A conceptual model for constructing high resolution gauge satellite merged precipitation analyses

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 116,, doi: /2011jd016118, 2011 A conceptual model for constructing high resolution gauge satellite merged precipitation analyses Pingping Xie 1 and An Yuan Xiong 2 Received 15 April 2011; revised 15 August 2011; accepted 16 August 2011; published 8 November [1] A conceptual model has been developed to create high resolution precipitation analyses over land by merging gauge based analysis and CMORPH satellite estimates using data over China for a 5 month period from April to September A two step strategy is adopted to remove the bias inherent in the CMORPH satellite precipitation estimates and to combine the bias corrected satellite estimates with the gauge analysis. First, bias correction is performed for the CMORPH estimates by matching the probability density function (PDF) of the satellite data with that of the gauge analysis using colocated data pairs over a spatial domain of 5 lat/lon centering at the target grid box and over a time period of 30 days, ending at the target date. The spatial domain is expanded wherever necessary over gauge sparse regions to ensure the collection of a sufficient number of gauge satellite data pairs. The bias corrected CMORPH precipitation estimates are then combined with the gauge analysis through the optimal interpolation (OI) technique, in which the bias corrected CMORPH is used as the first guess while the gauge data are used as the observations to modify the first guess over regions with station coverage. Error statistics are computed for the input gauge and satellite data to maximize the performance of the high resolution merged analysis of daily precipitation. Cross validation tests and comparisons against independent gauge observations demonstrate feasibility and effectiveness of the conceptual algorithm in constructing merged precipitation analysis with substantially removed bias and significantly improved pattern agreements compared with those of the input gauge and satellite data. Citation: Xie, P., and A. Y. Xiong (2011), A conceptual model for constructing high resolution gauge satellite merged precipitation analyses, J. Geophys. Res., 116,, doi: /2011jd Introduction [2] Despite the critical importance of precipitation in meteorology, hydrology, agriculture and many other scientific and societal applications, it remains a challenging task to accurately document precipitation over land at high temporal and spatial resolutions. Gauge observations provide accurate measurements of precipitation at the gauge location. The quality of gauge based analyses, however, relies on both the density and configuration of the gauge network [Morrissey et al., 1995; Villarini and Krajewski, 2008]. Great efforts have been made in recent decades by national agencies around the world to install manned and automatic rain gauges. The density of gauge networks, however, is still insufficient to resolve mesoscale systems over many regions over land, especially in developing countries [Chen et al., 2008]. 1 NOAA Climate Prediction Center, Camp Springs, Maryland, USA. 2 National Meteorological Information Center, China Meteorological Administration, Beijing, China. This paper is not subject to U.S. copyright. Published in 2011 by the American Geophysical Union. [3] Weather radars monitor precipitation over an area of 300 km radius surrounding the radar site. A composite of estimates based on observations of individual radars provides precipitation maps as spatially and temporally finite as 1 km and 5 min over a wide region [e.g., Zhang et al., 2009]. However, radar based precipitation estimates are restricted by technical and geographic problems, including mountain blockage, beam height, and uncertainties in the Z R relationship [Krajewski et al., 2010]. Generally speaking, radar observations are only available over populated areas and/or in developed countries. Estimates derived from satellite observations are capable of detecting spatial patterns and temporal variations of precipitation with high time and space resolutions over most of the global regions [Ferraro, 1997]. Satellite estimates of precipitation, however, contain regionally, seasonally, and diurnally varying biases and random errors. In general, the quality of precipitation data sets derived from individual sources is restricted by a combination of shortcomings, including (1) inappropriate coverage and/or sampling, (2) unnegligible bias, and (3) large random errors [Xie and Arkin, 1995; Ebertetal., 1996; McCollum et al., 2000, 2002; Adler et al., 2001; Ebert et al., 2007; Xie et al., 2007; Hong et al., 2007b; Shen et al., 2010; Tian et al., 2009; Xu and Xie, 2010]. 1of14

2 [4] The notion of combining information from multiple platforms was first adopted by hydrologists in constructing gauge radar merged analyses of precipitation around the mid 1980s [Krajewski, 1987]. Since then, extensive research efforts have yielded an improved understanding and refined quantification of the input gauge measurements and radar precipitation error structures. Thereby, several algorithms have been developed to optimize the information from the gauge and radar data. While these algorithms differ in the selected statistical methods, they are designed to perform two functions: to remove radar rainfall estimation bias through comparisons against gauge observations [e.g., Seo and Breidenbach, 2002], and to reduce the random error by further combining bias corrected radar estimates with gauge data [e.g., Seo, 1999; Gupta et al., 2006]. Several gaugeradar systems that generate high resolution merged precipitation maps over regional domains have been constructed and operationally implemented [Makihara, 2000;Lin and Mitchell, 2005; Nelson et al., 2010; Zhang et al., 2009; Amitai et al., 2011]. [5] Early efforts toward the development of gauge satellite merging algorithms were largely made in association with the Global Precipitation Climatology Project (GPCP), which aims at the construction of relatively coarse time and space resolution global precipitation analyses [Adler et al., 2001]. Efforts along this line focused on the construction of global precipitation analyses at relatively coarse time and space resolutions through merging gauge based analyses and satellite