scenarios were considered: (a) soil model parameter uncertainty only; (b) precipitation input uncertainty, based on the spatially-correlated radar pixel uncertainty model; and (c) combined uncertainty in soil model parameters, routing model parameters, and precipitation input. For each uncertainty scenario, discretization scale, and event, an ensemble of streamflow simulations was produced which included 100 ensemble members. The events selected were based on the observance of a distinct peak in the discharge record at the catchment outlet. Tables 3 and 4 present the dates and peak flow characteristics of the 25 events selected for the Blue River basin and 28 events selected for the Illinois River basin. For the ensemble generation, the events were simulated with uncertainty incorporated starting approximately two months prior to

Discretization scale dependencies of the ensemble flow range

249

Figure 6 Histograms of error distributions of the generated field shown in Fig. 7, of a Gaussian field with the same number of values, and of the theoretical Gaussian distribution discretized as indicated on the n-axis on the plot.

Figure 7 Linear spatial correlation function of the error field shown in the figure. An exponential correlation function with scale length of 3 pixels is also shown for reference.

the event, so that a stable distribution of soil moisture conditions was established at the beginning of the event. This methodology and the selected events were used in Carpenter and Georgakakos (2004b). As a validation of the ensemble simulations, a probabilistic analysis of the ensemble simulations is presented for the case of both parametric and radar-rainfall (i.e., combined)

uncertainty, and for the different discretization scales. This analysis examines the capability of the ensemble simulations to reliably reproduce selected ‘‘target events’’ with good resolution and sharpness. Reliability diagrams (Fig. 9) are used to summarize the analysis as they compactly display the full frequency distribution of the ensemble simulations and are thus more informative then single, scalar

250

Figure 8

T.M. Carpenter, K.P. Georgakakos

Illustration of the subcatchment precipitation scaling relationship for the (a) Blue River and (b) Illinois River catchments.

performance measures. The theoretical basis and development of the reliability diagram is discussed by Wilks (1995) and an example of the application in water resources management is given in Carpenter and Georgakakos (2001). In this study and to be able to combine event flows, we consider residual event flows (flow at given time step minus

the event mean flow). The ‘‘target events’’ are defined as (a) residual event flow in the upper tercile of the observed residual flow distribution, ðqr > qUro Þ and (b) residual event flow in the lower tercile of the observed residual flow distribution, ðqr < qLro Þ. Fig. 9 gives the reliability diagrams for all three discretization scales together for flows at the outlet

Discretization scale dependencies of the ensemble flow range Table 3 Summary of events selected for the Blue River catchment with statistics of simulated flows at Blue, OK #

Dates

# Days

Qpeak (cms)

Qmean (cms)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

2/27–3/07/1994 4/25–5/10/1994 10/19–31/1994 11/12–19/1994 3/11–20/1995 4/02–09/1995 5/06–13/1995 11/05–12/1996 11/15–23/1996 11/22–12/5/1996 2/18–25/1997 3/11–18/1997 3/23–29/1997 4/02–10/1997 4/09–17/1997 5/07–14/1997 6/08–16/1997 12/18–29/1997 1/03–14/1998 1/20–31/1998 3/05–14/1998 3/14–27/1998 12/02–10/1998 4/01–09/1999 5/08–16/1999

9 16 13 8 10 8 8 8 9 14 8 8 7 9 9 8 9 12 12 12 10 14 9 9 9

130 249 29 320 130 69 468 550 60 294 202 31 45 122 82 32 95 96 166 90 110 301 36 120 146

36 60 8 41 29 19 115 68 18 79 68 12 13 29 28 12 20 30 68 33 29 73 11 22 31

of (a) the Blue River catchment and (b) the Illinois River catchment. The reliability diagrams each present two forms of information. The bar plots show the unconditional ensemble frequency distributions. These plots describe the frequency of use of different frequency of occurrence ranges by the ensemble simulations, and allow for the assessment of ensemble simulation sharpness. Concentrated probability mass near 0 and 1, as in Fig. 9 for all discretization scales, suggest the ensemble simulations exhibit high sharpness. This implies that the ensemble simulations behave nearly as deterministic simulations. The scatter plots of Fig. 9 present the conditional distribution and reliability. Reliability is an indicator of whether the simulated frequencies of the target event correctly estimate the observed frequency of occurrence. Thus, these plots compare the simulated frequency to the sample observed frequency of occurrence of the target event given the simulation frequency within a specified range. In this analysis, the simulations frequencies are divided into deciles. The 1:1 diagonal line signifies the case of ‘‘perfect reliability’’ and the error bars represent the 95% sample confidence limits (see Carpenter and Georgakakos, 2001). Frequency values outside the confidence limits are deemed unreliable. These plots also allow for evaluation of the ensemble simulation frequency resolution. Resolution addresses the ability of the ensembles to discern differences in the conditional frequency for different simulations frequency ranges of the target event. Hence, if the conditional

