## Introduction

The basic input for hydrological modeling is the precipitation field. Accurate rainfall estimates are fundamentally important to the success of hydrologic simulation (Wyss et al. 1990; Michaud and Sorooshian 1994) regardless of the numerical complexity of the water balance formulation. The accuracy of hydrologic simulations always has been limited by this basic input, though not exclusively so (Duan et al. 1992; Farajalla and Vieux 1994, 1995; Hoos et al. 1989). A hydrometeorologic forecast system (HFS, see Fig. 1) has been developed to take advantage of new technology and new scientific methods. This manuscript focuses primarily on the precipitation analysis and its impact on the hydrology. The remaining components of this HFS are in Part II of this manuscript series (Pereira Fo. et al. 1999).

High-resolution rainfall fields can be obtained from Weather Surveillance Radar 1988-Doppler (WSR-88D) on Next-Generation Weather Radar (NEXRAD) weather radar systems (Crum and Alberty 1993) and surface mesonetworks such as the Oklahoma Mesonet (Brock et al. 1995). Even so, many sources of errors have been identified in estimating precipitation from volumetric measurements of radar reflectivity (Austin 1987), and controversy still surrounds the *Z–R* calibration (Seed et al. 1995; Atlas et al. 1995). On the other hand, surface measurements lack spatial representativeness, while the exposure of rain gauges can contribute to errors (Groisman and Legates 1994; Legates and DeLiberty 1993). With these new data sources, it is possible to combine the measurements to produce an analysis field containing less observational error than in each measurement set alone. Several procedures have been developed to integrate both radar and rain gauge measurements (e.g., Brandes 1975; Wilson and Brandes 1979; Crawford 1979; Krajewski 1987; Messaoud and Pointin 1990; Bhargava and Danard 1994).

The statistical objective analysis (SOA) scheme (Gandin 1963), which incorporates the statistical properties of a given variable to reduce analysis and observational errors, has been applied to storm total rainfall and rainfall over long time intervals due to the lack of information at high spatial and temporal resolutions (Creutin and Obled 1982). Crawford (1979) used the technique to design a hydrologic data network. While his focus was on the design of a surface gauge network, he combined rain gauge and radar measurements using statistical objective analysis.

A modernized SOA scheme is formulated and developed in section 2. The Dry Creek watershed (section 3) is used to evaluate the hydrologic impact of using both unadjusted and adjusted radar estimations of rainfall (section 4). Conclusions are presented in section 5.

## Precipitation analysis

The critical element for hydrologic simulations or forecasts is the precipitation field at past, present, and future time intervals. Consequently, it is appropriate to begin the development of a hydrometeorologic forecast system with an evaluation of the precipitation field.

### Data

The WSR-88D at Twin Lakes (TLX) near Oklahoma City (Crum and Alberty 1993) measured storm reflectivities for all events in this research. Details of available products from the WSR-88D can be found in Klazura and Imy (1993). Raw level II data (in a polar coordinate, azimuth/range format) were used to produce Constant Altitude Plan Position Indicator (CAPPI) maps of rainfall rates at 2 km AGL. A map of Oklahoma, its counties, the locations of the 111 meteorological stations in the Oklahoma Mesonet, and the area of coverage by the Twin Lakes WSR-88D are shown in Fig. 2. Mesonet stations within an enlarged analysis square of 440 km × 440 km also are shown. This fully automated network provides real-time data at 5-min intervals. A technical overview of the Mesonet is given by Brock et al. (1995).

#### Twin Lakes WSR-88D

*Z–R*relationship of

*Z*

*R*

^{1.4}

*Z*is the radar reflectivity (mm

^{6}m

^{−3}) and

*R*is the rainfall rate (mm h

^{−1}).

Reflectivities from the WSR-88D (spatial resolution of 1 km × 0.95°) were converted into rainfall rates. Mean rainfall rates for the entire radar umbrella were obtained by averaging the rates from all radar bins (including zero precipitation) within individual 2 km × 2 km pixels. The flat topography of Oklahoma allows 2-km CAPPIs to be devoid of most ground clutter close to the radar site. In most cases, a 2-km CAPPI remains below the bright band associated with the melting layer. Events 1–14 were used to develop population covariances required by the statistical objective analysis scheme, while events 15–18 were used as independent datasets for testing the scheme.

#### Mesonet observations

Precipitation datasets from the Mesonet were accumulated over the corresponding 15-, 30-, 60-, and 120-min periods for events 4 and 15–18 of Table 1. The network was fully operational during this period. Each tip of the tipping-bucket rain gauge in the Mesonet measures 0.254 mm (0.01 in.). Mesonet stations are separated by an average distance of 30.7 km, while the average distance from the geometric center of each radar pixel to the nearest Mesonet station is 23.9 km.

Although point measurements of rainfall suffer from errors of spatial representativeness and exposure (Duchon et al. 1995), these errors are greatly minimized in the Mesonet due to the density of stations and strict guidelines used for exposure of the equipment. Furthermore, Morrissey et al. (1995) classified the Oklahoma Mesonet as a quasi-uniform network, a characteristic that greatly reduces the standard error of simple spatial averages. Thus, Mesonet rain gauge measurements are assumed to have a smaller error component than occurs in more common, low-density rain gauge networks.

### The SOA scheme

*P*_{a}analyzed precipitation,*P*_{r}radar estimation of the precipitation (background),*P*_{g}rain gauge precipitation measurement (observation),*r*_{o}location of the radar pixel,*r*_{n}location of the rain gauge,*N*number of rain gauges, and*W*_{n}a posteriori weight.

*ρ*

_{nm}is the background error correlation at locations

*n*and

*m*;

*ρ*

_{no}is the background error correlation between locations

*n*and

*o*of the rain gauge and radar pixel, respectively; and

^{2}

_{n}

The normalization is achieved using the background error covariances. A detailed derivation and discussion of the algorithm are given in Daley (1991). The expression above provides the normalized expected analysis error variance for each analysis point (pixel). The two-dimensional structure of this quantity can be estimated if the background error covariance/correlation structure is known.

