## 1. Introduction

Satellite-based precipitation estimates are becoming popular for use in monitoring large-scale tropical systems such as hurricanes and typhoons, as well as fast-evolving and intense convective storms on a global scale. Such information can help the prediction of flood events, assessment of the extent of inundation of the impact regions, and estimation of financial losses associated with extreme weather conditions (Smith and Katz 2013; Ross and Lott 2003). The satellite provides imagery over mountainous and oceanic regions, where radar and gauge network coverage is poor or nonexistence. On a global scale, remote sensing (RS) precipitation products are also preferred for their higher spatial resolution (0.25°) and finer sampling intervals (3 h) than the ground-based precipitation datasets (Huffman et al. 2007; Joyce et al. 2004; Hsu et al. 1997).

The satellite-based precipitation measurement error is the difference between the RS-derived precipitation estimate and the collocated ground measurement (hereafter, the error), and it limits the potential application of RS-derived precipitation products (Sorooshian et al. 2011; Turk et al. 2008). The error can be quantified through a number of indices such as bias ratio, variance, correlation coefficient, and prediction skills such as probability of detection (Buarque et al. 2011; Gebremichael et al. 2005). The error is sensitive to rainfall intensity, spatiotemporal resolution, location, climate, elevation, season, and other factors (Adler et al. 2003; Buarque et al. 2011; Huffman 1997; Sorooshian et al. 2000; Steiner et al. 2003; Yuan et al. 2005). The relationships between the error and these factors have been studied by several error models (Gebremichael and Krajewski 2005; Gebremichael et al. 2006, 2011; Hossain and Anagnostou 2006; Yan and Gebremichael 2009).

The error is more related to precipitation intensity than to the other factors, and the relationship is determined using parametric and nonparametric models (Gebremichael et al. 2006, 2011; AghaKouchak et al. 2010; Ciach et al. 2007; Gebremichael and Krajewski 2005, 2007; Hossain and Anagnostou 2006; Moazami et al. 2014; Yan and Gebremichael 2009). In this paper, heteroscedasticity refers to the changing variance of the errors (i.e., between satellite-based precipitation estimates and ground-based precipitation observations) associated with the intensity of precipitation measured by ground radar. In some cases, modelers have assumed a linear relationship (AghaKouchak et al. 2010) between the errors as a function of intensity and others have preferred the assumption of nonlinearity (Gebremichael et al. 2011; Hossain and Anagnostou 2006) to represent the heteroscedasticity. To further investigate the nature of heteroscedasticity, the error’s probability density function (pdf) and confidence interval CI are determined by a nonparametric joint distribution (Gebremichael et al. 2011). Such information provides a probabilistic form of satellite-based precipitation measurements. However, it is unclear whether such error models are suitable for different regions or different seasons. As we know, the type and amount of precipitation can shift dramatically in different regions and in different seasons, and that shift should be reflected in the error model. Therefore, we need a model that can be adaptively adjusted to different scenarios.

To investigate the spatial and seasonal variation of heteroscedasticity, we first set up an adaptive parametric error model. We use a mixed joint pdf model to determine the shape of the radar’s pdf and the CI that is conditional at any precipitation intensity. Thus, we obtain a parametric joint pdf for the random pair (*X*, *Y*), where *X* represents the precipitation obtained from a satellite-based estimation and *Y* is the ground measurement (i.e., radar). The pair (*X*, *Y*) is discrete because of zero values; therefore, a discrete joint cumulative distribution function (cdf) is needed. Previous work suggests that the discrete joint cdf can be a combination of four continuous joint cdfs (Serinaldi 2009a,b; Villarini et al. 2008, 2014; Herr and Krzysztofowicz 2005). To focus on the continuous part of the joint cdf and to make the model adaptable, we applied a threshold to the variable pairs. After the variable pairs are modeled by a continuous joint cdf, we derive the conditional pdf for a range of rainfall intensities from satellite-based precipitation measurements. Then, we analyze three different RS-derived precipitation products during three seasons over three locations in the contiguous United States (CONUS) using our proposed parametric distribution.