estimates. Huffman et al. [1997] devised a procedure to adjust the precipitation estimates derived from infrared (IR) observations frequently sampled from geostationary satellites against relatively accurate retrievals based on less frequent passive microwave (PMW) observations aboard low Earth orbit platforms. Combined satellite precipitation estimates are then defined as the weighted mean of the PMW and the adjusted IR estimates. The combined satellite estimates are further calibrated against colocated monthly GPCC gauge analysis [Schneider, 1993]. The final merged analysis of global monthly precipitation, used by the GPCP [Arkin and Xie, 1994] as its official precipitation product, is computed as a weighted mean of the GPCP gauge analysis and the calibrated combined satellite estimates, with the weights inversely proportional to the error variances [Huffman et al., 1997]. [6] The Climate Prediction Center (CPC) Merged Analysis of Precipitation (CMAP), in the meantime, was developed with a slightly different approach [Xie and Arkin, 1996, 1997]. Satellite estimates derived from several IR and PMW observations are first combined through the maximum likelihood estimation (MLE) method in which weighting coefficients are inversely proportional to the individual error variances computed through comparisons with concurrent gauge analyses. Combined satellite estimates are then blended with gauge data using the method of Reynolds [1988], in which the satellite and gauge data are used to determine the shape and magnitude of the precipitation fields, respectively. Intercomparsion studies showed improved quality of the GPCP and CMAP merged analyses with substantially reduced biases and random errors compared with those of the individual gauge and satellite data sets [Ebert et al.,1996; Adler et al., 2001]. The GPCP and CMAP merged analyses have been constructed on a 2.5 lat/lon grid over the globe with monthly and pentad time resolutions for a 32 year period from 1979 to the present. Both merged analyses have been utilized in a wide range of applications, including weather and climate monitoring, climate analysis, numerical model verifications, and hydrological studies [e.g., Janowiak et al. 1998; Dai and Wigley, 2000; Trenberth and Caron, 2000; Lau and Wu, 2001; Janowiak and Xie, 2003; Roads et al., 2001; Xue et al., 2005; Wang et al., 2010]. [7] Over the past decade, substantial progress has been made to generate precipitation estimates of higher spatial and temporal resolutions through the combined use of IR and PMW observations from multiple satellite platforms. In their pioneering approach, Hsu et al. [1997] developed a sophisticated system to convert the IR brightness temperatures observed by geostationary satellites into instantaneous rain rates through an artificial neural network trained carefully using concurrent IR and PMW based precipitation estimates. Called the Precipitation Estimation from Remotely Sensed Information using Artificial Neural Network (PERSIANN), this technique is further improved by including additional information on cloud types [Hong et al., 2007a]. Turk et al, [2004] and Huffman et al. [2007, 2009] developed their algorithms to derive IR based precipitation estimates through matching the probability density function (PDF) of the geostationary IR data with that of the precipitation intensity from colocated PMW estimates and then combining these IR based estimates with PMW data to form maps of high spatial and temporal resolutions. Joyce et al. [2004], meanwhile, took a very different approach. Precipitation analysis is defined in 30 min intervals and on an 8 km 8 km grid over the globe through propagating the precipitating cloud clusters observed by the instantaneous PMW estimates along the advection vectors computed from consecutive geostationary IR images. A similar Lagrangian approach is adopted by Ushio et al. [2009], who developed a Kalman filtering based method to construct maps of hourly precipitation on a 0.1 lat/lon grid over the globe. [8] While these high resolution precipitation estimates are used widely in applications in meteorology, hydrology, and other related fields, they are constructed using mostly satellite observations and contain biases and random errors of substantial magnitude [Ebert et al., 2007; Xie et al., 2007; Liang and Xie, 2007; Shen et al., 2010; Tian et al., 2009; Boushaki et al., 2009; Xu and Xie, 2010]. One way to improve the quality of satellite based high resolution precipitation analysis is to incorporate information from additional sources. Pioneering works by Huffman et al. [2004], Smith et al. [2006], Chiang et al. [2007], and Boushaki et al. [2009] have demonstrated the effectiveness of reducing biases in satellite estimates with gauge measurements. A comprehensive examination of precipitation data sets by Ebert et al. [2007] showed consistently superior performance of the precipitation fields produced by the current generation operational numerical models over extratropical areas during cold seasons, suggesting potential improvements of the merged precipitation analysis, including information derived from numerical models. [9] The objective of this paper is to develop a conceptual model to construct high quality, high resolution precipitation analyses over land by merging information from gauge 2of14

3 observations and CMORPH satellite estimates using data over China. The algorithm is designed based on the assumptions that the gauge based analysis is unbiased over regions with reasonable station coverage and that satellite estimates are generally biased but contain useful information on the spatial patterns of precipitation. A two step strategy is adopted to remove the bias in the satellite estimates and to combine the gauge analysis with the bias corrected satellite estimates. Similar two step approaches have succeeded in constructing merged analyses of sea surface temperatures (SSTs) over both the global [Reynolds and Smith, 1994] and regional domains [Wang and Xie, 2007] and in combining gauge measurements with radar estimates of regional precipitation [e.g., Krajewski, 1987; Seo, 1999; Seo et al., 2000; Seo and Breidenbach,2002; Gupta et al. 2006]. [10] This paper is organized into five sections. Section 2 provides a brief description of the CMORPH satellite estimates and the gauge data used in the development. Sections 3 and 4 document the development of algorithms to remove the satellite bias through calibration against the gauge data and to combine the bias corrected satellite estimates with gauge data, respectively, Finally a summary is given at the end of the paper. 