The first assumption used to derive the a posteriori weights is certainly satisfied. Errors in radar measurement are independent of the rain gauge errors. The second assumption is not completely satisfied. The tipping-bucket gauge underestimates rainfall at all rainfall rates, especially so for light (<10 mm h^{−1}) and heavy (>200 mm h^{−1}) rainfall rates. The effect of wind and wetting losses also contribute to an underestimate of rainfall (Groisman and Legates 1994). Radar errors are not spatially independent, but the errors due to the physical characteristics of the precipitation systems (e.g., *Z–R*) are an important component of the total error. As a result, the bias can be neglected for a long time series.

### Background error variance

*d*

_{ij}is the distance between pixels

*i*and

*j,*

*s*

_{ij}is the estimated covariance between the precipitation estimates at pixels

*i*and

*j,*and

*s*

_{i(j)}is the estimated standard deviation of precipitation at pixel

*i*(

*j*).

^{4}), it produces the best fit possible to satisfy constraints of the SOA scheme, and it provides a final analysis that is more efficiently derived and accurate (Crawford 1979).

## Hydrologic modeling

The accuracy of the analyzed precipitation field can be tested by converting it into runoff in a watershed with at least one available stream gauge. The observed stream flow then is compared with the one simulated by a hydrologic model.

A hydrologic model with adequate hydrodynamics, simple soil moisture accounting, and its availability was selected to test the impact of the objective analysis scheme. The hydrologic model selected is known as Technical Release 20 (TR-20). It was developed by the U.S. Soil Conservation Service (1982, hereafter SCS82) and has been used over the past 30 years for water resource projects to determine the level of protection and prevention of floods required for small watersheds. Michaud and Sorooshian (1994) compared TR-20 simulations against a complex distributed runoff model and concluded that when they performed calibration, the accuracy of the complex distributed model was similar to that of the TR-20, while without calibration, the complex distributed model was more accurate than the TR-20. A lumped version of TR-20 did very poorly. A brief description of TR-20 is presented in the next subsection.

The hydrological analysis was confined to a small watershed in north-central Oklahoma (Dry Creek), whereas the precipitation analysis covered almost all of Oklahoma (Fig. 2). A single basin was selected rather than modeling the whole state to guarantee that at least one basin had an adequate calibration and good hydrological data.

### The TR-20 hydrologic model

TR-20 is a single-event hydrologic model that uses detailed physical characteristics of watersheds to estimate peak discharges in the analysis of water resource projects. The initial abstraction and infiltration are not retrievable during periods of no rainfall, while the time to peak, the rising limb, and peak flow are the main parameters considered by TR-20.

The input data are accumulated precipitation; cross sections and hydraulic structures along the main stream and its tributaries; length of the channels between cross sections and hydraulic structures; drainage area, time of concentration, and infiltration rate of the subwatersheds;and base flow and integration time step of the model. The principal elements of TR-20 are illustrated in Fig. 3. The schematic watershed illustrates a single drainage area upstream from a reservoir, followed by a channel and a downstream cross section. TR-20 estimates the respective runoff, reservoir and channel routing, and generates the resulting hydrographs. Other hydraulic and hydrologic elements can be added easily to TR-20 to form a more complex watershed system. A complete description of TR-20 is given by the Technical Release 20 (SCS82).

### Hydrology and hydraulics of TR-20

Once precipitation starts over a watershed, the resulting water volume is partitioned into various compartments where it is stored and infiltrated into deeper layers of soil. From there, this water can be reevaporated into the atmosphere or flow to the streams.

*Q*runoff (mm);*P*precipitation (mm);*I*_{a}initial water abstraction (mm), which is water totally absorbed by the soil; and*S*potential water retention (mm) or watershed storage.

^{3}s

^{−1}) of runoff is given by (SCS72)where

*t*

_{p}is the time to peak (h),

*A*is the area of the watershed (km

^{2}), and

*Q*

_{r}is the runoff (mm).

*t*

_{p}

*t*

_{r}

*t*

_{l}

*t*

_{l}is the watershed lag time (h) and

*t*

_{r}is the half time of unit excess rainfall duration (h).

The average watershed lag time is 60% of the time of concentration, while the unit hydrograph time increment (2*t*_{r}) is 13.3% of the time of concentration. The parameters and constants of the dimensionless unit hydrograph above were derived from a large number of natural unit hydrographs in watersheds of variable size and geographical locations (SCS72).

Since the unit hydrograph method is very simple, and the only physical parameters available are the drainage area and time of concentration, TR-20 requires large watersheds to be subdivided into uniformly shaped hydrologic units and drainage patterns with homogeneous subwatersheds to be no greater than 50 km^{2} (SCS72). The composite flood hydrograph for a stream section or a reservoir is given by the summation of all unit hydrographs developed for each unit hydrograph time increment. The time increment for the composite flood hydrograph is specified arbitrarily to add flexibility to the calibration of TR-20.

Therefore, time of concentration, drainage area, curve number, and rainfall accumulation for each subwatershed are required to develop the composite hydrographs. The composite hydrographs are then routed through stream sections and reservoirs and combined with other hydrographs along the streams to produce the final hydrograph at the outlet or forecast point.

- Δ
*V*change in volume of water storage (m^{3}) in the watershed, - Δ
*t*time interval (s), - 〈
*I*〉 average rate of inflow (m^{3}s^{−1}) to the watershed, and - 〈
*O*〉 average rate of outflow (m^{3}s^{−1}) to the watershed.

Inflow rates are given by the upstream hydrograph, and the outflow rates are determined by the physical characteristics of the reservoir. It is assumed that there is an instantaneous response in outflow to an inflow.

The reach routing of a hydrograph through a channel is an unsteady hydrodynamic process. As the flood wave propagates through the channel, distortion and attenuation occurs. The phenomenon is analogous to a shallow water wave propagating through a straight channel. The wave amplitude is in general larger than the linear limit approximation, so the nonlinear steeping of the wave, caused by velocity variations within the wave, will eventually produce a hydraulic jump if the wave is not dispersive. If the wave is dispersive, due to variations in the depth of the channel, the nonlinear steeping will be partially compensated by the dispersion and it will not break. The degree of distortion and attenuation of the wave is a function of the amount of dispersion and nonlinear effects.