251 Table 4 Summary of events selected for the Illinois River catchment with statistics of simulated flows at Tahlequah, OK #

Dates

# Days

Qpeak (cms)

Qmean (cms)

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

11/12–20/1993 3/06–18/1994 3/25–4/1/1994 4/10–18/1994 4/28–5/7/1994 11/03–13/1994 12/07–14/1994 1/12–18/1995 5/06–13/1995 6/08–16/1995 9/24–10/1/1996 11/05–12/1996 11/15–23/1996 11/22–12/5/1996 2/19–25/1997 3/08–21/1997 4/07–16/1997 6/15–24/1997 1/03–14/1998 3/05–15/1998 3/14–27/1998 10/3–12/1998 2/05–12/1999 3/10–19/1999 4/02–13/1999 5/02–10/1999 5/09–17/1999 12/19/97–1/02/98

9 13 8 9 10 11 8 7 8 9 8 8 9 14 7 14 10 10 12 11 14 10 8 10 12 9 9 15

194 335 148 178 129 194 123 484 497 543 591 620 164 578 447 178 120 146 1130 170 297 347 276 390 166 320 187 163

74 98 55 92 53 57 42 128 192 170 90 93 76 155 133 75 66 51 240 65 127 80 95 158 67 119 95 58

frequency maintains constant or similar values over a range of simulated frequencies, low resolution is exhibited. The results show highest reliability and resolution for the calibration (medium) scale of discretization for both catchments as expected. Blue River results exhibit less variability in reliability between discretization scales, with residual flows in the lower tercile being more reliable than those in the high tercile. For both catchments and both terciles (except perhaps for the upper tercile for the Illinois River) reliability results for the near zero and near one frequency ranges show good reliability and low sensitivity to discretization scale. As noted earlier, these frequency ranges contain the largest number of simulations. Large deviations from the 95% error bars for the upper tercile case are shown for the high resolution model in low to intermediate frequency ranges. The low-resolution model exhibits deviations from the 95% error bars in the mid to high frequency range for the upper tercile case on the Illinois River. These cases represent a small number of simulations and it is concluded that the model ensemble simulations exhibit comparable reliability for different discretization scales and for both catchments. The upper tercile target-flow event in the Illinois River catchment may present an exception for the near-one frequency range ensembles, which tend to

252

T.M. Carpenter, K.P. Georgakakos

Figure 9 Reliability diagrams for ensemble residual flows at the outlet in the upper (right column) and lower (left column) tercile of the observed event distribution.

Discretization scale dependencies of the ensemble flow range overestimate the actual frequency especially for high resolution discretization scale.

Results To facilitate the development of sensitivity measures of ensemble simulations and following previous work (Carpenter and Georgakakos, 2004b), a normalized measure of the range in the flow ensemble was computed for each subcatchment. At each time step, the difference between the 90th and 10th percentile ensemble flow values normalized by the median ensemble flow value was computed: RQ ðtÞ ¼

Q 90 ðtÞ  Q 10 ðtÞ Q 50 ðtÞ

ð18Þ

The value of RQ ðtÞ at the time of maximum range (i.e., maximum difference between Q90 and Q10) was reported for

253 each subcatchment and each event and is denoted simply by RQ. Further, the average value and standard deviation of RQ were computed over all events for each subcatchment. The results of the ensemble simulation analysis are presented in terms of these statistics of RQ. The average and standard deviation of the range measure, RQ, are plotted as functions of subcatchment cumulative drainage area in Fig. 10. This figure presents the case of precipitation input uncertainty on the Blue River basin with high model discretization scale, and the case of combined uncertainty with low model discretization scale for the Illinois River basin. This combination of uncertainty scenario and model discretization scale was selected to illustrate that there consistently was found a well-defined relationship between the statistics of the ensemble flow range measure and subcatchment area. A decreasing log-linear trend is found in the average and the standard deviation of the ensemble flow range measure as the subcatchment drainage

Figure 10 Relationship between subcatchment drainage area and (a) average sensitivity measure (upper plots) and (b) standard deviation of the sensitivity measure (lower plots) for the Blue River watershed (left hand column) and Illinois River watershed (right hand column). The case of precipitation uncertainty for high resolution discretization is shown for the Blue River, while the low resolution discretization and combined uncertainty case is shown for the Illinois River.

