Satellite multisensor precipitation products (SMPPs) have a variety of potential uses but suffer from relatively poor accuracy due to systematic biases and random errors in precipitation occurrence and magnitude. The censored, shifted gamma distribution (CSGD) is used here to characterize the Tropical Rainfall Measurement Mission Multisatellite Precipitation Analysis (TMPA), a commonly used SMPP, and to compare it against the rain gauge–based North American Land Data Assimilation System phase 2 (NLDAS-2) reference precipitation dataset across the conterminous United States. The CSGD describes both the occurrence and the magnitude of precipitation. Climatological CSGD characterization reveals significant regional differences between TMPA and NLDAS-2 in terms of magnitude and probability of occurrence. A flexible CSGD-based error modeling framework is also used to quantify errors in TMPA relative to NLDAS-2. The framework can model conditional bias as either a linear or nonlinear function of satellite precipitation rate and can produce a “conditional CSGD” describing the distribution of “true” precipitation based on a satellite observation. The framework is also used to “merge” TMPA with atmospheric variables from version 2 of the Modern-Era Retrospective Analysis for Research and Applications (MERRA-2) to reduce SMPP errors. Despite the coarse resolution of MERRA-2, this merging offers robust reductions in random error due to the better performance of numerical models in resolving stratiform precipitation. Improvements in the near-real-time version of TMPA are relatively greater than for the higher-latency research version.
Precipitation data are critical in a variety of subjects, including climate studies, meteorology, hydrology, and natural hazards. While precipitation is relatively easy to measure at a single point using a rain gauge, measurement over large regions at high spatial and temporal resolution is a major challenge. A “constellation” of Earth-observing satellite missions, including the now-defunct Tropical Rainfall Measuring Mission (TRMM) and the follow-on Global Precipitation Measurement (GPM) mission, co-led by the National Aeronautics and Space Administration (NASA) and the Japan Aerospace Exploration Agency, provides a mix of direct measurements of precipitation and related processes using active radar and indirect measurements using passive microwave (PMW) and infrared (IR). Satellite multisensor precipitation products (SMPPs) merge these various observations to create near-global precipitation records that approach two decades in length. Examples include the 3-hourly, 0.25° TRMM Multisatellite Precipitation Analysis (TMPA; Huffman et al. 2010, 2007); the 30-min, 8-km Climate Prediction Center (CPC) morphing technique (CMORPH; Joyce et al. 2004); and the hourly, 4-km Precipitation Estimation from Remotely Sensed Information Using Artificial Neural Networks (PERSIANN; Sorooshian et al. 2000). Most SMPPs are available in near–real time (with latency on the order of several hours), and some have non-real-time variants that utilize ground-based rain gauge information for bias correction. Launched in 2014, the GPM mission builds on TRMM’s legacy with an advanced active and passive instrument package. NASA’s 30-min, 0.1° Integrated Multisatellite Retrievals for GPM (IMERG; Huffman et al. 2014) dataset builds on more than a decade of experience with SMPPs, combining the strengths of TMPA, CMORPH, and PERSIANN and incorporating additional improvements.
Despite widespread interest in SMPPs, these datasets often exhibit considerable errors, both systematic (i.e., bias) and random, stemming from a variety of sources. Observation quality varies within the satellite constellation, with active radar being the most accurate, followed by PMW and IR. Sensor technology and resolution varies with age and mission. The current constellation of satellites provides a PMW observation for most locations on Earth approximately every 3 h, while radar observations are much less frequent. Between PMW measurements, algorithms typically use spatiotemporal interpolation of PMW or “infilling” using lower-accuracy IR. PMW observations tend to be more accurate nearer the tropics and for convective than for stratiform storm systems (Ebert et al. 2007) and are influenced by the underlying land or water surface, and microwave emissions from snow or ice-covered ground can be difficult to distinguish from emissions due to ice scatter in precipitating clouds (Ferraro et al. 2013; Ringerud et al. 2014; Tian and Peters-Lidard 2007). IR and PMW instruments have difficulties with orographic precipitation systems because of their shallow nature (Shige et al. 2013) and high variability in microscale and macroscale dynamics (Anders et al. 2007).
Given the potential usefulness of SMPPs, it is natural to want to characterize SMPP errors using an error model that compares SMPP against “ground truth,” that is, more reliable reference data (typically rain gauges or ground-based weather radar). Systematic error is usually heteroscedastic (i.e., depends on precipitation observation magnitude), a phenomena known as conditional bias (Ciach et al. 2000). Such errors tend to be multiplicative (Tian et al. 2013), with a magnitude that increases with precipitation observation intensity. Error models can be used to identify and thus remove systematic errors. They can also describe the statistical distribution of random errors, which can be understood as the residuals once the systematic error has been removed. Using this approach, individual random errors are irreducible without some sort of additional explanatory information.
