Empirically-based modeling of spatial sampling uncertainties associated with rainfall measurements by rain gauges

In the quantitative evaluation of radar-rainfall products (maps), rain gauge data are generally used as a good approximation of the true ground rainfall. However, rain gauges provide accurate measurements for a specific location, while radar estimates represent areal averages. Because these sampling discrepancies could introduce noise into the comparisons between these two sensors, they need to be accounted for. In this study, the spatial sampling error is defined as the ratio between the measurements by a single rain gauge and the true areal rainfall, defined as the value obtained by averaging the measurements by an adequate number of gauges within a pixel. Using a non-parametric scheme, the authors characterize its full statistical distribution for several spatial (4, 16 and 36 km²) and temporal (15 min and hourly) scales.     

Estimation of radar-rainfall error spatial correlation

The study presents a theoretical framework for estimating the radar-rainfall error spatial correlation (ESC) using data from relatively dense rain gauge networks. The error is defined as the difference between the radar estimate and the corresponding true areal rainfall. The method is analogous to the error variance separation that corrects the error variance of a radar-rainfall product for gauge representativeness errors. The study demonstrates the necessity to consider the area–point uncertainties while estimating the spatial correlation structure in the radar-rainfall errors. To validate the method, the authors conduct a Monte Carlo simulation experiment with synthetic fields with known error spatial correlation structure. These tests reveal that the proposed method, which accounts for the area–point distortions in the estimation of radar-rainfall ESC, performs very effectively. The authors then apply the method to estimate the ESC of the National Weather Service’s standard hourly radar-rainfall products, known as digital precipitation arrays (DPA). Data from the Oklahoma Micronet rain gauge network (with the grid step of about 5 km) are used as the ground reference for the DPAs. This application shows that the radar-rainfall errors are spatially correlated with a correlation distance of about 20 km. The results also demonstrate that the spatial correlations of radar–gauge differences are considerably underestimated, especially at small distances, as the area–point uncertainties are ignored.     

Rainfall spatial variability observed by X-band weather radar and its implication for the accuracy of rainfall estimates

The main objective of this paper is to estimate the error in the rainfall derived from a polarimetric X-band radar, by comparison with the corresponding estimate of a rain gauge network. However the present analysis also considers the errors inherent to rain gauge, in particular instrumental and representativeness errors. A special emphasis is addressed to the spatial variability of the rainfall in order to appreciate the representativeness error of the rain gauge with respect to the 1 km square average, typical of the radar derived estimate. For this purpose the spatial correlation function of the rainfall is analyzed.     

Detection of mean-field bias of the radar rain rate using rain gauges available within a small portion of radar umbrella: a case study of the Donghae (East Sea) radar in Korea

Observation of a storm approaching from the ocean to the in-land area is very important for the flood forecasting. Radar is generally used for this purpose. However, as rain gauges are mostly located within the in-land area, detection of the mean-field bias of radar rain rate cannot be easily made. This problem is obviously different from that with evenly-spaced rain gauges over the radar umbrella. This study investigated the detection problem of mean-field bias of radar rain rate when rain gauges are available within a small portion of radar umbrella. To exactly determine the mean-field bias, i.e. the difference between the radar rain rate and the rain gauge rain rate, the variance of the difference between two observations must be small; thus, a sufficient number of observations are indispensable. Therefore, the problem becomes determining the number of rain gauges that will satisfy the given accuracy, that being the variance of the difference between two observations. The dimensionless error variance derived by dividing the expected value of the error variance by the variance of the areal average rain rate was introduced as a criteria to effectively detect the mean field bias. Here, the variance of the areal average rain rate was assumed to be the climatological one and the expectation for the error variance could be changed depending one the sampling characteristics. As an example, this study evaluated the rainfall observation over the East Sea by the Donghae radar. About 6.8 % of the entire radar umbrella covered in-land areas, where the rain gauges were available. It was found that, to limit the dimensionless error variance to 2 %, a total of 26 rain gauges are required for the entire radar umbrella; whereas, a total of 24 rain gauges would be required within the in-land area with available for the rain gauge data.     

雷达测雨误差及其对淮河流域径流模拟的影响

将雷达测雨数据与分布式水文模型相耦合进行径流过程模拟,分析雷达测雨误差及其径流过程模拟效果,研究雷达测雨误差对径流过程模拟的影响效应.在对淮河流域气象中心业务化的5种淮河流域雷达测雨数据进行误差分析的基础上,采用雷达测雨数据驱动HEC-HMS水文模型,模拟分析淮河息县水文站以上流域2007年7月1-10日强降雨集中期的径流过程.结果表明:利用雷达测雨数据的径流模拟结果与实测资料的模拟结果基本吻合,各种雷达测雨数据误差经过HEC-HMS水文模型传递后,误差明显减小.联合校准法对应的模拟效果最好,过程流量相对误差NBs'和洪峰流量相对误差Z'分别为-20.2%和-13.3%.     