2. Data [11] The goal of this paper is to develop a conceptual model to combine gauge observations with satellite estimates for the construction of high quality, high resolution precipitation analyses over various regions over global land. The development work here therefore needs to take into account the varying space and time variabilities of the target precipitation fields as well as the differences in the surface station network used to measure the precipitation. To this end, daily precipitation fields over China are selected as the target of the test work. Produced by a variety of weather systems, including typhoons, the East Asian monsoons, and frontal systems, precipitation over China exhibits variations of different time and space scales. The density of the gauge network, however, differs greatly from more than two stations per 0.25 lat/lon in the populated eastern coast to less than one in a region of 500 km radius over the western part of the country. Experimental results using data over China provide us with insights into the algorithm performance in constructing gauge satellite merged precipitation analyses over many other global regions with various combinations of target precipitation field variabilities and gauge network densities. [12] Gauge observations and satellite estimates of daily precipitation are two primary inputs to the objective merged analysis to be developed in this paper. The gauge data used here are analyses of daily precipitation defined by interpolating reports at over 2400 stations over China [Shen et al., 2010] through an optimal interpolation (OI) based algorithm with consideration of orographic effects [Xieetal., 2007]. Figure 1 presents an example of the daily precipitation analysis (top) and gauge station distributions (middle) over China for 2 August The fine spatial structure in the precipitation field is well captured over eastern China, where the gauge network is relatively dense, while unrealistic distribution patterns are observed in the analysis over gauge sparse western China because of the extrapolation of heavy rainfall at isolated stations. [13] Satellite estimates of precipitation used in this study are those generated by the CPC morphing technique (CMORPH) [Joyce et al., 2004]. CMORPH defines 30 min mean precipitation analysis on an 8 km 8 km grid over the globe (60 S 60 N) by propagating estimates of instantaneous precipitation from PMW observations along the cloud system advection vectors computed from consecutive IR cloud images. Intercomparison studies have shown superior performance of the CMORPH compared with other estimation techniques in representing spatial temporal variations of precipitation over most of the global regions, including China [Ebertetal., 2007; Xie et al., 2007; Liang and Xie, 2007; Shen et al., 2010]. CMORPH precipitation data at their raw resolution are accumulated to daily/0.25 lat/lon resolution and used in this work (Figure 2, top). [14] A conceptual model is developed using the daily gauge and the CMORPH precipitation data on a 0.25 lat/lon grid over China (70 E 140 E; 15 N 55 N) for a 5 month period from 1 May to 30 September Removing Bias in the Satellite Estimates [15] As discussed in the introduction, virtually all of the satellite based precipitation products present spatially varying, temporally changing, and range dependent biases [Chiang et al., 2007;Hong et al., 2007b; Tian et al., 2009; Xu and Xie, 2010]. While these biases may not be a significant issue for some applications such as weather and climate monitoring, accurate documentation of the absolute magnitude of precipitation is critical for many other areas, especially in hydrometeorological and oceanographic applications [Tian et al., 2007; Large and Yeager, 2009]. [16] Among the five sets of routinely generated highresolution satellite precipitation estimates, only the Tropical Rainfall Measuring Mission (TRMM) Multisatellite Precipitation Analysis (TMPA) [Huffman et al., 2004] explicitly includes processing for bias removal. Once the real timeversion precipitation estimates, TRMM 3B42RT, are postprocessed with updated inputs, the estimates based purely on satellite observations are adjusted against the GPCP monthly precipitation analysis through a locally defined adjustment factor, creating a new precipitation product, TRMM 3B42. Boushaki et al. [2009] proposed a method to correct the bias in the PERSIANN high resolution satellite precipitation estimates by removing the weighted mean difference between the satellite estimates and the CPC daily gauge analysis computed over 3 3 grid boxes centered at the target grid box. Neither technique described above takes into account the range dependence of the bias in satellite precipitation estimates. In addition, use of monthly gauge data as their reference field by Huffman et al. [2004] makes it impossible to reduce the bias of submonthly time scales shown in many published studies [e.g., Shen et al., 2010]. [17] Chiang et al. [2007], meanwhile, experimented with a technique to reduce the influence of satellite estimate bias on their hydrological model by defining a weighted mean of gauge and satellite precipitation and tuning the weighting coefficients to get hourly runoff predictions with minimum error. No bias correction is performed in an explicit way. [18] Different from Huffman et al. [2004], in our approach to correct the bias in the CMORPH satellite estimates, we use a gauge based analysis of daily precipitation as the reference. 3of14