254

T.M. Carpenter, K.P. Georgakakos

area increases. A line of best linear fit is included on the plots as well. For the average RQ, regression correlation coefficients (R) of 0.98 and 0.84 are computed for the cases presented on the Illinois and Blue Rivers, respectively. The relationship between drainage area and the standard deviation of RQ is not as well defined, as shown in the lower plots of Fig. 10, but there is a tendency of the variability about the average RQ to scale with the logarithm of the area. The regression correlation coefficients for the line of best fit are 0.55 and 0.90 for the standard deviation of RQ and for the presented cases on the Blue and Illinois Rivers, respectively. For both study catchments, similar relationships between the statistics of RQ and drainage area are found for all uncertainty cases. The parameters of the linear regression equation and the regression correlation coefficients for all uncertainty cases and discretization scales are listed in Table 5. For the Illinois River catchment, the regression correlation coefficients for the average RQ relationship are high (>0.94) for all cases. The correlation coefficients are also high (>0.85) for the standard deviation of RQ relationships, with the exception of the cases of soil model parameter uncertainty only. For the Blue River catchment, the regression correlation coefficients for the average RQ relationship are >0.78 for all uncertainty and discretization cases. There is a weaker relationship between standard deviation of RQ and drainage area for the Blue River, with relatively low correlation coefficients (<0.6). There is evidence of heteroscedasticity in the standard deviation of RQ with greater variation for small drainage areas (see Fig. 10). The finding of scale-dependent ensemble flow range due to input and parametric uncertainty is consistent across all uncertainty cases, discretization scales, and for both study

catchments. This supports the results reported in Carpenter and Georgakakos (2004b) and Carpenter et al. (2001) corresponding to only a few subcatchments in the study catchments and a single discretization scale. Such well-defined relationships suggest that the uncertainty in flow simulations at a given subcatchment may be estimated if the uncertainty is quantified at another location (e.g., where streamflow observations are available) and these relationships have been estimated for the catchment of interest. Fig. 11 inter compares the dependence of the average RQ on area for the high and the medium model-discretization resolution cases and for the Blue (a) and Illinois (b) River catchments. The case of combined uncertainty is presented for the Blue River, while the case of soil parameter uncertainty only is shown for the Illinois River. This figure highlights several findings and differences among the study catchments and uncertainty scenarios. First, the figure shows the difference in the range in subcatchment drainage area and number of subcatchments in the two study basins. It also underscores the lower values of average RQ for the case of soil model parameter uncertainty. Finally, and significantly, the figure illustrates a consistent finding: the average ensemble flow range RQ for both combined (radarrainfall and parameter uncertainty) and parametric uncertainty is lowest for the high discretization level for catchments with area that is substantially greater than the average model discretization area. That is, higher discretization produces a smaller ensemble flow range for such catchments. For the case of combined uncertainty (Fig. 11a) where radar-rainfall uncertainty is dominant, this suggests that the smoothing effect of the soil water accounting on smaller scales in conjunction with higher resolution channel flow routing on the input uncertainty is greater than the ef-

Table 5 Best fit regression equations and regression correlation coefficients (R) of the relationship between drainage area and the statistics of RQ Case Low resolution Combined Precipitation Soil model parameter Medium resolution Combined Precipitation Soil model parameter High resolution Combined Precipitation Soil model parameter

Blue River

Illinois River

Avg RQ = 1.67  0.14 * ln(A) R = 0.78 SD(RQ) = 0.75  0.08 * ln(A) R = 0.56 Avg RQ = 1.52  0.16 * ln(A) R = 0.85 SD(RQ) = 0.75  0.09 * ln(A) R = 0.65 Avg RQ = 0.86  0.06 * ln(A) R = 0.84 SD(RQ) = 0.28  0.03 * ln(A) R = 0.42

Avg RQ = 1.63  0.15 * ln(A) R = 0.98 SD(RQ) = 0.89  0.10 * ln(A) R = 0.91 Avg RQ = 1.39  0.14 * ln(A) R = 0.98 SD(RQ) = 0.77  0.08 * ln(A) R = 0.91 Avg RQ = 0.94  0.08 * ln(A) R = 0.94 SD(RQ) = 0.35  0.03 * ln(A) R = 0.73