Natural streams have variable depths, roughness, width, and are not straight. The full dynamic equations would be required to study the true flood-wave propagation, a scenario that is not feasible. An alternative to full physics models is the parameterization of the dispersion and attenuation of floodwaves by much simpler procedures. The propagation downstream of a flood wave in a channel can be described in terms of translation and reservoir effects. The TR-20 model uses a modified attenuation–kinematic procedure; the wave is routed by considering channel storage effects that reduce the amplitude of the wave and conserve the water volume and kinematic translation.

*t*− 1, instead of the average of inflows at time steps

*t*and

*t*− 1. This approximation is valid if the time step is small, so large variations do not occur in the inflow at consecutive time steps. Furthermore, the storage in the channel, a function of its length and cross section, is assumed to be proportional to the outflow. Therefore, the outflow floodwave can be explicitly determined by

*O*

_{t+1}

*C*

_{r}

*I*

_{t}

*C*

_{r}

*O*

_{t}

*O*outflow (m^{3}s^{−1}),*I*inflow (m^{3}s^{−1}),*C*_{r}coefficient for routing, and*t*time step (h).

*K*comes from the assumption that the storage in the channel is proportional to the outflow (

*V*=

*KO*) and is obtained byThe subscripts

*p*and

*b*refer to peak and base flow, respectively. Thus, a cross section and a channel length are required to estimate the coefficient of routing; it is similar to a stability parameter and, for a stable solution, it should be

*C*

_{r}⩽ 1.

The resulting attenuated flood-wave hydrograph is kinematically translated in time. The difference in peak times for the inflow and outflow hydrograph is estimated by the ratio of variation in the channel storage at inflow and outflow peaks and the respective peaks of inflow and outflow. Physically, the temporal variation of maximum storage is equal to the variation of maximum peak discharge between the inflow and outflow hydrographs.

Further discussion and derivations of model parameters can be found in SCS72 and SCS82. The hydrology of the TR-20 model, although simplistic, can nicely reproduce the rising limb of the hydrograph at the outlet. The recession part of the hydrograph is always underestimated due to the lack of feedback from baseflow and interflow contributions. Routing procedures are based on the hydraulic properties of the watershed such as reach lengths, cross sections, and hydraulic structures. TR-20 has been applied successfully in water resource planning over the past 30 years. With new precipitation measuring systems, such as the WSR-88D and Mesonet, it is possible to make short-term hydrologic forecasts for small watersheds already studied by the U.S. Soil Conservation Service. The critical parameters in flood forecasting are the rising limb of the hydrograph, the time of peak, and its magnitude. Due to the formulation of TR-20, the level of the stream at each cross section along the watershed also can be estimated. Therefore, forecasting floods in small watersheds across Oklahoma and the United States represents a potential application of the precipitation analysis and quantitative precipitation forecast (QPF) techniques developed in this manuscript.

### The Dry Creek watershed

The selection of the Dry Creek watershed (Fig. 4) was based on the quality of physical parameters, availability of stream gauge data, and the presence of a natural stream with a minimum of hydraulic structures. The cross sections and the single reservoir are indicated in Fig. 4. Dashed lines delineate subbasin areas for which average precipitation was determined. The stream gauge is located upstream from the outlet used in the model. The drainage area covered by the stream gauge and outlet cross section are equal to 178.7 and 186.4 km^{2}, respectively.

Soil in the Dry Creek watershed is of the Darnell–Stephenville association (U.S. Soil Conservation Service 1970). These soils develop from red, acidic, weathered sandstone in wood uplands and are characterized by a shallow layer of reddish brown silt loam with low to moderate infiltration rates between 20 and 60 mm h^{−1}. A layer of sandstone below occurs at a depth of 0.3–0.9 m, which severely limits water storage in the soil. The soil-cover complex is predominantly of the shallow Savannah range type.

### Hydrologic data

The U.S. Geological Survey measured streamflow at the location indicated in Fig. 4 until July 1994 when the gauge in the Dry Creek watershed was terminated. The observed hydrographs in Fig. 5 accompany the respective WSR-88D and Mesonet datasets used in this research. Water level was measured at 1-h intervals. Rising limbs of the hydrographs increase rapidly due to the low infiltration rates of soils in the watershed.

Rainfall rates greater than 20 mm h^{−1} produce some runoff, so even small precipitation accumulations can produce runoff if the rainfall rates are high at any time during the event. Dry Creek is an ideal watershed to study with TR-20 because it satisfies most of the limitations of the model and has a quick response to precipitation pulses (much like a linear system having a small time constant).

## Results

The methodology developed in section 2 is applied to the datasets obtained during events 1–14 (Table 1) that are equivalent to 38 000 time series of 250 h each. The first step is to estimate the correlation structures as a function of distance for rainfall accumulations of 15, 30, 60, and 120 min. Subsequently, the SOA scheme is tested with an independent dataset (events 15–18, Table 1), and hydrologic simulations are performed.

### Background error correlation function

Spatial correlation coefficients calculated with Eq. (5) were averaged and modeled by a ninth-order polynomial given by Eq. (6). These spatial correlation structures are shown in Fig. 6 for various rainfall-accumulation time intervals. The four curves correspond to the isotropic component of the spatial correlation function averaged over all precipitation systems (i.e., the 14 events).

It is likely that individual precipitation systems have an anisotropic structure (Huff and Shipp 1969). Moreover, Thiebaux (1976) has shown that the anisotropy of the correlation function is a significant source of errors to the interpolation scheme in regions of low data density or when station configurations are irregular. She used a two-dimensional correlation function to improve the accuracy of interpolations when using data from the radiosonde network of North America (Thiebaux 1977). However, the anisotropic nature of the correlation function is neglected in this work for ease of calculation.

The curves of Fig. 6 decrease monotonically with distance, which satisfies the positive definiteness constraint of the SOA methodology (Weber and Talkner 1993). Extrapolation of these curves to the origin yields a correlation coefficient of 0.975, 0.983, 0.998, and 0.998 for 15-, 30-, 60-, and 120-min accumulation intervals, respectively. This result indicates that the quality of the estimated accumulated rainfall increases with increasing accumulation time interval, as expected (i.e., small-scale features representing “meteorological noise” are averaged out in the longer time accumulations).

These 14 events were separated into those of a convective rainfall nature and those of a more stratiform rainfall nature to investigate the frequency distribution of various reflectivity values. The frequency distribution of equivalent reflectivity values derived directly from the raw radar data for the convective and the stratiform systems as well as the probability distribution for both types of events combined are shown in Fig. 7. There were 32 × 10^{6} radar bins in the combined frequency distribution with stratiform systems, representing 56.3% of the data used in this paper.