4 of 5 lat/lon is expanded wherever necessary, especially over gauge sparse areas, until more than 500 pairs of data are collected to ensure stable statistics. Only data pairs with nonzero CMORPH estimates and/or gauge observations are included. Cumulative PDFs (CPDFs), defined as the PDF for precipitation equal to or larger than a threshold, are then computed for the satellite and gauge data, respectively. Assuming that precipitation intensity at a percentage point in the CPDF table for the satellite estimates should be the Figure 1. Example of (top) gauge based analysis (mm/d), (middle) number of reporting gauge stations in a 0.25 lat/lon grid box, and (bottom) estimated error (mm/d) in the gauge analysis for 2 August The use of daily data enables us to remove submonthly time scale bias and to ensure the quantitative accuracy not only in the overall magnitude but also in the frequency of precipitation events with different intensities. To this end, the bias correction is performed for the CMORPH estimates by matching the PDF of the satellite data with that of the daily gauge analysis. [19] Colocated pairs of the gauge and satellite data are collected over 0.25 lat/lon grid boxes with at least one reporting station over a spatial domain of 5 lat/lon centered over the target grid box and over a time period of 30 days, ending at the target date. The initial data collection domain Figure 2. Example of (top) daily precipitation estimates (mm/d) from the original CMORPH, (middle) the biascorrected CMORPH, and (bottom) estimated error (mm/d) in the bias corrected CMORPH for 2 August of14

5 same as that in the CPDF table for the gauge analysis, the bias in the satellite estimates is finally identified and removed by matching the CPDF of the satellite estimates with that of the gauge analysis. For example, if the 75 percentile values at the CPDF for the satellite estimates and gauge analysis are 25 mm/d and 20 mm/d, respectively, satellite estimates of 25 mm/d will be reduced to 20 mm/d so that the CPDF of the bias corrected satellite estimates will be the same as that for the gauge analysis. This biascorrection procedure is performed for each 0.25 lat/lon grid box and for each day to resolve the spatial and temporal variations of the CMORPH bias. As illustrated in Figure 2 (top), the original CMORPH estimates exhibit lower or higher values of precipitation over southern or eastern China, respectively, compared with the gauge analysis (Figure 1, top). The PDF matching procedure described above reduced the regional biases successfully, producing a precipitation field with improved accuracy over China (Figure 2, middle). [20] To quantitatively assess the effectiveness of the PDF matching procedures in reducing biases, cross validation tests are conducted over the time period from 1May to 30 September To do this cross validation exercise, gauge data over 10% of the gauge grid boxes are withdrawn randomly and the gauge data for the remaining 90% of the grid boxes are used to perform the bias correction for the CMORPH satellite estimates. This procedure is repeated 10 times so that the gauge data at each grid box are withdrawn once. The bias corrected CMORPH is then compared against the withdrawn gauge analysis to examine its quantitative accuracy. Only data over grid boxes with at least one reporting gauge were included in the comparison to ensure reasonable quality of the gauge analysis as the ground truth. [21] In general, bias in the original CMORPH precipitation has been removed substantially and the pattern correlation for the bias corrected CMORPH is improved compared with that for the original CMORPH, a result of refined precipitation distribution attributable to the reduction of regional biases (Table 1). Bias of the original CMORPH estimates averaged over the entire space domain presents variations of submonthly time scales, suggesting potential limitations for procedures to adjust the bias based on a comparison of monthly data (Figure 3, bottom). Applying PDF matching of daily data, our procedure is capable of removing the bias in the original CMORPH at various time scales (Figure 3, bottom) and increasing the pattern correlation consistently throughout the test period (Figure 3, top). [22] The original CMORPH tends to generate precipitation distributions with reduced overall raining areas, reduced or elevated frequencies for heavy or intermediate rainfall events compared with those captured by the gauge analysis (Figure 4, red line). Our bias corrected CMORPH estimates present frequencies of rainfall intensity much closer to those of the gauge analysis (Figure 4, green line) than to those of the original satellite estimates (Figure 4, red line), ensuring improved applications of the satellite estimates in documenting hydrological processes. [23] One hidden assumption of this PDF bias correction procedure is that the gauge based analysis of daily precipitation is unbiased. Prior work showed negligible bias in the gauge based analysis over grid boxes with at least one reporting station [Xie et al., 2007; Chen et al., 2008]. We therefore construct PDF tables with colocated CMORPH and gauge data only over those grid boxes with one or more gauge, as described early in this section. 4. Combining Gauge Data With Bias Corrected Satellite Estimates 4.1. Basic Framework [24] The bias corrected CMORPH precipitation estimates are combined with the gauge based analysis through the OI technique of Gandin [1965] to produce a gauge satellite merged analysis of daily precipitation with further improved quality. Under the OI framework, the final analysis value (A k ) at a target grid box (k) is defined through modifying the first guess (F k ) at the grid box using observations at and near the target grid box [Gandin, 1965; Japan Meteorological Agency, 1990; Daly, 1991]: Ak ¼ Fk þ Xn i¼1 WkiðOi FiÞ; ð1þ where O i, F i, and W ki are the observation, first guess, and weighting, respectively, at grid box i where observations are available, and n is the number of observation grid boxes used in the interpolation. In our applications here, the biascorrected CMORPH is used as the first guess, while the gauge analysis is used as the observations to modify the first guess field over regions with station observations. [25] The weighting coefficients in equation (1) are determined by minimizing the analysis error variance (E k ) 2 : ðe k Þ ¼ hða k T k Þ 2 i; ð2þ where T k is the truth at grid point (k) and angle brackets represent an ensemble process. [26] Under the assumptions that there is (a) no bias in the observations and the first guess, (b) no correlation between error in the first guess and error in the observations, and (c) no correlation between errors of the gauge data at two different grid boxes, the weighting coefficients W ki in equation (1) can be defined by solving the following linear equation group: P n f ij þ o ij i j W kj ¼ f ki ; j¼1 ð3þ i ¼ 1; 2;...; n; where m f ij (m o ij ) is the first guess (observation) error correlation f at two grid boxes i and j and m ki is the first guess error correlation between the target grid box k and the observation grid box i, respectively. Under assumption (c), m o ij is set to 1fori = j and to 0 for i j. i ¼ o is the ratio between the standard deviation of the observation f error and that of the first guess error at grid box i. [27] Once the weight coefficients (W ki ) are determined in equation (3), the analyzed value (A k ) can be defined from the first guess and observations through equation (1). Expressing the truth (T k ) as the summation of the first guess (F k ) and the first guess error (s f k ), error variance (E k ) 2 defined in equation (2) is estimated as! 2 ðe k Þ 2 ¼ f k 1 Xn W ki f ki : ð4þ i¼1 5of14