Avg RQ = 1.71  0.17 * ln(A) R = 0.90 SD(RQ) = 0.64  0.06 * ln(A) R = 0.54 Avg RQ = 1.40  0.15 * ln(A) R = 0.87 SD(RQ) = 0.61  0.06 * ln(A) R = 0.68 Avg RQ = 0.87  0.08 * ln(A) R = 0.90 SD(RQ) = 0.21  0.01 * ln(A) R = 0.33

Avg RQ = 1.71  0.18 * ln(A) R = 0.97 SD(RQ) = 0.80  0.08 * ln(A) R = 0.86 Avg RQ = 1.44  0.16 * ln(A) R = 0.98 SD(RQ) = 0.75  0.08 * ln(A) R = 0.91 Avg RQ = 0.98  0.10 * ln(A) R = 0.94 SD(RQ) = 0.30  0.02 * ln(A) R = 0.52

Avg RQ = 1.56  0.16 * ln(A) R = 0.86 SD(RQ) = 0.53  0.05 * ln(A) R = 0.42 Avg RQ = 1.25  0.13 * ln(A) R = 0.84 SD(RQ) = 0.50  0.05 * ln(A) R = 0.55 Avg RQ = 0.88  0.09 * ln(A) R = 0.91 SD(RQ) = 0.19  0.01 * ln(A) R = 0.45

Avg RQ = 1.69  0.19 * ln(A) R = 0.95 SD(RQ) = 0.79  0.08 * ln(A) R = 0.86 Avg RQ = 1.37  0.16 * ln(A) R = 0.96 SD(RQ) = 0.65  0.07 * ln(A) R = 0.87 Avg RQ = 1.02  0.12 * ln(A) R = 0.94 SD(RQ) = 0.32  0.03 * ln(A) R = 0.65

Discretization scale dependencies of the ensemble flow range

Figure 11 Comparison of the dependence of average RQ on area for the high and medium resolution discretization scales for (a) the Blue River and for the combined uncertainty scenario and (b) the Illinois River catchment and for the case of soil model parameter uncertainty.

fect of spatial averaging of the radar pixel uncertainty and soil water accounting on coarser discretization scales; and this is true when one compares the uncertainty of catchments with area substantially greater than the discretization scale. Even when only parametric uncertainty is considered, Fig. 11b indicates that for large enough catchments higher discretization restricts the range of the flow ensemble members primarily due to the effect of higher resolution channel routing. Fig. 12 shows the range regression relationships for the different uncertainty scenarios at a single discretization scale (the highest) for both study catchments. This further emphasizes the result indicated above, that the average range measure is smallest for the case of soil model parameter uncertainty and largest for the case of combined uncertainty. This confirms the findings of Carpenter and Georgakakos (2004b). Also indicated is steeper slope in the regression relationships for cases involving precipitation uncertainty. This finding suggests that there is greater im-

255

Figure 12 Comparison of the average RQ relationship for various uncertainty scenarios. The high resolution discretization scale is presented for the (a) Blue River basin and (b) Illinois River catchments.

pact in terms of uncertainty in flow simulations at smaller subcatchments for the case of precipitation uncertainty as compared to the case of model parameter uncertainty. Lastly, the Illinois River catchment shows steeper relationships than the Blue River catchment. That is, higher small scale variability is constrained more effectively for larger scales in the Illinois River, which is characterized by a broader channel network (see Fig. 2).

Conclusions and recommendations This paper presents findings from an extensive Monte Carlo numerical simulation experiment undertaken to explore the sensitivity of ensemble flow simulations from a distributed hydrologic model to input and parameter uncertainty at varying scales of catchment delineation resolution. The overall goal of this research has been the identification of the impact of parametric and precipitation uncertainty on streamflow simulations in an operational flow forecasting