## 2. Method and materials

### a. Study locations and datasets

Three satellite-based precipitation products are included in this study: PERSIANN from the University of California, Irvine (UCI); the real-time TRMM product 3B42, version 7 (TRMM-3B42-RTV7) from NASA; and the Climate Prediction Center (CPC) morphing technique (CMORPH) from NOAA/CPC. The NEXRAD Stage IV product is used as our ground reference, and a 9-yr (~2003–11) overlapping period of satellite–radar rainfall record is obtained for evaluation. Because NEXRAD Stage IV datasets have a finer spatiotemporal resolution than satellite-based datasets, they were transformed to satellite resolution using the arithmetic mean method. A brief summary and the corresponding references are given in Table 1.

Satellite-derived algorithms and ground radar products used in this study, along with their coverage, resolution, and primary investigators.

For the study area, we selected three climatically diverse locations based on the Köppen climate classification as reference. For example, Köppen’s classification considers most of Oklahoma and Florida as “humid subtropical climate” and Montana as “semiarid climate” (Peel et al. 2007). The three selected locations are tested in the warm season (April–September), the cold season (October–March), and over the entire year (both warm and cold together). The uncertainties in the radar-derived measurements increase with distance away from the radar base because of range degradation (Villarini and Krajewski 2010). Therefore, we chose study sites (Fig. 1) that are either close to radar stations (in Montana and Florida) or have dense gauge networks available (in Oklahoma). For each site and season, individual uncertainty models are obtained for each of the RS-derived precipitation products.

### b. The bivariate mixed model

Random pairs (*X*, *Y*) denote nonnegative random variables, of which *X* represents the precipitation estimate belonging to one of the RS-derived products and *Y* is the ground measurement belonging to the same grid. The joint pdf for (*X*, *Y*) is formulated to obtain the pdf of a conditional distribution for *Y* given *X* = *x*.

The framework that formulates the joint pdf consists of three steps. First, a threshold is chosen to account for minimum errors in the datasets in preparation for making the dataset ready for obtaining the joint pdf. Second, a copula-based method is applied to the truncated positive pairs in order to determine the suitable formulation of the joint cdf. Third, the parameters of the joint pdf are calibrated by minimizing a multiobjective criterion that measures the differences between the model and the data. The criterion includes the four indices correlation coefficient CORR, bias ratio BIAS, mean absolute error MAE, and mean-square error MSE.

*X*and

*Y*, and

*X*,

*Y*) is discrete and is defined as

In our study, the NEXRAD Stage IV rainfall observation system has a minimum error of 0.2 mm day^{−1} (Seo and Krajewski 2010; Xie et al. 2006), and ~3%–5% of the data pairs fall into the ~0–0.2 mm day^{−1} range (Fig. 2), while the total positive pairs account for ~10%–20% of all pairs (see Table B1 in appendix B). These variable pairs, which only account for less than 5% of the total rainfall amount, will hamper the adaptability of a joint cdf. After a threshold of 0.2 mm day^{−1} is applied to truncate the data pairs, less than 5% of total rainfall amount is discarded from the positive pairs (see Tables B1–B3 in appendix B), and the GoF tests are passed for all the products.

*X*=

*x*, is

*p*is the conditional cumulative probability) in Eq. (8) as

## 3. Results and discussion

We evaluated three sets of RS-derived precipitation products over three study locations and during three different periods. These datasets were then used to test the validity of our joint pdf models for each of the scenarios. We evaluated the probabilities for each of the four categories

For the positive pairs, we also tested the GoFs of the copula and the marginal distributions (see Tables A1–A3 in appendix A) and fine-tuned the parameters of the joint pdfs. To validate the accuracy of the joint pdfs, 10^{6} samples of positive pairs are generated, and the generation is repeated 100 times for each parameter set to estimate the correlation index and difference-based indices of the models (CORR, BIAS, MSE, and MAE; see Tables B1–B3 in appendix B). We compared the correlation index and difference-based indices between the observed and the model-generated data. The indices from model-generated samples fall within the ±5% range of those from the observed data. The results suggest that 1) the joint pdf models sufficiently capture the error characteristics of the satellite-derived precipitation, as measured by difference-based indices, and 2) the differences in uncertainty associated with the location, climate, and seasons are also captured by the different parameter sets.