6 Table 1. Cross Validation Statistics for the Original and Gauge Adjusted CMORPH Satellite Estimates of Daily Precipitation Over China CMORPH Bias (%) Correlation Original Adjusted Key to the development of an OI based gauge satellite combined algorithm is the quantification of error structures for the input first guess and the observations. Under the assumptions described above, three error parameters need to be quantified to calculate the OI based analysis. These include (a) the error estimation for the gauge based analysis (s o ), (b) the error estimation for the bias corrected CMORPH (s f ), and (c) the error correlation for the bias corrected CMORPH at two different grid boxes (m f ij ). All errors described above are for estimates and measured precipitation averaged over a grid box of 0.25 lat/lon and for a daily time accumulation period in this study Gauge Analysis Error [28] Quantifying gauge analysis error requires the comparison of the target gauge analysis against the truth defined by observations from a very dense gauge network. Unfortunately, no data from a network of appropriate density over China are available to the investigators of this research. Instead, reports of daily precipitation from a dense station network over South Korea are used to define the gauge analysis error, assuming that the daily gauge analysis defined using the same interpolation algorithm presents similar error structures over Korea and China. Since precipitation over Korea is often caused by the same or similar weather systems (e.g., Meiyu and typhoons) as those over the eastern China, the spatial and temporal structures of the target precipitation, and thereby the error characteristics of a gauge based analysis for the precipitation, are expected to be similar. [29] The daily gauge reports used here are collected and quality controlled by the Korea Meteorological Agency (KMA) from its automatic weather stations (AWSs) (I. C. Shin, personal communication, 2010). The KMA/AWS is a network of over 600 AWSs, covering the entire South Korea with an average station distance of 13 km. The station density is particularly dense around Seoul, the capital city of the nation, where a total of 28 stations is installed inside a 0.25 lat/lon grid box ( E; N). In this study, an arithmetic mean of the daily precipitation values at the 28 stations is computed and used as the truth for the daily precipitation. Analyzed fields of daily precipitation are calculated through the same OI algorithm in defining the analysis over China using reports from subsets of stations over the target grid box and adjacent regions for a 3 year period from 2005 to A total of 1000 sets of Figure 3. Time series of (top) pattern correlation and (bottom) bias (%) for the (dashed line) original and (solid line) gauge adjusted CMORPH satellite estimates against gauge based analyses of daily precipitation over China. The statistics are computed through cross validations in which satellite estimates are compared with gauge analysis over grid boxes withdrawn from the processes of the development of the bias correction procedures. 6of14

7 Figure 4. Probability density function (PDF, %) of daily precipitation averaged on a 0.25 lat/lon grid over China, derived from the (black) daily gauge analysis, (red) the original, and (green) gauge adjusted CMORPH satellite estimates. Original data over a 0.25 lat/lon grid box with one or more reporting gauge are included in the calculations. PDFs for daily precipitation from (top) 0 to 0.4 mm, (middle) 0.1 to 8 mm, and (bottom) 8 to 60 mm and more are plotted to better represent the dynamic range of the PDFs for different precipitation intensities. subset station combinations is simulated through random selection to mimic the gauge network density and configurations over China. In particular, the maximum number of gauges inside any 0.25 lat/lon grid box is restricted to fewer than five. [30] A proxy index, called number of equivalent gauges (N eg ), is defined to measure the density of the local station network over and near the target grid box: N eg ¼ N g0 þ N g1 =8 þ N g2 =32; where N g0, N g1, and N g2 are the numbers of gauges inside the target grid box, the grid boxes neighboring the target grid box, and the grid boxes one layer farther away, respectively. Coefficients for N g0,n g1 and N g2 in equation (5) are assigned values of 1, 1/8, and 1/32, respectively, to approximate the weights for the station reports at the corresponding grid boxes in defining the daily gauge analysis through the OI algorithm. Further work is required to quantify the coefficients for a better representation of the local network density in relation to the analysis quality. Adaptation of the proxy ð5þ index is to simplify the relationship between gauge analysis uncertainty and station network density examined by prior studies [e.g., Morrissey et al., 1995; Villarini and Krajewski, 2008; Wang, 2010]. [31] The OI based daily precipitation analysis at the target grid box is computed for a 3 year period from 2005 to 2007 through the interpolation of station reports from the 1000 combinations of subset stations and compared against the truth defined as the arithmetic mean of all 28 stations. To quantify the random error in the daily gauge analysis, the 3 year data are stratified into 20 classes in the precipitation intensity and into 5 classes based on the gauge network density (N eg ). The definition of the class boundaries is determined to ensure that the range for the precipitation or gauge network density is represented appropriately