256 environment. The methodology was applied to two study catchments in the south-central United States using operational quality data and model components. The catchments are typical of catchment areas for which operational forecasts are issued by the US National Weather Service. For each study catchment, ensemble streamflow simulations were produced for three scales of catchment discretization, representing low-, medium-, and high-resolution distributed model applications, and for various scenarios of input and parametric uncertainty. The generated streamflow ensembles for selected events were summarized in terms of the maximum range in the flow ensembles (90th minus the 10th percentile flows), normalized by the median ensemble flow of the event for each subcatchment. One of the innovative aspects of this work is the development of a pixel-to-catchment error model for radar-rainfall errors that are assumed correlated in space. A spatially correlated radar-pixel error field was generated as a moving average of uniform variates and it is assigned point variance that is dependent on the point observation of radar rainfall. A linear decrease of the coefficient of variation of pixel error with pixel rainfall is assumed up to a maximum rainfall threshold beyond which the coefficient of variation remains constant. Pixel radar errors are assumed free of systematic bias in this analysis. Through Monte Carlo sampling and spatial averaging, ensembles of mean areal-rainfall error estimates were generated for each subcatchment of the model domain, and were used to determine the influence of radar-rainfall errors on model-flow simulation uncertainty. For correlation coefficients that are expected in radar pixel errors, the procedure followed yields catchment mean areal precipitation errors that have coefficients of variation dependent on catchment mean rainfall in a manner similar to the pixel scale dependence except for different relationship parameters. Larger catchments yield lower error coefficients of variation. The study consistently finds well-defined scaling relationships between average ensemble flow range and subcatchment drainage area for all discretization scales and for all uncertainty scenarios. The relationships are quantified in terms of a linear fit between average ensemble flow range and the logarithm of drainage area, with regression correlation coefficients exceeding 78% for the Blue River and exceeding 94% for the Illinois River. These results strengthen the findings of Carpenter and Georgakakos (2004b) by including all subcatchments of the study catchments rather than a few selected subcatchment locations and allowing for three different model discretizations. The results suggest that if a distributed model is calibrated with observations at a given scale and with a resultant estimate of the average uncertainty in the simulated flow, these measures of simulation uncertainty at higher resolutions may be obtained with reasonable accuracy. This may be particularly valuable in providing distributed model flow estimates at small ungauged subcatchments within a given calibrated catchment. Comparison of the scaling relationships for ensemble flow range at different scales of catchment discretization indicates that for subcatchments greater than the average discretization area, higher resolution in catchment discretization yields lower average flow range. This reduction in flow sensitivity varies with the input uncertainty definition

T.M. Carpenter, K.P. Georgakakos (i.e., parameter, precipitation or combined), and generally with subcatchment size. Such subcatchment-size dependent decrease in flow sensitivity may be related to the relative importance of different model components at different scales (i.e., runoff generation versus streamflow routing). These issues should be explored further to examine (a) whether there is a limit in the reduction in flow sensitivity with higher resolution catchment discretization scales, and (b) the subcatchment scales at which different model components dominate the ensemble flow sensitivity. The study catchments are located in a similar geographic and hydroclimatic region. Similar magnitudes of average ensemble flow range are found for the subcatchment sizes included in this study for these two catchments. One may question whether these results hold for other catchments in different regions with different hydroclimatic forcing (but with available radar data), leading to a sort of ‘‘similarity condition’’ for ensemble flow simulations from distributed models. Thus, a natural extension of this work is the application of this methodology to additional catchments with different hydroclimatic forcing. In addition, analysis for mountainous and other regions where snowfall and snowmelt are significant sources of streamflow should be undertaken. Further study would also be necessary to explore any relationship between the uncertainty models implemented and the parameters of the regression relationships between the statistics of the ensemble flow range measure and catchment drainage area. Finally, this analysis involved characterization of the sensitivity of streamflow simulations from a distributed model to input and parametric uncertainty. Another important extension of this work is to examine the sensitivity of streamflow forecasts to the same input and parametric uncertainty definitions. Uncertainty in precipitation forecasts is expected to be the main determinant of ensemble flow range in such an undertaking.