Spatial reconstruction of rainfall fields from rain gauge and radar data

Rainfall is a phenomenon difficult to model and predict, for the strong spatial and temporal heterogeneity and the presence of many zero values. We deal with hourly rainfall data provided by rain gauges, sparsely distributed on the ground, and radar data available on a fine grid of pixels. Radar data overcome the problem of sparseness of the rain gauge network, but are not reliable for the assessment of rain amounts. In this work we investigate how to calibrate radar measurements via rain gauge data and make spatial predictions for hourly rainfall, by means of Monte Carlo Markov Chain algorithms in a Bayesian hierarchical framework. We use zero-inflated distributions for taking zero-measurements into account. Several models are compared both in terms of data fitting and predictive performances on a set of validation sites. Finally, rainfall fields are reconstructed and standard error estimates at each prediction site are shown via easy-to-read spatial maps.     

Microwave link rainfall estimation: Effects of link length and frequency,temporal sampling,power resolution,and wet antenna attenuation

Issues associated with microwave link rainfall estimation such as the effects of spatial and temporal variation in rain, the nonlinearity of R–k$R''–''k$ relations, temporal sampling, power resolution, and wet antenna attenuation are investigated using more than 1.5 years of data from a high-resolution X-band weather radar. Microwave link signals are generated for different link frequencies and lengths from these radar data, so that retrieved path-averaged rainfall intensities can be compared to true path-averaged values. Results of these simulations can be linked to the space–time structure of rain. A frequency-dependent relation between the rainfall intensity at an antenna and the attenuation caused by its wetting is derived using microwave link and rain gauge data. It is shown that if the correct temporal sampling strategy is chosen, the effects of the degradation of power resolution and of wet antenna attenuation (if a correction is applied) are minor (i.e., MBE and bias-corrected RMSE are >−20% and <20% of the mean rainfall intensity, respectively) for link frequencies and lengths above ∼20 GHz and ∼2 km, respectively.     

Spatial rainfall estimation by linear and non-linear co-kriging of radar-rainfall and raingage data

The feasibility of linear and nonlinear geostatistical estimation techniques for optimal merging of rainfall data from raingage and radar observations is investigated in this study by use of controlled numerical experiments. Synthetic radar and raingage data are generated with their hypothetical error structures that explicitly account for sampling characteristics of the two sensors. Numerically simulated rainfall fields considered to be ground-truth fields on 4×4 km grids are used in the generation of radar and raingage observations. Ground-truth rainfall fields consist of generated rainfall fields with various climatic characteristics that preserve the space-time covariance function of rainfall events in extratropical cyclonic storms. Optimal mean areal precipitation estimates are obtained based on the minimum variance, unbiased property of kriging techniques under the second order homogeneity assumption of rainfall fields. The evaluation of estimated rainfall fields is done based on the refinement of spatial predictability over what would be provided from each sensor individually. Attention is mainly given to removal of measurement error and bias that are synthetically introduced to radar measurements. The influence of raingage network density on estimated rainfall fields is also examined.     

Spatial rainfall estimation by linear and non-linear co-kriging of radar-rainfall and raingage data

The feasibility of linear and nonlinear geostatistical estimation techniques for optimal merging of rainfall data from raingage and radar observations is investigated in this study by use of controlled numerical experiments. Synthetic radar and raingage data are generated with their hypothetical error structures that explicitly account for sampling characteristics of the two sensors. Numerically simulated rainfall fields considered to be ground-truth fields on 4×4 km grids are used in the generation of radar and raingage observations. Ground-truth rainfall fields consist of generated rainfall fields with various climatic characteristics that preserve the space-time covariance function of rainfall events in extratropical cyclonic storms. Optimal mean areal precipitation estimates are obtained based on the minimum variance, unbiased property of kriging techniques under the second order homogeneity assumption of rainfall fields. The evaluation of estimated rainfall fields is done based on the refinement of spatial predictability over what would be provided from each sensor individually. Attention is mainly given to removal of measurement error and bias that are synthetically introduced to radar measurements. The influence of raingage network density on estimated rainfall fields is also examined.     