while the sufficient number of cases is secured for each of the 20 5 classes to stabilize the statistics. The root mean square (RMS) difference between the OI analysis and the 28 station mean is then computed for each class using the data over the 3 year period (Figure 5). [32] As expected, the error variance for the gauge based analysis of daily precipitation increases linearly with the precipitation intensity (Figure 5, top), implying a decrease in the relative error as the precipitation becomes stronger. The error variance, meanwhile, decreases linearly with the local gauge density approximated by the number of equivalent gauges, N eg (Figure 5, middle). The error variance is reduced to about half when N eg increases from one to two. [33] Based on these results, the random error for the gauge analysis is assumed to be proportional to the precipitation intensity and inversely to the number of equivalent gauges (N eg ): o 2¼ k a þ b R= Neg þ 1 ; ð6þ where (s k o ) 2 and R are the error variance and the daily precipitation intensity, respectively, and a ( = 0.15 (mm/d) 2 ) and b ( = 4.09 mm/d) are empirical coefficients determined through least squares fitting using the error variance data computed for the 20 5 classes described in the previous paragraph. The extra 1 in the denominator of equation (6) is set, partially arbitrarily, to ensure that the error estimation does not tend to an infinite value when there is no station report available within two layers of grid boxes. The gauge interpolation algorithm would extend the searching distance over gauge sparse regions to get gauge reports from at least four stations [Xie et al., 2007]. [34] Gauge analysis error depends heavily on the spatial and temporal scales of the target precipitation fields. Error estimation simplified by equation (6), however, takes into account only the target precipitation field intensity (R) and the local network density represented by an index, the equivalent number of gauges (N eg ). Future work to develop this conceptual model into an operational algorithm for gauge based analysis over other regions needs to consider the scale dependence of the gauge analysis error. Coefficients a and b in equation (6) will represent regional and seasonal changes. [35] The estimated error for the daily gauge analysis (Figure 5, bottom) increases with the square root of the precipitation intensity, showing a larger relative error for weak precipitation. For precipitation of the same intensity, 7of14

8 Figure 5. Error variance of daily gauge analysis as a function of (top) precipitation intensity and (middle) gauge network density represented by a parameter called the equivalent gauge number (N eg ) together with (bottom) the error estimation (mm/d) defined from the precipitation intensity and gauge network density through an empirical function based on the simulation results using data over a dense gauge network around Seoul, South Korea. analysis based on gauge reports from a denser network contains an error of reduced magnitude, as expected. [36] The empirical equation derived from the data over Korea is applied to estimate the daily gauge analysis error over China. Figure 1 (bottom) illustrates an example of the estimated gauge analysis error for 2 August While maximum precipitation over eastern and southern China presents a similar magnitude, the estimated error for the gauge analysis is much smaller over eastern China, where the station network is denser. The error for the gauge analysis is pretty large over Tibet and northwestern and northeastern China, where relatively weak precipitation is observed by sparse local networks. [37] As stated in section 4.1, the correlation between the gauge analysis (observation) error at two different grid boxes is assumed to be zero in developing this conceptual model. Although preliminary research showed less sensitivity of the final OI based analysis to the gauge error correlation than other parameters (e.g., gauge analysis error itself), further work is desirable to quantify the gauge error correlation as a function of the space and time scales of the target precipitation field [Ciach et al., 1997; Ciach and Krajewski, 1999]. No such error correlation quantification was performed in this study because high density gauge observations are available only at one single grid box of 0.25 lat/lon grid box over South Korea Error in the Bias Corrected CMORPH [38] Many published theoretical, simulation, and observational studies have shown that random error variance for satellite based precipitation estimates is proportional or inversely proportional to the precipitation intensity or sampling size, respectively [e.g., Huffman, 1997; Li et al., 1998; Bell and Kundu, 2000]. To quantify the random error, biascorrected CMORPH satellite estimates are compared with the gauge based analysis of daily precipitation over China for the 5 month period from May to September Only gauge data over 0.25 lat/lon grid boxes with one or more reporting station are included in the comparison to reduce aliasing that may be caused by gauge analysis error [Ciach and Krajewski, 1997]. As we will see from the examination of the results later, error variance for gauge analysis over grid boxes with at least one reporting station is about 10% of that for the CMORPH satellite estimates. Although restricting the comparison to grid boxes with two or more reporting gauges would yield a more accurate truth of precipitation, the number of the multigauge grid boxes is very small and they all are distributed in the eastern half of the country (Figure 1, middle). Similar to the gauge analysis, comparisons between the bias corrected CMORPH and the gauge analysis are conducted for 20 different data groups stratified according to the intensity of precipitation as estimated by the biascorrected CMORPH. [39] As shown in Figure 6 (top), the error variance for the bias corrected CMORPH satellite estimates increases linearly with the intensity of precipitation. A least squares fitting yields the following empirical relationship between the error variance (s f k ) 2 and precipitation intensity (R): 2¼ f k c þ d R; ð7þ where linear coefficients c and d are 2.93 (mm/d) 2 and mm/d, respectively. [40] The empirical equation is then applied to estimate the random error for the bias corrected CMORPH over each grid box for the entire 