The distribution of reflectivities is significantly different at the low end of the reflectivity scale. Reflectivities below 20 dB*Z* (0.5 mm h^{−1}) account for 45% and 48% of the convective and stratiform system, respectively, and reveal the great sensitivity of the WSR-88D system in detecting weak atmospheric phenomena. These low-end precipitation reflectivities do not produce any accumulated rainfall of consequence, but they do serve to define the spatial coherence of individual events.

The difference also can be explained in part by the strength of the vertical motion associated with both types of systems. Convective systems have stronger updrafts that increase growth processes of condensation and coalescence for the cloud droplets, while for stratiform systems, cloud droplet growth occurs at a lower rate. Another factor is the available liquid water content per unit area. Convective systems have higher water content per unit area than do stratiform systems.

The cumulative probability distribution of various rainfall amounts during 30-min accumulation periods for stratiform and convective systems (not shown) indicates that rainfall accumulations below 5 mm account for 90% and 98% of precipitation area measurements in convective and stratiform systems, respectively. Consequently, values of rainfall accumulation less than 5 mm have a large effect on the correlation structure of precipitation systems. In addition, distance correlation curves of stratiform and convective systems for 30-min accumulations (not shown) are identical for distances less than 18 km; they have a maximum difference of 0.08 at a spatial lag of 40 km. This result implies that the use of the average correlation curve (averaged over all 14 events) should not introduce large errors in the interpolation scheme.

The cumulative probability distribution of rainfall accumulations over intervals of 15, 30, 60, and 120 min for all cases combined (stratiform and convective) is given in Fig. 8. More than 80% of the total accumulated precipitation (as estimated by radar over a 120-min period) is less than or equal to 5 mm. Aside from the inherent inaccuracies of radar precipitation estimates, these curves of cumulative probability distributions are logarithmic.

Thus, it is clear that the spatial correlation coefficients in Fig. 6 are influenced the most by the very large number of low precipitation values. Fortunately, a stratification of the correlation structure by system type does not appear necessary. Therefore, the curves of Fig. 6 are used in subsequent tests of the analysis algorithm.

### Analysis error variance × number of gauges

Before applying the algorithm, it is necessary to choose the “best” number of radar–rain gauge pairs (*N*-dimensional innovation vector) that will be used for each interpolation point as given by Eq. (2). A major strength of the optimum interpolation methodology is that the normalized expected analysis error variance (NEXERVA) can be determined even before the analysis scheme is used. The spatial representation of NEXERVA for 30-min accumulations that used a three–rain gauge configuration is shown in Fig. 9. This plot of NEXERVA reaches a maximum in areas beyond Mesonet stations, primarily north and south of Oklahoma (Fig. 2). The closer a radar pixel is to a nearby rain gauge, the lower the NEXERVA (meaning a greater confidence in the analysis accuracy). Each rain gauge has a limited influence radius that is predetermined by the statistical behavior of the precipitating system rather than subjectively determined as in most objective analysis schemes.

For this accumulation time interval, the overall NEXERVA could be decreased if additional rain gauges were installed at locations where the analysis error variance is maximized. Thus, the analysis scheme also can be used to determine the optimum configuration of a gauge network given constraints imposed by geographic and economic factors as studied by Crawford (1979).

Individual values of NEXERVA within the analysis area (Fig. 2) were averaged and plotted (Fig. 10) as a function of the number of rain gauges used in the interpolation procedure. This plot reveals that the average NEXERVA decreases with an increase in the precipitation accumulation time interval. For 15- and 30-min accumulations, the average NEXERVA reaches its minimum value with a three–rain gauge configuration.

When accumulations are computed over 60-min windows, a two–rain gauge configuration produces the“best” analysis results. For 120-min accumulations, the analysis error increases asymptotically as more and more rain gauges are used in the analysis of long-time accumulations. This nonapparent situation results from a slow decrease of the correlation coefficient with distance. Each additional rain gauge that is added to the analysis actually reduces the information content of the dataset used to produce the analysis. This situation is a result of a high cross correlation (or a lack of independence) among nearby data values, which increases the uncertainty of the interpolation.

The curves reveal that, in general beyond the three–rain gauge configuration, the average NEXERVA is constant. Therefore, additional gauge information does not improve the quality of the final analysis; it also is computationally more expensive. Hence the three–rain gauge configuration is used in this paper.

The next step is to test the analysis algorithm with an independent dataset (Table 1, events 15–18). A common procedure in objective analysis is to measure the rms error of the analysis against the observations (here rain gauges). The rainfall analysis field is interpolated back to rain gauge locations through bilinear interpolation of the four nearest analysis grid points to the rain gauge location. We subtract the analyzed rainfall at the rain gauge location from the rain gauge measurements to calculate the rms error. We assumed that rain gauge measurements were free of error, so the rms error shows how close the analysis is to the “true” rainfall amount at the rain gauge location. The cross-validation procedure was not used in this work since one can infer from Fig. 9 that if we take any rain gauge from the analysis for verification, the expected analysis error variance at that point will be larger, that is, the rms error will be large. All analyses were performed using rainfall accumulation time intervals of 15, 30, 60, and 120 min.

### Application of the SOA scheme

The time evolution of rms errors (before and after the analysis) is given in Fig. 11; each analysis represents one time step (Δ*t*) through the individual datasets. The precipitation events are plotted in chronologic order and the rainfall accumulation interval is indicated on each plot.

The analysis scheme reduces the rms error for all cases. Large fluctuations of the rms error occurred when 15-min accumulations were used, primarily because large discrepancies existed between radar estimates and rain gauge measurements of precipitation on short time intervals. The amplitude of the fluctuations decreases as the rainfall accumulation time intervals become longer. The effect of longer time integrations is to reduce the sampling error discrepancies. In addition, differences between radar estimates and rain gauge measurements of rainfall become more evident for longer time integrations. These differences could be caused by the use of an improper *Z–R* relationship, calibration of the equipment, rainfall attenuation, or a range effect. Other sources of error are associated with physical characteristics of the precipitating systems. They include the presence of strong updrafts, radar beamfilling, overshooting the bright band, hail contamination, and anomalous propagation. It is believed that not every source of error is present in any given case and that no single source of error is present in every case.