SMPP characterization efforts (e.g., AghaKouchak et al. 2011; Behrangi et al. 2011; Tian et al. 2009) often distinguish between three error “cases”: false alarms, in which the SMPP reports precipitation while the reference data does not; misses, in which the reference reports precipitation while the SMPP does not; and hits, in which both report precipitation, but not necessarily of the same magnitude. Most error models that have been developed in the context of precipitation estimation using ground-based radar (AghaKouchak et al. 2010; Ciach et al. 2007; Germann et al. 2009) and SMPPs (Gebremichael et al. 2011; Sarachi et al. 2015; Yan and Gebremichael 2009) have tended to focus on hit cases only.
Several previous SMPP error models have considered false alarms, misses, and hits separately, and then recombine these separate descriptions to create an overall estimated distribution of true precipitation. For example, the Precipitation Uncertainties for Satellite Hydrology framework (PUSH) introduced by Maggioni et al. (2014) uses a gamma distribution to describe the precipitation intensity associated with misses, exponential decay, and linear regression models to describe the probability and intensity associated with false alarms, and a generalized linear model is used to generate a gamma distribution of precipitation magnitude associated with hits. PUSH also uses a uniform distribution to describe possible trace precipitation associated with cases where neither the SMPP nor reference data report precipitation. For any zero or nonzero SMPP observation, a probability distribution can be generated by combining these cases. The two-dimensional satellite rainfall error model (SREM2D) introduced by Hossain and Anagnostou (2006) takes a somewhat similar approach, but incorporates spatial and temporal autocorrelation functions to construct ensembles of correlated precipitation fields.
This study applies a new censored, shifted gamma distribution (CSGD) methodology to characterize precipitation and create an SMPP error model that produces a “best guess” distribution of the true precipitation by considering hits, misses, and false alarms. The CSGD technique presented in this paper is arguably simpler than most, and comparison with the PUSH error model suggests that this relative simplicity is advantageous.
Previous precipitation error model studies have generally focused on relatively small geographic areas where spatial stationarity of rainfall and model parameters can be assumed; however, these approaches have not explored spatial variability in these parameters or in model performance. This study is one of the few, along with Maggioni et al. (2016), that applies an error model over a large region to better understand SMPP performance characteristics and how they are tied to physiographic and climatological features.
This study moves beyond the traditional notions of precipitation error modeling toward error correction by allowing the incorporation of additional information to reduce random errors. Previous researchers have suggested that topography and other land surface characteristics as well as other atmospheric variables such as humidity could help understand and, in principle, correct SMPPs (Gebregiorgis and Hossain 2013; Gebremichael et al. 2011). As far as we are aware, this study is the first to explore the potential benefit of incorporating atmospheric variables such as humidity and precipitation from numerical weather models (specifically, atmospheric reanalysis) in a satellite precipitation error model to reduce SMPP random errors. This is a promising approach since the complementary performances of numerically simulated and remotely sensed precipitation estimates provide the opportunity to produce merged datasets with smaller systematic and random errors.
The SMPP, ground reference, and atmospheric reanalysis datasets utilized in this study are described in section 2. The CSGD and the CSGD-based precipitation error modeling and correction frameworks are introduced in section 3. Results for precipitation characterization and SMPP error modeling are provided in section 4. A summary and closing discussion follow in section 5.
This study focuses on daily-scale, 0.25° (approximately 25 km) precipitation over the conterminous United States (CONUS; see Fig. 1). This large geographic extent allows us to robustly demonstrate not only how the CSGD can be used to characterize precipitation and how the CSGD-based error modeling framework can correct for biases and characterize remaining uncertainties, but also how these features vary with climatic and physiographic controls.
We examine two variants of TMPA version 7.0 (also known as TRMM 3B42). TMPA merges PMW, active radar, and IR observations from multiple satellites to create a near-global (±50° latitude) rainfall dataset with 3-hourly, 0.25° resolution. The “research” version includes a monthly rain gauge–based bias correction and is available approximately 2 months after real time. In this study, analyses using this version cover 1998–2014. Several analyses consider TMPA-RT, which is available approximately 8 h after real time and only includes a gauge-based climatology correction. Such near-real-time analyses cover 2000–14, since the pre-2000 TRMM orbit precludes near-real-time analysis. “TMPA” is used to refer to the research version and “TMPA-RT” for the near-real-time version. The TRMM satellite ceased operations in April 2015, but the TMPA product is continuing to be produced by leveraging other satellites in the constellation. NASA’s recent IMERG SMPP was not used in this study, since at the time of writing it was only available for 2014 onward.