Radar rainfall estimation for flash flood forecasting in small urban watersheds

Radar rainfall estimation for flash flood forecasting in small, urban catchments is examined through analyses of radar, rain gage and discharge observations from the 14.3 km² Dead Run drainage basin in Baltimore County, Maryland. The flash flood forecasting problem pushes the envelope of rainfall estimation to time and space scales that are commensurate with the scales at which the fundamental governing laws of land surface processes are derived. Analyses of radar rainfall estimates are based on volume scan WSR-88D reflectivity observations for 36 storms during the period 2003–2005. Gage-radar analyses show large spatial variability of storm total rainfall over the 14.3 km² basin for flash flood producing storms. The ability to capture the detailed spatial variation of rainfall for flash flood producing storms by WSR-88D rainfall estimates varies markedly from event to event. As spatial scale decreases from the 14.3 km² scale of the Dead Run watershed to 1 km² (and the characteristic time scale of flash flood producing rainfall decreases from 1 h to 15 min) the predictability of flash flood response from WSR-88D rainfall estimates decreases sharply. Storm to storm variability of multiplicative bias in storm total rainfall estimates is a dominant element of the error structure of radar rainfall estimates, and it varies systematically over the warm season and with flood magnitude. Analyses of the 7 July 2004 and 28 June 2005 storms illustrate microphysical and dynamical controls on radar estimation error for extreme flash flood producing storms.     

Review of the Different Sources of Uncertainty in Single Polarization Radar-Based Estimates of Rainfall

It is well acknowledged that there are large uncertainties associated with radar-based estimates of rainfall. Numerous sources of these errors are due to parameter estimation, the observational system and measurement principles, and not fully understood physical processes. Propagation of these uncertainties through all models for which radar-rainfall are used as input (e.g., hydrologic models) or as initial conditions (e.g., weather forecasting models) is necessary to enhance the understanding and interpretation of the obtained results. The aim of this paper is to provide an extensive literature review of the principal sources of error affecting single polarization radar-based rainfall estimates. These include radar miscalibration, attenuation, ground clutter and anomalous propagation, beam blockage, variability of the Z–R relation, range degradation, vertical variability of the precipitation system, vertical air motion and precipitation drift, and temporal sampling errors. Finally, the authors report some recent results from empirically-based modeling of the total radar-rainfall uncertainties. The bibliography comprises over 200 peer reviewed journal articles.     

Variability of the spatial structure of intense Mediterranean precipitation

Intense Mediterranean precipitation can generate devastating flash floods. A better understanding of the spatial structure of intense rainfall is critical to better identify catchments that will produce strong hydrological responses. We focus on two intense Mediterranean rain events of different types that occured in 2002. Radar and rain gauge measurements are combined to have a data set with a high spatial (1 × 1 km²) and temporal (5 min) resolution. Two thresholds are determined using the quantiles of the rain rate values, corresponding to the precipitating system at large and to the intense rain cells. A method based on indicator variograms associated with the thresholds is proposed in order to automatically quantify the spatial structure at each time step during the entire rain events. Therefore, its variability within intense rain events can be investigated. The spatial structure is found to be homogeneous over periods that can be related to the dynamics of the events. Moreover, a decreasing time resolution (i.e., increasing accumulation period) of the rain rate data will stretch the spatial structure because of the advection of rain cells by the wind. These quantitative characteristics of the spatial structure of intense Mediterranean rainfall will be useful to improve our understanding of the dynamics of flash floods.     

A stochastic framework for downscaling processes of spatial averages based on the property of spectral multiscaling and its statistical diagnosis on spatio-temporal rainfall fields

Spectral multi-scaling postulates a power-law type of scaling of spectral distribution functions of stationary processes of spatial averages, over nested and geometrically similar sub-regions of the spatial parameter space of a given spatio-temporal random field. Presently a new framework is formulated for down-scaling processes of spatial averages, following naturally from the postulate of spectral multi-scaling, and key ingredients required for its implementation are described. Moreover, results from an extensive diagnostic study are presented, seeking statistical evidence supportive of spectral multi-scaling. Such evidence emerges from two sources of data. One is a 13 year long historical record of radar observations of rainfall in southeastern UK (Chenies radar), with high spatial (2 km) and temporal (5 min) resolution. The other is an ensemble of rain rate fields simulated by a spatio-temporal random pulse model fitted to the historical data. The results are consistent between historical and simulated rainfall data, indicating frequency-dependent scaling relationships interpreted as evidence of spectral multi-scaling across a range of spatial scales.     