5 month test period. Figure 2 (bottom) presents an example of the estimated random error of biascorrected CMORPH for 2 August Overall, random errors for the daily CMORPH satellite estimates are much larger than those for the gauge analysis (Figure 1, bottom) over regions covered by a reasonable gauge network, implying that the final merged analysis will be dominated by the gauge analysis over those regions. [41] Coefficients in equation (7) are empirical in nature and determined in this study through comparison against a gauge analysis over China without consideration of the physical and statistical characteristics of the target precipitation. Examinations of the error variance of bias corrected CMORPH are not performed for data groups with different sampling sizes because of the relatively small changes in the input PMW availability over the test period. A comprehensive simulation test, however, showed degraded performances for the CMORPH satellite estimates with limited 8of14

9 sampling sizes of the input PMW retrievals available over the early period of the PMW observations [Joyce and Xie, 2011]. Future work to extend the merged analysis will require examinations of the random error as a function of satellite sampling size as well. [42] Regional and seasonal changes of the proportional coefficients are not investigated either. Preliminary work by Xu and Xie [2010] showed that the local gauge network density needs to be considered to account for the small scale biases left in the bias corrected CMORPH. Since the bias correction for the original CMORPH at a target grid box of 0.25 lat/lon is performed using PDF tables derived from colocated CMORPH and gauge data over a spatial domain centering at the target grid box, the correction procedure can ensure the removal of only the bias averaged over the data collection domain. Bias at a spatial scale smaller than the size of the PDF table data collection domain remains in the bias corrected CMORPH. The magnitude of the remaining error therefore is a function of the size of the spatial domain from which the colocated gauge satellite data pairs are collected to construct the PDF tables used in the bias correction. Future work in developing the operational system of this conceptual model needs to refine the error definition to achieve better quantitative accuracy Error Correlation for the Bias Corrected CMORPH [43] The last error statistics that need to be defined for the implementation of the OI based combining algorithm are the correlation between errors of the first guess, i.e., the bias corrected CMORPH, at two separated grid boxes. The error of the original CMORPH at a grid box exhibits correlation with that over a nearby location that is due to the combined effects of (1) precipitation systems, which are often organized as a system with a spectrum of spatial scales; and (2) satellite observations, which tend to overestimate or underestimate precipitation at the edge or the center of a weather system. [44] The error correlation model is developed using the daily gauge and bias corrected CMORPH data over China for the 5 month period from May to September First, time series of errors are defined as the difference between the bias corrected CMORPH and the gauge analysis over grid boxes with one or more reporting station. The correlation is then calculated for each pair of grid boxes where the error is defined. The calculated correlation coefficients are averaged for data pair groups at different separation distances, binned in a 20 km interval. As shown in Figure 7 (dots), the error correlation decreases sharply from 0.8 for neighboring grid boxes to almost zero for a separation of 200 km. Error correlation becomes negative for distance of Figure 6. (top) Error variance and (bottom) random error for the bias corrected CMORPH as a function of precipitation intensity measured by the CMORPH. Results in Figure 6 (top) are derived from a comparison of bias corrected CMORPH against the gauge analysis over 0.25 lat/lon grid boxes with one or more reporting stations over the test period from April to September. The line in Figure 6 (bottom) is based on least squares fitting of the results in Figure 6 (top) onto an empirical equation. Figure 7. Error correlation for the bias corrected CMORPH as a function of separation between two grid boxes, derived from daily data over China for a 5 month period from May to September The correlation is calculated for the CMORPH error at two separate grid boxes averaged for data classes with different distances (dots). The averaged correlationisthenfittedtoanexponential function (solid line) to model the correlation distance relationship. 9of14

10 [47] As part of the development of the conceptual model for blending the gauge and CMORPH satellite estimates, no regional changes, seasonal variations, or anisotropy of the error correlation are considered in this study. Further work is desirable to improve the modeled error correlation with refined representations for various environmental situations The OI Based Combining Procedure [48] Finally, a conceptual model is developed to combine the gauge analysis and the bias corrected CMORPH satellite estimates through the optimal interpolation (OI) algorithm of Gandin [1965]. The bias corrected CMORPH is used as the first guess, while the gauge analysis is used as the observations to modify the first guess field over regions with station observations. The error statistics for the gauge analysis and bias corrected CMORPH are defined in subsections , while assumptions are made that there is no correlation between gauge observation errors at two different locations and between errors from gauge observations and bias corrected CMORPH satellite estimates. [49] As illustrated in Figure 8, the gauge satellite combined analysis (Figure 8, top) exhibits much more reasonable patterns of daily precipitation over China. Precipitation distribution in the combined analysis over the gauge network in dense eastern China is very close to that in the gauge analysis (Figure 1, top), while the bull s eye in the gauge based analysis over the Tibetan high plain is diminished in the combined analysis with the inclusion of information from the satellite estimates. In addition to the analyzed value of daily precipitation, error estimation is also Figure 8. (top) Example of daily