We use the “File A” precipitation forcing from phase 2 of NASA’s National Land Data Assimilation System (NLDAS-2; Xia et al. 2012a,b) as the reference. NLDAS-2 precipitation has hourly, 0.125° resolution, disaggregated from daily CPC-Unified gauge analysis (Chen et al. 2008; Xie et al. 2007) and features a statistical topographic correction based on the PRISM climatology by Daly et al. (1994). NLDAS-2 was selected rather than the Stage IV bias-corrected radar rainfall dataset that has been used in some SMPP validation studies (AghaKouchak et al. 2011; Qiao et al. 2014) since visual inspection of Stage IV revealed very poor performance in mountainous regions. We have aggregated NLDAS-2 from its hourly 0.125° resolution to the same daily 0.25° resolution as the TMPA data. Thus, the NLDAS-2 precipitation values used in this study are very similar, but not exactly identical, to CPC-Unified, which has been used in several previous SMPP error characterizations (Maggioni et al. 2016, 2014; Tian et al. 2013). The reader is referred to Ferguson and Mocko (2017) for a detailed explanation of the data sources utilized to create the NLDAS-2 precipitation forcing.
Though there is likely overlap in terms of the rain gauges used to create NLDAS-2 and those used to bias correct the research version of TMPA, the CSGD-based framework does not require strict independence of SMPP and reference data. This study assumes that NLDAS-2 is free of errors, which is of course never the case for any dataset, let alone a continental-scale one such as NLDAS-2. Rain gauge undercatch errors in gridded rain gauge datasets can be substantial, particularly for snowfall and for extreme rainfall (Adam and Lettenmaier 2003). NLDAS-2 does not use a gauge undercatch correction and thus probably underestimates true precipitation. It should be noted that the monthly gridded rain gauge data used to bias correct TMPA does use an undercatch correction. Thorough investigation of the role of gauge undercatch errors in satellite precipitation validation is beyond the scope of this study.
We also present analyses that utilize surface precipitation rate and vertically integrated total precipitable water (TPW) from Version 2 of the Modern-Era Retrospective analysis for Research and Applications (MERRA-2; Bosilovich et al. 2015; Rienecker et al. 2011) from NASA. MERRA-2 is generated using an atmospheric model that assimilates a range of surface and atmospheric observations including satellite PMW. MERRA-2 outputs have hourly, 0.5° latitude × 0.625° longitude resolution. It is unnecessary to regrid the MERRA-2 datasets to the 0.25° resolution of TMPA for this study, but the same daily temporal resolution is used. Though MERRA-2 provides several surface-level precipitation outputs, including a version primarily based on rain gauges, we use model internally generated precipitation to ensure greater independence from TMPA and NLDAS-2 and to illustrate the value of numerically generated precipitation and other atmospheric variables for reducing SMPP errors.
The precipitation datasets utilized in this study consider all seasons and precipitation phases (i.e., rain, snow, hail, etc.), represented in terms of depth of liquid water. Determination of precipitation phase is a challenge in gridded precipitation datasets, whether the underlying data come from rain gauge networks, satellites, ground-based radar, or numerical models.
We treat data prior to 2014 as the “training period,” that is, used for model parameter estimation as well as error analysis. Data from 2014 are used as “validation,” to assess model robustness when used outside of the training period. Though this training period is much longer than the validation period, this typifies many settings in which an error model might be used, since many reference datasets date at least as far back as most or all SMPPs.