### a. Results about the error–intensity relationship

According to previous studies, the shapes of the conditional pdfs of radar-derived precipitation change across the intensity range (Gebremichael et al. 2011; Hossain and Anagnostou 2006). To see how the shape varies with increasing precipitation intensity, we plot the radar’s conditional pdfs for RS-derived estimates of 10, 20, 40, and 60 mm day^{−1} (Fig. 3) for three locations and for datasets during the whole year. To evaluate the sensitivity of the difference-based indices as a function of rain intensity, we also calculated the mean values and variances for these pdfs (Fig. 3). Doing so allowed us to quantify the accuracy of the RS-derived products as a function of rain intensity. Less difference between the mean value of the pdf and RS-derived estimates implies fewer over- and underestimations, and smaller variance indicates better confidence for the RS-derived estimates.

The results show when the three satellite-based algorithms estimate rainfall at ~10–20 mm day^{−1} intensity (Figs. 3a–f), the conditional pdfs of radar-derived measurements are similar and are concentrated in the ~0–30 mm day^{−1} range and skewed toward zero. At the intensity range from 40 to 60 mm day^{−1} (Figs. 3g–l), the shapes of the conditional pdfs become bell-like, but they start deviating from each other.

### b. Results about seasonal and spatial variation and intercomparison among RS-derived products

To further investigate the heteroscedasticity of satellite-based precipitation errors as a function of intensity, CIs are estimated at various rain rates for the three products. This is achieved by obtaining the confidence range from the conditional pdfs driven from the joint pdf of simultaneous radar- and satellite-derived precipitation values. For each individual rain estimate of a satellite-derived product, the CI represents the range in which 90% of the corresponding radar-observed intensities fall. We calculate the upper and lower CLs for the three RS-derived products over the three locations and three seasons listed above, at the confidence level of 90%

The results (Fig. 4) are described separately for the three aspects considered (i.e., seasons, regions, and different RS-based algorithms). To facilitate clarity for the interpretation of results, it is important to describe the information depicted in Fig. 4. The thick black lines give the unbiased situation across the intensity range. The colored curves represent the expected values of the conditional pdfs across the intensity range from 5 to 55 mm day^{−1}, and they are associated with symbols. If the colored curves are below the black line, it means their corresponding satellite-based precipitation measurements generally estimate rainfall intensity higher than what ground radar does. In this case, a flatter (i.e., less inclined) curve means more positively biased. The colored vertical lines represent the CIs that are associated with discretely sampled intensities (every 5 mm day^{−1}). The shorter the vertical lines are, the more confident their associated satellite-derived precipitation is. In summary, the colored curve lines of mean values are used to evaluate how accurately these satellite-derived precipitation measurements tend to estimate, and the colored vertical lines of CIs are used to evaluate how much these estimates disperse*.*

#### 1) Discussion of seasonal and spatial results

In regard to variations from warm (April–September) to cold seasons (October–March), each satellite-derived product has distinct features within a season. These features are also sensitive to the regional variations (Figs. 4d–i, see Tables B1–B3 in appendix B).

CMORPH overestimated somewhere between 17% and 65% of total precipitation in the warm season and underestimated ~13%–21% of total precipitation in the cold season (except for Montana). However, its CIs were ~31%–33% smaller in the warm season than in the cold season.

PERSIANN’s seasonal variation depends on the region under study. It overestimated (9%) in Florida’s warm season and underestimated (18%) in the cold season. In the Oklahoma region, it overestimated (60%) in the warm season and underestimated (2%) in the cold season. In both Florida and Oklahoma, it also showed 13%–30% smaller CI in the warm season than in the cold season.

TRMM-3B42-RTV7 shows consistent performance from warm to cold season in the Oklahoma region (−4% change in bias ratio and +2% change in CI). However, over Florida it exhibited similar behavior to other RS-derived products. It showed more bias in the warm season (34%) than in the cold season (6%).