References Beven, K., 2002. Towards an alternative blueprint for a physically based digitally simulated hydrologic response modelling system. Hydrological Processes 16, 189–206. Burnash, R.J., Ferral, R.L., McGuire, R.A., 1973. A generalized streamflow simulation system: conceptual modeling for digital computers. Technical Report, Joint Federal-State River Forecast Center, US National Weather Service and California Department of Water Resources, Sacramento, CA, 204 pp. Carpenter, T.M., Georgakakos, K.P., 2001. Assessment of Folsom lake response to historical and potential future climate scenarios: 1. Forecasting. Journal of Hydrology 249, 148–175. Carpenter, T.M., Georgakakos, K.P., 2004a. Continuous streamflow simulation with the HRCDHM distributed hydrologic model. Journal of Hydrology 298 (1–4), 61–79. Carpenter, T.M., Georgakakos, K.P., 2004b. Impacts of parametric and radar rainfall uncertainty on the ensemble streamflow simulations of a distributed hydrologic model. Journal of Hydrology 298 (1–4), 202–221. Carpenter, T.M., Georgakakos, K.P., Sperfslage, J.A., 2001. On the parametric and NEXRAD-radar sensitivities of a distributed hydrologic model suitable for operational use. Journal of Hydrology 254, 169–193. Ciach, G.J., Krajewski, W.F., 1999. On the estimation of radar rainfall error variance. Advances in Water Resources 22 (6), 585–595.

Discretization scale dependencies of the ensemble flow range Georgakakos, K.P., Carpenter, T.M., 2003. A methodology for assessing the utility of distributed model forecast applications in an operational environment. In: Tachikawa, Y., Vieux, B.E., Georgakakos, K.P., Nakakita, E. (Eds.), Weather Radar Information and Distributed Hydrological Modeling, IAHS Publ. No. 282. IAHS Press, Wallingford, UK, pp. 85–92. Georgakakos, K.P., Krzysztofowicz, R. (Eds.), 2001. Probabilistic and ensemble forecasting. Journal of Hydrology 249 (1–4), 196. Gurnell, A.M., Montgomery, D.R. (Eds.), 2000. Hydrological Applications of GIS. John Wiley, New York, p. 184. Kitchen, M., Blackall, R.M., 1992. Representativeness errors in comparisons between radar and gauge measurements of rainfall. Journal of Hydrology 134, 13–33. Koren, V.I., Finnerty, B.D., Schaake, J.C., Smith, M.B., Seo, D.-J., Duan, Q.-Y., 1999. Scale dependencies of hydrologic models to spatial variability of precipitation. Journal of Hydrology 217, 285–302. Koren, V., Reed, S., Smith, M., Zhang, Z., Seo, D.-J., 2004. Hydrology Laboratory Research Modeling System (HL-RMS) of the US National Weather Service. Journal of Hydrology 291 (3–4), 297–318. Mantoglou, A., Wilson, J.L., 1982. The turning bands method for simulation of random-fields using line generation by a spectral method. Water Resources Research 18 (5), 1379–1394. NRCS (National Resource Conservation Service), 1994. State Soil Geographic (STATSGO) Database. Misc. Publ. 1492, US Department of Agriculture, Fort Worth, TX. 37pp + Appendices.

257 Owe, M., Brubaker, K., Ritchie, J. (Eds.), 2001. Remote Sensing and Hydrology 2000. IAHS Publication No. 267. IAHS Press, Centre for Ecology and Hydrology, Wallingford, UK, 610 pp. Reed, S., Koren, V., Smith, M., Zhang, Z., Moreda, F., Seo, D.-J. DMIP Participants, 2004. Overall distributed model intercomparison project results. Journal of Hydrology 298 (1–4), 27–60. Refsgaard, J.S., Knudsen, J., 1996. Operational validation and intercomparison of different types of hydrological models. Water Resources Research 32 (7), 2189–2202. Seo, D.-J., 1998. Real-time estimation of rainfall fields using radar rainfall and raion gage data. Journal of Hydrology 208, 37–52. Seo, D.-J., Breidenbach, J., Fulton, R., Miller, D., O’Bannon, T., 2000. Real-time adjustment of range-dependent biases in WSR88D rainfall estimates due to non-uniform vertical profile of reflectivity. Journal of Hydrometeorology 1 (3), 222–240. Singh, V.P., Frevert, D.K., 2002. Mathematical Models of Small Watershed Hydrology and Applications. Water Resources Publications, LLC, Highlands Ranch, CO, 972 pp. Smith, J.A., Seo, D.J., Baeck, M.L., Hudlow, M.D., 1996. An intercomparison study of NEXRAD precipitation estimates. Water Resources Research 32 (7), 2035–2045. Smith, M.B., Seo, D.-J., Koren, V.I., Reed, S.M., Zhang, Z., Duan, Q., Moreda, F., Cong, S., 2004. The distributed model intercomparison project (DMIP): motivation and experiment design. Journal of Hydrology 298 (1–4), 4–26. Wilks, D.S., 1995. Statistical Methods in the Atmospheric Sciences. Academic Press, New York, 467 pp.