Plots similar to those of Fig. 11 are given (Fig. 12) for the volume of water that each event produced (before and after the analysis). The volume of water measured during these events has a direct impact on the hydrologic response of the basins under the radar umbrella. A striking result is apparent in these plots. The differences between the observed and analyzed volumes of water are small when the event produces minimum areal rainfall. However, as the rainfall event becomes more significant, large differences persist for all time integration intervals. In other words, the value of the analysis algorithm and of the Mesonet rain gauges becomes more significant as the rainfall event itself becomes more significant.

Small differences between the raw radar and the analyzed radar fields during events do not mean that the radar measurements are correct or have small errors; instead, it is possible that short-lived convective cells occurred between the rain gauge locations. In these situations and without gauge information to adjust the radar estimates, the analysis algorithm relies exclusively on the background value given by the radar estimate. Equation (2) shows that if *P*_{g}(*r*_{n}) and *P*_{r}(*r*_{n}) values are zero, then *P*_{a}(*r*_{o}) is equal to *P*_{r}(*r*_{o}), or the analyzed rainfall is equal to the background field. If the gauge accumulation is zero and the radar is greater than zero at the gauge location, the analysis at that location will be close to zero [see Eq. (2)]. Figures 11 and 12 show that, in general, when the gauge accumulation is zero, the radar accumulation is also zero. Radar rainfall estimates and rain gauge measurements differ most during periods of rain. Thus, if the analysis grid point is surrounded by gauge measurements equal to zero and the respective radar rainfall estimates are equal to zero, the rainfall analysis will be equal to the background field at the grid point. That might be a problem to analyze localized storms if the gauge network is not dense enough.

Rainfall volumes accumulated over 120 min were used to investigate the discrepancies more closely. The first and the third events of Fig. 12 were considered representative of storm events that produced relatively small water volumes. Similarly, the fifth and sixth peak water volumes were considered to be representative of events that produced relatively large water volumes. The frequency distribution of rainfall accumulation (before and after the analysis) of the first, third, fifth, and sixth peak water volumes (Fig. 12) are shown in Fig. 13. The spectrum of rainfall accumulation during the minimal water-volume events underwent only minor adjustments during the analysis procedure. In the more significant rainfall events, the spectrum shifted toward higher values of accumulation (following the analysis), indicating the positive impact of the analysis algorithm in adjusting the original radar estimates upward.

An apparent spurious peak in the observed frequency centered at 9 mm on 29 May (Fig. 13) was “smeared” by the objective analysis algorithm into a distribution that seemed more realistic. The frequency distribution of reflectivities in Fig. 7 indicates that the algorithm also successfully returned the precipitation spectrum to one that closely resembled those normally observed.

Precipitation maps ending at 1200 UTC 29 May 1994 (Fig. 14) have been produced using 1) a Mesonet rain gauge-only analysis derived from a two-pass Barnes objective analysis (Koch et al. 1983), 2) a raw-radar estimate, 3) the statistical objective analysis results, and 4) a difference map between the SOA scheme and the raw radar estimates. This has strong implications for the resulting hydrologic forecast. Heavy rainfall occurred over the radar site and attenuated the signal of some targets (Fig. 14d). This situation is not unique. Ryzhkov and Zrnic (1994) used a dual-polarized, S-band radar to document significant attenuation from rainfall through a squall line in Oklahoma. Mesonet rain gauges detected at least four convective cells within the precipitation system (Fig. 14a) that are clearly identified in Fig. 14b. The internal structure of the precipitation system was unaffected by the statistical analysis scheme (Figs. 14b,c), although significant adjustments were made (Fig. 14d). This is another positive characteristic of the SOA scheme when used for situations of organized convection.

The overall impact of the analysis scheme is shown in Figs. 15 and 16 for 15-min accumulations from the independent dataset. A plot of the mean precipitation (determined by including radar pixels where the estimated precipitation exceeded 0.1 mm) estimated by the radar-only and by the objective analysis is given in Fig. 15. Seemingly, it appears that the analyzed mean precipitation field has smaller rainfall totals than those in the observed mean precipitation. This scenario results because of the large number of observed radar pixels (initial rainfall estimates <0.1 mm) whose values have been adjusted upward and now barely exceed 0.1 mm. The dispersion about the ideal fit in the plot increases for higher mean precipitation associated with convective cells. The outliers on the right side of Fig. 15 were related to hail contamination that was at least partially corrected by the analysis scheme. Another cluster of points lies to the left of the ideal fit; these pairs were associated with analyzed mean precipitation that was greater than the observed. This latter group of points was associated with widespread precipitation (such as in Fig. 14) when the range effect and attenuation by intervening rain were important factors. Overall, the analysis scheme tended to perform better during widespread precipitation when the number of rain gauges sampling a large system increased linearly with the size of the system.

The volume of water estimated by the radar is 28% less than the volume produced by the analysis (Fig. 16). In fact, the dispersion in water volumes about the ideal fit increased as a function of the water volume, a feature that is highly correlated to the widespread nature of these particular systems. Calheiros (1993) noted similar results. As precipitation systems grow larger, precipitation at longer ranges from the radar likely is present;the effect is to reduce the performance of the radar (Kitchen and Jackson 1993). Similar results were obtained by Kelly (1994) and Seo and Johnson (1995). The later identified two major sources of error associated with the Twin Lakes WSR-88D radar: range effect and improper electronic calibration. Attenuation by intervening rain also is a possible source of error that can lead to larger biases (Fig. 16). Also, a small drift in the transmitting frequency of WSR-88D could cause the loss of the signal power due to transmitter mismatch (Massambani and Pereira 1991).

When one compares Figs. 15 and 16, it is clear that the analysis scheme reasonably corrects for the higher reflectivities due to hail contamination, for attenuation and for range effects, which are the most important sources of error in the events analyzed. From a hydrologic viewpoint, the analysis algorithm performs better during the more critical cases, especially when water volumes are falling at rates above 10 × 10^{6} m^{3} h^{−1}. In addition, the scheme has proven effective in reducing the analysis error.