A comparison of gauge and radar precipitation data for simulating an extreme hydrological event in the Severn Uplands,UK

This paper provides a comparison of gauge and radar precipitation data sources during an extreme hydrological event. November–December 2006 was selected as a time period of intense rainfall and large river flows for the Severn Uplands, an upland catchment in the United Kingdom. A comparison between gauge and radar precipitation time‐series records for the event indicated discrepancies between data sources, particularly in areas of higher elevation. The HEC‐HMS rainfall‐runoff model was selected to assess the accuracy of the precipitation to simulate river flows for the extreme event. Gauge, radar and gauge‐corrected radar rainfall were used as model inputs. Universal cokriging was used to geostatistically interpolate gauge data with radar and elevation data as covariates. This interpolated layer was used to calculate the mean‐field bias and correct the radar composites. Results indicated that gauge‐ and gauge‐corrected radar‐driven models replicated flows adequately for the extreme event. Gauge‐corrected flow predictions produced an increase in flow prediction accuracy when compared with the raw radar, yet predictions were comparative in accuracy to those using the interpolated gauge network. Subsequent investigations suggested this was due to an adequate spatial and temporal resolution of the precipitation gauge network within the Severn Uplands. Results suggested that the six rain gauges could adequately represent precipitation variability of the Severn Uplands to predict flows at an approximately equal accuracy to that obtained by radar. Temporally, radar produced an increase in flow prediction accuracy in mountainous reaches once the gauge time step was in excessive of an hourly interval. Copyright © 2010 John Wiley & Sons, Ltd.     

Validation and use of rainfall radar data to simulate water flows in the Rio Escondido basin

This paper presents a combined validation method of radar-sensed rainfall, using rain gauge data and hydrologic closure, with an application to the Rio Escondido basin (North-East of Mexico). The space–time scaling behavior of rainfall between rain gauge and radar scales is compared with the intrinsic variability of rainfall, for a statistical validation of space–time variability. For hydrological validation purposes, the CEQUEAU model is used to perform rainfall-runoff routing. It provides a basin-wide water balance, to be compared with the measured water flow at the Villa de Fuentes hydrometric station, for mean-value gauging closure. A good qualitative agreement in terms of hydrograph shape and timing is obtained between the simulated and the observed water flows, and a multiplicative correction factor of an initially proposed Z–R relationship is adopted for the watershed under study, which agrees approximately with other authors’ findings about that relationship. The results are considered particularly useful as a validation-and-correction methodology of radar rainfall estimates for areas sparsely covered by rain gauges.     

Conditional simulation of remotely sensed rainfall data using a non-Gaussian v-transformed copula

Quantification of rainfall and its spatial and temporal variability is extremely important for reliable hydrological and meteorological modeling. While rain gauge measurements do not provide reasonable areal representation of rainfall, remotely sensed precipitation estimates offer much higher spatial resolution. However, uncertainties associated with remotely sensed rainfall estimates are not well quantified. This issue is important considering the fact that uncertainties in input rainfall are the main sources of error in hydrologic processes. Using an ensemble of rainfall estimates that resembles multiple realizations of possible true rainfall, one can assess uncertainties associated with remotely sensed rainfall data. In this paper, ensembles are generated by imposing rainfall error fields over remotely sensed rainfall estimates. A non-Gaussian copula-based model is introduced for simulation of rainfall error fields. The v-transformed copula is employed to describe the dependence structure of rainfall error estimates without the influence of the marginal distribution. Simulations using this model can be performed unconditionally or conditioned on ground reference measurements such that rain gauge data are honored at their locations. The presented model is implemented for simulation of rainfall ensembles across the Little Washita watershed, Oklahoma. The results indicate that the model generates rainfall fields with similar spatio-temporal characteristics and stochastic properties to those of observed rainfall data.     

Flood simulation using the gauge‐adjusted radar rainfall and physics‐based distributed hydrologic model

This paper reports the results of an investigation into flood simulation by areal rainfall estimated from the combination of gauged and radar rainfalls and a rainfall–runoff model on the Anseong‐cheon basin in the southern part of Korea. The spatial and temporal characteristics and behaviour of rainfall are analysed using various approaches combining radar and rain gauges: (1) using kriging of the rain gauge alone; (2) using radar data alone; (3) using mean field bias (MFB) of both radar and rain gauges; and (4) using conditional merging technique (CM) of both radar and rain gauges. To evaluate these methods, statistics and hyetograph for rain gauges and radar rainfalls were compared using hourly radar rainfall data from the Imjin‐river, Gangwha, rainfall radar site, Korea. Then, in order to evaluate the performance of flood estimates using different rainfall estimation methods, rainfall–runoff simulation was conducted using the physics‐based distributed hydrologic model, Vflo^?. The flood runoff hydrograph was used to compare the calculated hydrographs with the observed one. Results show that the rainfall field estimated by CM methods improved flood estimates, because it optimally combines rainfall fields representing actual spatial and temporal characteristics of rainfall. Copyright © 2008 John Wiley & Sons, Ltd.     