With respect to Montana, the three RS-derived products showed very poor correspondence with ground-based observations in the cold season. In the analytical result of observed datasets (see Table B2 in appendix B), PERSIANN and TRMM-3B42-RTV7 had a CORR of 0.09 and −0.12, respectively. In addition, CMORPH showed only 2.82% out of 744 days (~21 days in the winter season over 9 years) when both CMORPH and radar estimate positive precipitation intensities. The number of rainy days captured by CMORPH was 86% less than PERSIANN and 78% less than TRMM-3B42-RTV7 during the same period. These problems are caused by a poor performance of PERSIANN and TRMM-3B42-RTV7 during the winter season over the Montana region, and CMORPH has serious detection problems as well.

#### 2) Discussion of intercomparison among RS-derived products

RS-derived products also show distinct features in terms of their bias and CIs. Here, we discuss the overall relative performance of the three RS-derived products under various scenarios (Fig. 4, see Tables B1–B3 in appendix B).

Over Oklahoma, CMORPH had 20% smaller CIs than TRMM-3B42-RTV7 and 9% smaller CIs than PERSIANN. But TRMM-3B42-RTV7 showed an 80% smaller bias than CMORPH and a 78% smaller bias than PERSIANN over the same region. PERSIANN outperformed TRMM-3B42-RTV7 in the wintertime over Oklahoma in both bias (PERSIANN’s −2% vs TRMM-3B42-RTV7’s −13%) and CIs (PERSIANN’s CIs are 3% less than TRMM-3B42-RTV7).

Over Florida, PERSIANN performed the best in term of bias (+2%) but had the largest CI among the three products. On the other hand, TRMM-3B42-RTV7 had the smallest CI but showed the most biased (+27%) estimates among the three.

The take-home message is that, at least during the warm season over Florida, neither product showed clear superiority over the others. The passive microwave (PMW)-based CMORPH- and TRMM-3B42-RTV7-derived products had ~2%–25% smaller CIs in the hurricane-dominated Florida region as opposed to Oklahoma (Figs. 4d, 4f, 4g, and 4i), while the infrared-based PERSIANN showed ~5%–21% larger CIs over Florida than over the convective-storm-dominated Oklahoma (Figs. 4d, 4f, 4g, and 4i).

## 4. Conclusions

The following conclusions can be drawn from our investigations.

The parametric joint pdf proposed for modeling the errors from three RS-derived products over three regions and during three seasons has proven to be an effective framework for studying the heteroscedasticity of the errors as a function of precipitation intensity. The forms of the pdfs are similar and can thus be easily parameterized and adapted to various situations such as different satellite-based algorithms, climates, and seasons.

The joint pdf form can be calibrated and validated using the indices such as CORR, BIAS, MSE, and MAE. After calibration, differences between these indices from the simulated samples and those from the observed datasets are less than 5%.

Using the joint pdf, one can observe that the shape of the conditional pdf shifts across the intensity ranges. At the ~10–20 mm day

^{−1}range the conditional pdf is L shaped, while in the ~40–60 mm day^{−1}range the conditional pdf becomes more bell shaped. This implies that the characteristics of errors in the satellite-derived precipitation vary over different intensity ranges.Our study’s results on seasonal variations are consistent with previously reported studies, such as Ebert et al. (2007), and provide additional information. In Ebert et al. (2007), the seasonal fluctuation in the accuracy of the satellite-derived precipitation is usually discussed using CORR and RMSE. These two indices are quite sensitive to the number of rainfall events. Therefore, the seasonal variation of accuracy could not be justified without compensating for the number of events. On the other hand, our parametric error model is insensitive to the number of samples; thus, it compensates for the seasonal fluctuations in rainfall events. Additionally, our model provides details about the degree of heteroscedasticity of the conditional pdfs with their associated mean values and CIs. This information provides the foundation for comparative and comprehensive studies related to RS-derived precipitations’ accuracy.

We also draw the following broader conclusion that, in general, no single satellite-derived precipitation product outperforms others with respect to the different test scenarios (i.e., seasons, regions, and climates) used in the study. Similarly, no individual product showed superior bias and CIs across the intensity ranges, even under the same test scenario. Our conclusion captures the essence of other reported studies that show that the accuracy rankings of the various satellite-derived precipitation products are highly dependent on the seasonal extreme precipitations, regions, elevations, and the evaluation metrics used (AghaKouchak et al. 2011; Gao and Liu 2013; Hirpa et al. 2010). Future work should focus on identifying and combining the strength of satellite-based precipitation measurements for various scenarios, rather than merely providing rankings for specific scenarios or with specific evaluation metrics.