### Hydrologic verification

Further verification of the analysis scheme was accomplished by means of hydrologic simulations. Observed river flow for the Dry Creek basin were compared with TR-20 simulations, which used analyzed rainfall. One of the difficulties in simulating hydrologic events for this particular watershed was not due to the hydrologic modeling itself, but rather due to the completeness, independency, and quantity of the datasets available. For the three independent datasets of Table 1, only event 17 had complete files of radar data that began when rainfall occurred over the watershed; unfortunately, this particular event but produced very little runoff (Fig. 5). While the data for event 4 are complete, it does not represent an independent dataset since it was used to derive the covariance structure of the statistical scheme. Unfortunately, events 4 and 17 were the only cases available that could be used to verify the impact of the statistical analysis scheme on hydrologic simulations. This is especially unfortunate since the stream gauge was discontinued in July of 1994.

Preliminary simulations were performed using event 17. A single value of basin average precipitation was used along with a very low value for soil infiltration [high curve number (CN) and AMC = 3] to evaluate the time to peak of the observed and simulated hydrographs. The results revealed an almost perfect match between the simulated (with the analyzed precipitation) and observed time to peak (not shown). Further simulations were made to analyze the impact of spatial resolution of the radar data in the basin average rainfall. The shape of the hydrograph improved as smaller rainfall area averages were used (not shown); it was deemed that six rainfall area averages of the Dry Creek basin would be enough to simulate the hydrographs adequately. While each of the 64 subwatersheds can be calibrated with an individual curve number, an average curve number for the whole basin was used. The time increment required by TR-20 was varied to test its impact on the resulting streamflow hydrographs; a Δ*t* ⩽ 30 min was found adequate.

Following this preliminary analysis, the impact of the rainfall accumulation interval on the hydrographs was evaluated. The results revealed that TR-20 is sensitive to the time increment of the rainfall accumulation in that the peak runoff increased for longer accumulation intervals (not shown). A detailed analysis of the TR-20 model outputs revealed that some reach lengths, such as the one just prior to the stream exiting the basin, are too long; the channel routing did not attenuate the simulated hydrograph and a streamflow response was sharper (narrower) than the one observed (not shown).

Having determined the sensitivity of TR-20 to input rainfall for the Dry Creek basin, the next step was to readjust the CN with respect to the field of analyzed precipitation. Radar-only simulations also were included to assess the impact of analyzed estimates upon the hydrologic simulations. Simulations for event 17 (11 April 1994) are shown in Fig. 17. Simulations for 15- and 30-min accumulations are shown in panels a and b, respectively. The first striking difference between using radar-only estimates and the analyzed accumulated precipitation is the resulting magnitude of the peak flows. The time evolution of the basin average precipitation (Fig. 18) revealed an accumulation difference of 23% between the raw radar estimates and the objective analysis, while respective differences in peak flows (Fig. 18) are on the order of 500%. In neither case did the radar-only simulations produce a streamflow forecast that even remotely resembled what actually occurred. Plausible explanations for other differences between the observed and simulated hydrographs are given below.

- The observed flow is higher than in the analysis simulation between 5 and 10 h. Either an underestimation of the CN or the basin average precipitation could have caused the difference. Note in Fig. 17 that the simulated flows during those time intervals (hours 5–10) increased slightly when longer time intervals were used to determine rainfall accumulation. The accumulation of radar-estimated rainfall were virtually unchanged when longer time intervals (Fig. 18) were used. Similar results were obtained for 1- and 2-h accumulation time intervals of rainfall.
- A secondary peak occurs at 7.5 h in the analysis simulation that does not occur in the observed flow due to a lack of basin feedback from baseflow and interflow that are not handled by TR-20. As the precipitating system moved upstream over the watershed, rainfall accumulation ended at downstream locations first, even though upstream subwatersheds apparently had not yet begun to respond. Notice that the recession curve decreases faster for the analysis hydrograph because of TR-20 modeling deficiencies explained above.
- The hydrograph peak generated by the analyzed precipitation field is higher and sharper than in the observed hydrograph because of a lack of channel attenuation. Some reach lengths may be too long, which results in a routing that is purely kinematic. Thus, the main factor for the difference in the observed and simulated hydrographs apparently results from the calibration of TR-20.

The resulting 23% difference between analyzed and radar estimates of rainfall accumulation (Fig. 18) occurred during the periods of larger accumulations (between *t* = 6.0 and *t* = 7.5 h). The passage of an upper-level trough generated several hours of stratiform precipitation with embedded areas of heavy precipitation over Oklahoma. The response of the Dry Creek basin to low accumulation rates is quite nonlinear. The initial abstraction in the basin seems to play an important role during periods of low rainfall accumulations. While minima in peak flow are very difficult to model in general, TR-20, although simple, was able to reproduce most of the important features of the rising limb of the hydrographs. Intermittent rainfall events such as the one above, also are more difficult to model with the current configuration of TR-20.

A similar plot of basin average rainfall over Dry Creek for event 4 (9 June 1993) is shown in Fig. 19. A squall line moved through the basin on this day and generated large precipitation accumulations in a short time interval. The 24% difference between radar estimates and the analysis-adjusted estimates is associated with the period of maximum rainfall rates. Precipitation began over the whole basin at about the same time.

Observed and simulated hydrographs are presented in Fig. 20. Surprisingly, the radar-only simulations underestimated the peak flow by a factor of 3, while the analysis-adjusted rainfall closely reproduced the peak flow. Aside from a small shift in the rising limbs, the analysis simulations were coincident with the observed one. The observed peak hydrograph probably occurred after the indicated time in Fig. 20. Thus, the time to peak of the simulated hydrographs are perhaps 30 min out of phase with the observed one. Although not important in this case, an incorrect attenuation of the wave was the main factor in producing a faster increase in flow, and consequently, an earlier peak flow. For this nearly instantaneous pulse of precipitation, basin infiltration was limited. Moreover, the initial baseflow in the basin was very low; the previous rainfall had occurred three weeks earlier on 18 May 1993. Long periods without rain in this Oklahoma basin makes the soil more compact and reduces the infiltration, which probably occurred for the event on 9 June 1993. As in the previous case, a higher peak flow was simulated when a longer accumulation time interval was used.