Adjustment of wind-drift effect for real-time systematic error correction in radar rainfall data

An effective bias correction procedure using gauge measurement is a significant step for radar data processing to reduce the systematic error in hydrological applications. In these bias correction methods, the spatial matching of precipitation patterns between radar and gauge networks is an important premise. However, the wind-drift effect on radar measurement induces an inconsistent spatial relationship between radar and gauge measurements as the raindrops observed by radar do not fall vertically to the ground. Consequently, a rain gauge does not correspond to the radar pixel based on the projected location of the radar beam. In this study, we introduce an adjustment method to incorporate the wind-drift effect into a bias correlation scheme. We first simulate the trajectory of raindrops in the air using downscaled three-dimensional wind data from the weather research and forecasting model (WRF) and calculate the final location of raindrops on the ground. The displacement of rainfall is then estimated and a radar–gauge spatial relationship is reconstructed. Based on this, the local real-time biases of the bin-average radar data were estimated for 12 selected events. Then, the reference mean local gauge rainfall, mean local bias, and adjusted radar rainfall calculated with and without consideration of the wind-drift effect are compared for different events and locations. There are considerable differences for three estimators, indicating that wind drift has a considerable impact on the real-time radar bias correction. Based on these facts, we suggest bias correction schemes based on the spatial correlation between radar and gauge measurements should consider the adjustment of the wind-drift effect and the proposed adjustment method is a promising solution to achieve this.     

Sampling error of areal average rainfall due to radar partial coverage

Areal average rainfall is important as it is used as an input for most rainfall-runoff analysis in Hydrology and Water Resources. Different from traditional methods of using rain gauge data, the use of radar rainfall for the estimation of areal average rainfall is very straightforward. However, in some cases with severe terrain blockages, the value of the incomplete radar information is of serious concern. This study investigated this problem and derived an equation for estimating the error involved in the areal average rainfall due to partial radar coverage of a basin or sub-basin. When only partial radar information is available, the sampling error decreases with increasing radar coverage and the number of radar bin clusters. As an application example, this study considered the Han River Basin with its rainfall observations using the Ganghwa rain radar. Among a total of 24 mid-sized sub-basins in the Han River Basin evaluated, only five sub-basins were fully covered by the radar and three were totally uncovered. The remaining 16 sub-basins were covered partially by radar leading to incomplete radar information. The results show that the sampling error ranged from several % to tens % of standard deviation of the areal average rainfall depending on the relative areal radar coverage.     

Estimation of rainfall from infrared‐microwave satellite data for basin‐scale hydrologic modelling

The infrared‐microwave rainfall algorithm (IMRA) was developed for retrieving spatial rainfall from infrared (IR) brightness temperatures (TBs) of satellite sensors to provide supplementary information to the rainfall field, and to decrease the traditional dependency on limited rain gauge data that are point measurements. In IMRA, a SLOPE technique (ST) was developed for discriminating rain/no‐rain pixels through IR image cloud‐top temperature gradient, and 243K as the IR threshold temperature for minimum detectable rainfall rate. IMRA also allows for the adjustment of rainfall derived from IR‐TB using microwave (MW) TBs. In this study, IMRA rainfall estimates were assessed on hourly and daily basis for different spatial scales (4, 12, 20, and 100 km) using NCEP stage IV gauge‐adjusted radar rainfall data, and daily rain gauge data. IMRA was assessed in terms of the accuracy of the rainfall estimates and the basin streamflow simulated by the hydrologic model, Sacramento soil moisture accounting (SAC‐SMA), driven by the rainfall data. The results show that the ST option of IMRA gave accurate satellite rainfall estimates for both light and heavy rainfall systems while the Hessian technique only gave accurate estimates for the convective systems. At daily time step, there was no improvement in IR‐satellite rainfall estimates adjusted with MW TBs. The basin‐scale streamflow simulated by SAC‐SMA driven by satellite rainfall data was marginally better than when SAC‐SMA was driven by rain gauge data, and was similar to the case using radar data, reflecting the potential applications of satellite rainfall in basin‐scale hydrologic modelling. Copyright © 2010 John Wiley & Sons, Ltd.