Our study also has its own limitations. For one, there is a scenario where a large number of observations that fell below 0.2 mm day^{−1} were not included in the process of modeling

In summary, our findings suggest that the proposed two-dimensional joint pdf can effectively quantify the conditional pdf and CIs of the satellite-derived precipitation’s error under different scenarios. Our proposed parametric error model can also be applied to other climate and hydrometeorological variables.

## Acknowledgments

We thank University of California, Irvine, for providing PERSIANN datasets. PERSIANN is available on the FTP server

## APPENDIX A

### Determine the Type of Marginal Distributions and Copula

Sklar’s theorem suggests that a unique two-dimensional copula exists when the there is a two-dimensional joint cdf and its two marginal cdfs. Thus, to ensure there is a continuous joint cdf for the positive pair (*X*, *Y*), the continuous marginal distributions that fit the marginal dataset need to be identified. There are a number of distributions reported to be fitted into the daily precipitation (Li et al. 2013). Five candidate distributions (Weibull distribution, exponential distribution, lognormal distribution, general extreme value distribution, and Gamma distribution) were tested using the Kolmogorov–Smirnov two-sample test for three RS-derived datasets over a region in Oklahoma (Fig. 1). Then, the negative log likelihoods for these distributions were estimated. Among these distributions, the Weibull distribution passed the GoF tests for all three products and ground observations (Table A1), and it also shows the smallest negative log likelihood (Table A2). Therefore, the Weibull distribution is the most suitable distribution among the candidates.

Results of KS test after fitting marginal data into candidate distributions; larger *p* values indicate better fitting. Selected candidate is in boldface.

Results of negative log likelihood after fitting marginal data into different distributions; smaller values indicate better fitting. Selected candidate is in boldface.

The copula function’s type is closely related to the two variables’ dependence. We selected eight copula functions, of which five functions are Archimedean copulas and four functions are extreme-value copulas. The Gumbel–Hougaard copula is an Archimedean copula and an extreme-value copula. All of these eight copulas only have one parameter (Nazemi and Elshorbagy 2012). We estimated these copulas’ parameters based on Kendall’s tau for three RS-derived datasets over a region in Oklahoma and then test their GoFs. We followed a KS-test-based GoF test procedure recommended by Genest et al. (2009). The most suitable copula function is chosen based on the

GoF test results for candidate copulas; larger *p* values indicate better fitting. Selected candidate is in boldface.

## APPENDIX B

### Sensitivity Analysis of the Parameters in the Joint CDF

To determine the efficiency of the proposed calibration approach, a sensitivity analysis is conducted for each of the five parameters of the joint CDF.

First, we change the value of each parameter while fixing the other parameters to see how those difference-based indices respond to the parameter. We calculate the indices when each of the parameters varies within a small range (±20% of their values, which are shown in the first column of Table B1, for the case of CMORPH in Oklahoma; Tables B2 and B3 show results for Montana and Florida, respectively). In this way, we can see the relationship between every difference-based index and the parameters.

Total rainfall amount, total rainfall events, and error models’ parameters for the entire year, warm season (April–September), and cold season (October–March) for the study site in Oklahoma (34.875°N, 98.125°W), which has a humid subtropical climate. Calibration and verification results for these models are also provided.

As in Table B1, but for the study area in Montana (47.875°N, 107.625°W), which has a semiarid climate.

As in Table B1, but for the study area in Florida (27.625°N, 82.125°W), which has a humid subtropical climate.

The results are summarized in Fig. B1. The increase in copula parameter

Second, we change the value of each parameter while fixing the other parameter to see how each of the multiobjective components in the OF responds to the parameter. To do so, we calculate the four components in Eq. (7)

We also analyzed the parameters’ response to wet and dry years in the region of Oklahoma. We calculated the annual rainfall amount measured by ground radar to determine the wettest and driest year (green lines in Fig. B3). The result shows an insignificant relationship between annual rainfall amount and parameters. Longer period of datasets are required to study how wet and dry years affect the parameters.

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