Average precipitation accumulations over the Dry Creek basin as a function of the accumulation time increment are given in Fig. 19. In spite of having only two events to use, the 8% difference in total rainfall accumulation from 15- and 120-min analysis in both events (not shown) indicates better accuracy of longer accumulation time intervals. Exposure-related errors in rain gauge measurement led to rainfall underestimation (Groisman and Legates 1994) on the order of 6% (Legates and DeLiberty 1993); they probably were present in the Mesonet observations, but because of much larger errors in radar estimates, these sources of error are secondary. Although radar and Mesonet observation errors exist and are unknown, the statistical analysis scheme effectively reduced them by combining both sets of information in an optimum fashion after due consideration of the statistical properties in the rainfall patterns.

Even though the radar underestimates the mean areal rainfall, the impact was limited when determining the spatial covariance structure in precipitation systems over Oklahoma; this occurs since the mean rainfall accumulation was removed from the time series. On the other hand, Mesonet rainfall measurements produce better estimates of the mean rainfall; even so, the Mesonet is unable to resolve the spatial covariance structure at fine spatial scales. Therefore, the statistical combination of radar estimates and Mesonet measurements is superior to using any field by itself.

## Conclusions

The proposed statistical objective analysis scheme integrates WSR-88D rainfall estimates with Mesonet rain gauge measurements. The former is used as a background field, while the later acts as a component of the innovation vector. Each pair of collocated WSR-88D and Mesonet precipitation data are filtered using the two-dimensional covariance structure of precipitating systems in Oklahoma. This procedure reduces the analysis error variance and maximizes the extractable precipitation signal.

Variable time intervals between 15–120 min for rainfall accumulation have been used to study the impact on the rainfall analysis by the accumulation time interval and by the number of WSR-88D and Mesonet pairs used at each analysis grid point. Results reveal a 30% reduction in the normalized expected analysis error variance when the rainfall accumulation time interval is increased from 15 to 120 min. However, no improvement occurred in the analysis quality when more than three point estimates of rainfall by Mesonet gauges were used in the innovation vector for rainfall accumulation time intervals of 15–60 min. As the accumulation interval increased to 120 min, the addition of more observational pairs increased the analysis error due to the strong cross correlations that existed over small distances between analysis grid points and the input precipitation locations. In general, a more accurate rainfall analysis becomes possible either by increasing the accumulation time interval or by increasing the density of Mesonet rain gauges.

Unfortunately, there is clear evidence that the Twin Lakes WSR-88D underestimates precipitation produced by long-lived, organized precipitating systems, sometimes by as much as 100% in water volume. Smith et al. (1996) noted that on average Twin Lakes WSR-88D rainfall estimates are 30% less than the Tulsa, Oklahoma, WSR-88D rainfall estimates for the overlap area of these sites. Isolated convective systems were less influenced by the proposed rainfall analysis procedure, due apparently to the finite resolution of the Mesonet, which is unable to resolve fine structures of short-lived, localized rainfall systems.

Mesonet observational errors, although not analyzed and applied to this research, can be incorporated easily into the statistical analysis equations. WSR-88D radar reflectivities and adjustable *Z–R* relationships can be stratified by precipitation type, duration, and spatial extent before incorporation into the analysis scheme. These improvements could greatly enhance the application of the proposed statistical analysis scheme. In fact, the proposed scheme is an important first step toward a neural system in which better precipitation estimates are obtained, which in turn improve the precipitation covariance structures. More importantly, analysis errors could be objectively identified and incorporated into the neural system. With completion of the WSR-88D network across the United States, the development of such analysis systems are not only possible but are vitally necessary due to the quantity and quality of the new datasets.

With this portion of the HFS developed, the subsequent step toward completing of the HFS is the QPF by short-term extrapolation and mesoscale modeling. These two types of QPF use the analyzed rainfall field to adjust rainfall rates for an extrapolation QPF and to initialize a mesoscale model as is shown in Part II of this research (Pereira Fo. et al. 1999).

The authors are grateful to Dale Morris, University of Oklahoma (OU), for providing us with WSR-88D data; to Keith Brewster (OU) for helping with the software to read level II data; to Howard Johnson and Billy McPherson from Oklahoma Climatological Survey for providing us with Mesonet data; to C. T. Haan, Oklahoma State University, and Ray Riley and Gary Utley, U.S. Soil Conservation Service, for helping with TR-20 and Dry Creek hydrologic parameters; and to David Adams, U.S. Geological Survey, for providing us with stream flow data; to the anonymous reviewers for their comments, suggestions, and corrections. This research was funded by Fundação de Amparo A Pesquisa do Estado de São Paulo—FAPESP, under Grant 91/1388-1. Partial support was provided by the U.S. Department of Commerce (NOAA) under Grant NA37RJ0203 III-19 and by the Oklahoma Climatological Survey.

## REFERENCES

Atlas, D., A. Ryzhkov, and D. Zrnic, 1995: Polarimetric tuned

*Z–R*relations and comparison of radar rainfall methods. Preprints,*Third Int. Symp. Hydrological Applications of Weather Radars,*São Paulo, Brazil, ABRH/IAHR, 386–395.Austin, P. M., 1987: Relation between measured radar reflectivity and surface rainfall.

*Mon. Wea. Rev.,***115,**1053–1070.Bhargava, M., and M. Danard, 1994: Application of optimum interpolation to the analysis of precipitation in complex terrain.

*J. Appl. Meteor.,***33,**508–518.Brandes, E., 1975: Optimizing rainfall estimates with aid of radar.

*J. Appl. Meteor.,***14,**1339–1345.Brock, F. V., K. C. Crawford, R. L. Elliott, G. W. Cuperus, S. J. Stadler, H. L. Johnson, and M. D. Eilts, 1995: The Oklahoma Mesonet: A technical overview.

*J. Atmos. and Oceanic Technol.,***12,**5–19.Calheiros, R. V., 1993: The area–time integral technique for tropical stratiform rainfall. Preprints,

*26th Conf. Radar Meteorology,*Norman, OK, Amer. Meteor. Soc., 768–770.Crawford, K. C., 1979: Considerations for the design of a hydrologic data network using multivariate sensors.

*Water Resour. Res.,***15,**1752–1762.Creutin, J. D., and C. Obled, 1982: Objective analysis and mapping techniques for rainfall fields: An objective comparison.

*Water Resour. Res.,***18,**413–431.Crum, T. D., and R. L. Alberty, 1993: The WSR-88D and the WSR-88D operational support facility.