precipitation (mm/d) and (bottom) estimated error (mm/d) for the merged analysis for 2 August km, before bouncing back to close to zero for farther separation. The negative error correlation is largely attributable to the opposite tendency of satellite precipitation in overestimating or underestimating weak or strong precipitation often located at the center or edge of a weather system that is often km in radius [Zepeda Arce and Foufoula Georgiou, 2000]. [45] Least squares fitting is applied to model the error correlation (r) as an exponential function of separation distance (h) using the bin averaged data plotted in Figure 7: f ¼ r 0 þ r 1 ½e ð h=h0þ Š; ð8þ where empirical coefficients r 0 and r 1 are and 1.196, respectively. [46] The e folding distance (h 0 ) is set to 60.0 km in this study based on an inspection of the scatterplots (Figure 7). The resulting curve fits the observations (dots) very well, especially for error correlations with distances of 300 km or shorter. The maximum correlation in equation (8) is capped to 1.0 for the situation of small h values. Although the error correlation model is unable to represent the bouncing of the correlation for distances longer than 400 km, this will not cause problems in most applications because the difference in correlation is quite small and the OI searching distance usually does not reach that far. Figure 9. Root mean square (RMS) differences (top) between the merged analysis and the gauge based analysis and (bottom) between the merged analysis and the bias corrected CMORPH. The statistics are computed over grid boxes with at least one reporting gauge. 10 of 14

11 Figure 10. Scatterplots of (top) the correlation for the biascorrected CMORPH versus that for the combined analysis and (bottom) correlation for the gauge analysis based on randomly selected subset station reports versus that for the combined analysis. The correlation for the subset gauge analysis, bias corrected CMORPH, and the combined analysis is calculated through a comparison against the ground truth of daily precipitation over a 0.25 lat/lon grid box over Seoul, South Korea, for the test period from April to September The ground truth is defined as the arithmetic mean of gauge reports from all 28 stations available inside the grid box. Dots of different colors represent results for cases based on different local gauge network densities, measured by the number of equivalent gauges. derived through equation (4) as an index of the resulting analysis uncertainty. The estimated relatively error is large or small over regions with weak or strong precipitation, respectively (Figure 8, bottom). [50] A three way comparison is performed among the gauge analysis, the bias corrected CMORPH, and the gauge CMORPH combined analysis to examine if the combining process works as we designed. To this end, root mean square (RMS) differences among the three data sets are calculated for grid boxes with various numbers of gauges and for the entire data period from 1 May to 30 September 30. Differences between the gauge analysis and the combined analysis are very large over grid boxes with no gauges and decrease as the local gauge network becomes dense (Figure 9, top). The differences between satellite estimates and the combined analysis, meanwhile, are minimized over grid boxes without gauge reports and increase substantially over regions with gauge stations (Figure 9, bottom). These results indicate that the combined analysis over gauge sparse regions is controlled primarily by satellite estimates while it is influenced more by gauge analysis where station reports are available, confirming that our objective algorithm works as designed Independent Validations [51] One straightforward way to verify the OI based combining procedure is to compare the gauge satellite merged analysis with independent gauge observations over a region of dense station networks. Unfortunately, at the time of this research, no such dense network is available over China from which a subset of the station reports may be used as inputs to the merging procedures, while gauge data from the remaining stations can be used to create an independent analysis with sufficiently high quality to define the truth of the daily precipitation. [52] Instead, gauge reports from the KMA/AWS network, described in section 4.2, are used to quantify the performance of the gauge satellite combining technique developed in this paper. As described in section 4.2, reports from 1000 combinations of subset stations are collected and gauge based analyses are constructed using the subset station reports. The gauge based analyses are then used to remove the biases in the CMORPH satellite estimates and are further combined with the bias corrected CMORPH. The merged analyses defined using the 1000 combinations of the station reports are finally compared against the ground truth of daily precipitation over the 0.25 lat/lon grid box over Seoul, Korea, defined as the arithmetic mean of the station reports from a total of 28 gauges. [53] The merged analysis presents substantial improvements on the bias corrected CMORPH satellite estimates. Correlations with the ground truth range between 0.70 and 0.98, compared with about for the bias corrected CMORPH (Figure 10, top). The performance of the merged analysis is very close to that of the gauge analysis over regions of dense station networks (N eg 2), while significant improvements are observed in the gauge analysis over regions with sparse gauge coverage (N eg 1, Figure 10, bottom). Because of the limitation in the spatial domain size of the KMA/AWS network, it is impossible to simulate and evaluate the performance of the blending procedure over regions of extremely sparse station networks (e.g., station distance > 500 km). An intuitive inference of the correlation differences over grid boxes with different numbers of equivalent gauges (N eg, Figure 10, bottom) suggests even greater improvements of the merged analysis over regions of sparser station networks, as illustrated in Figures 1, 2, and Summary and Future Work [54] A conceptual model has been developed to create high resolution precipitation analyses over land by merging gauge based analysis and satellite estimates using daily data over China for a 5 month period from April to September The gauge based analysis used here [Shen et al., 11 of 14