*Bull. Amer. Meteor. Soc.,***74,**1669–1687.Daley, R., 1991:

*Atmospheric Data Analysis.*Cambridge University Press, 457 pp.Duan, Q., S. Sorooshian, and V. Gupta, 1992: Effective and efficient global optimization for conceptual rainfall–runoff models.

*Water Resour. Res.,***28,**1015–1031.Duchon, C. E., T. M. Renkevens, and W. L. Crosson, 1995: Estimation of daily area-average rainfall during CaPE experiment in central Florida.

*J. Appl. Meteor,***34,**2704–2714.Farajalla, N. S., and B. E. Vieux, 1994: Capturing the essential spatial variability in distributed hydrological modelling: Hydraulic roughness.

*Hydrol. Process.,***8,**221–236.——, and ——, 1995: Capturing the essential spatial variability in distributed hydrological modelling: Infiltration parameters.

*Hydrol. Process.,***9,**55–68.Gandin, L. S., 1963:

*Objective Analysis of Meteorological Fields.*English translation by R. Harding, Israel Program for Scientific Translation, 242 pp.Groisman, P. Y., and D. R. Legates, 1994: The accuracy of the United States precipitation data.

*Bull. Amer. Meteor. Soc.,***75,**215–227.Hoos, A. B., A. D. Koussis, and G. O. Beale, 1989: A channel dynamic model for real-time flood forecasting.

*Water Resour. Res.,***25,**691–705.Huff, F. A., and W. L. Shipp, 1969: Spatial correlations of storm, monthly and seasonal precipitation.

*J. Appl. Meteor.,***8,**542–550.Kelly, D. S., 1994: A comparison of high resolution precipitation accumulation estimates from the WSR-88D precipitation algorithm with rain gage data. Preprints,

*First WSR-88D User’s Conf.,*Norman, OK, WSR-88D OSF and NEXRAD Joint System Program Office, 185–192.Kitchen, M., and P. M. Jackson, 1993: Weather radar performance at long range—Simulated and observed.

*J. Appl. Meteor.,***32,**975–985.Klazura, G. E., and D. A. Imy, 1993: A description of the initial set of analysis products available from the NEXRAD WSR-88D system.

*Bull. Amer. Meteor. Soc.,***74,**1293–1311.Koch S., M. Desjardins, and P. Kocin, 1983: An interactive Barnes objective map analysis scheme for use with satellite and conventional data.

*J. Climate Appl. Meteor.,***22,**1487–1503.Krajewski, W. F., 1987: Cokringing radar-rainfall and rain gage data.

*J. Geophys. Res.,***92,**9571–9580.Legates, D. R., and T. L. DeLiberty, 1993: Measurement biases in the United Sates raingage network,

*Water Resour. Bull.,***29,**855–861.Massambani, O., and A. J. Pereira Fo., 1991: The diurnal evolution of the ground return intensities and its use as a radar calibration procedure.

*Hydrological Applications of Weather Radars,*Clukie I. D. and C. G. Collier, Eds., Ellis Horwood, 143–150.Messaoud, M., and Y. B. Pointin, 1990: Small time and space measurement of the mean rainfall rate made by a gauge network and by a dual-polarization radar.

*J. Appl. Meteor.,***29,**830–841.Michaud, J., and S. Sorooshian, 1994: Comparison of simple versus complex distributed runoff models on a midsized semiarid watershed.

*Water Resour. Res.,***30,**593–605.Morrissey, M. L., J. A. Maliekal, J. S. Greene, and J. Wang, 1995: The uncertainty of simple spatial averages using rain gauge networks.

*Water Resour. Res.,***31,**2011–2017.Pereira Fo., A. J., K. C. Crawford, and D. J. Stensrud, 1999: Mesoscale precipitation fields. Part II: Hydrometeorologic modeling.

*J. Appl. Meteor.,***38,**102–125.Ryzhkov, A. V., and D. S. Zrnic, 1994: Precipitation observed in Oklahoma mesoscale convective systems with a polarimetric radar.

*J. Appl. Meteor.,***33,**455–464.Seed, A. W., J. Nicol, G. L. Austin, C. D. Stow, and S. G. Bradley, 1995: A physical basis for parameter selection for

*Z–R*relationships. Preprints,*Third Int. Symp. on Hydrological Applications of Weather Radars,*São Paulo, Brazil, ABRH/IAHR, 100–108.Seo, D.-J., and E. R. Johnson, 1995: The WSR-88D precipitation processing subsystem: An overview and performance evaluation. Preprints,

*Third Int. Symp. on Hydrological Applications of Weather Radars,*São Paulo, Brazil, ABRH/IAHR, 222–231.Smith, J. A., D. J. Seo, M. L. Baeck, and M. D. Hudlow, 1996: An intercomparison study of NEXRAD precipitation estimates.

*Water Resour. Res.,***32,**2035–2045.Thiebaux, H. J., 1976: Anisotropic correlation function for objective analysis.

*Mon. Wea. Rev.,***104,**994–1002.——, 1977: Extending estimation accuracy with anisotropic interpolation.

*Mon. Wea. Rev.,***105,**691–699.U.S. Soil Conservation Service, 1970:

*Soil Survey of Lincoln County,*57 pp.——, 1972:

*National Engineering Handbook; Hydrology.*Water Resources, 589 pp.——, 1982:

*TR-20 Computer Program for Project Formulation Hydrology.*Northeast NTC and Hydrology Unit Soil Conservation Service, 168 pp.Weber, R. O., and P. Talkner, 1993: Some remarks on spatial correlation function models.

*Mon. Wea. Rev.,***121,**2611–2617.Wilson, J. W., and E. A. Brandes, 1979: Radar measurement of rainfall—A summary.

*Bull. Amer. Meteor. Soc.,***60,**1048–1058.Wyss, J., E. R. Williams, and R. L. Bras, 1990: Hydrologic modeling of New England river basins using radar rainfall data.

*J. Geophys. Res.,***95,**2143–2152.

WSR-88D datasets used in this paper and listed in chronological order. Also listed are the starting (ST) and ending (ET) date and time, the number of the volume scans (VS) available, the number of individual rainfall events (EV), and the rainfall type (IS = isolated, SL = squall line, SF = stratiform).