a. The CSGD
The two-parameter gamma distribution has been used in precipitation modeling since at least Das (1955). Like precipitation itself, the gamma distribution is left-bounded at zero and can take many possible “shapes,” in terms of its density and cumulative distribution function (CDF). Generally, a precipitation process can be modeled in two steps using a total of three parameters. First, the probability of occurrence is modeled via a Bernoulli trial with the “success” parameter equal to the probability of precipitation (POP). Second, the nonzero precipitation magnitude is modeled via the two-parameter gamma with shape parameter and scale parameter expressed using the distribution mean and standard deviation by
The CSGD is an alternative formulation presented in Scheuerer and Hamill (2015) in which the CDF is “shifted” left and subsequently left-censored at zero, meaning all negative values are replaced by zero. Thus, the density to the left of zero represents the probability of zero precipitation (), while the density to the right of zero represents the likelihood of a particular nonzero value. To achieve this, a shift parameter , is introduced such that, if denotes the CDF of a gamma distribution, then the CDF of the CSGD model is defined by
where x is rainfall depth. In this way, the CSGD eliminates the initial Bernoulli trial from the precipitation modeling process, though the introduction of means the total number of parameters remains at three. Thus, while the conventional gamma distribution has the property that (i.e., the CDF is equal to zero at zero rainfall depth), the CDF of a CSGD has the property (see Fig. 2). Scheuerer and Hamill (2015) provide details for CSGD parameter estimation based on minimization of the continuous ranked probability score, which essentially minimizes the integrated quadratic distance between the empirical and theoretical CSGD distribution functions.
CDFs for “climatological CSGDs” (to distinguish from conditional CSGDs, described in section 3b) are shown for the 0.25° grid cells nearest to Charlotte, North Carolina, and Denver, Colorado (top panel of Fig. 3). These demonstrate good fit to the empirical CDFs while highlighting the differences between locations and between TMPA and NLDAS-2.
b. CSGD-based error modeling and correction framework
The climatological CSGD is insufficient for generating a distribution of estimated “true” precipitation values (or, equivalently, a distribution of SMPP errors) based on a given observation at time , since the mean , standard deviation , and perhaps shift depend on the magnitude of . Thus, we use a CSGD-based error modeling framework to reduce systematic SMPP biases and to model and reduce SMPP random errors. The framework was first introduced in Scheuerer and Hamill (2015) and further explored in Báran and Nemoda (2016) for statistical postprocessing of ensemble numerical precipitation forecasts. The CSGD-based approach uses a statistical regression model “trained” using a past record of contemporaneous satellite and reference observations. The regression model is then conditioned using a satellite observation for time t to generate “conditional CSGD” parameters , , and from the climatological CSGD parameters , and .
In the simplest version, increases linearly with and increases proportionally to the square root of . Allowing to vary offers little benefit and can lead to parameter estimation difficulties (M. Scheuerer 2017, personal communication). We will refer to this version as the “linear model,” since it models conditional bias linearly with precipitation rate. It has the form
where denotes the mean of the SMPP time series. Example CDFs of conditional CSGDs are shown in the lower panel of Fig. 3 for values of 2.5 and 25 mm day−1 for the 0.25° grid cells nearest to Charlotte, North Carolina, and Denver, Colorado. These show that as increases, the probability of the true precipitation being zero decreases [approaching zero for = 25 mm day−1] while the probability of higher true values increases. The value of will always be nonzero and greater than the conditional median at time t, which will be equal to zero when the conditional POP is less than 0.5.
Scheuerer and Hamill (2015) also present a more complex version that can account for nonlinearity in conditional bias. This model, hereinafter called the nonlinear model, has the form
where and .
The regression framework can also accommodate an arbitrary number n of additional contemporaneous covariates such as TPW, temperature, or humidity from atmospheric observations or simulations. In this case, Eq. (3) expands to
and is the mean of the time series of the ith covariate. A similar variant of the nonlinear model [Eq. (6)] could be written to include covariates. The inclusion of covariates allows for additional information to be introduced to the SMPP–reference intercomparison, allowing the explanation of some of the residual (i.e., random) error. We use the techniques described in Scheuerer and Hamill (2015) to estimate the parameters of the CSGD correction framework.
The models described above are consistent with the notions that satellite errors are multiplicative (Tian et al. 2013) and that error magnitude grows with . They bear passing resemblance to the PUSH model of Maggioni et al. (2014), in that the conditional distribution of estimated true precipitation given is assumed to be gamma distributed, though we use the three-parameter CSGD rather than the conventional two-parameter gamma used to model precipitation hits in PUSH. This allows for the possibility of the estimated true precipitation to be zero, even if (i.e., a false alarm) or vice versa (missed precipitation). PUSH, in contrast, accounts for false alarms and misses using separate models, making it impossible to construct a theoretical distribution for estimated true precipitation and involves additional parameters. Like PUSH, the CSGD framework has the advantage of being parametric, which can be helpful in conditions of very low or very high precipitation rates (Gebremichael et al. 2011; Zhang et al. 2013).
4. Results and discussion
a. CSGD-based precipitation characterization
Estimates of , , and for 1998–2013 for NLDAS-2 and TMPA are compared for every grid cell over the CONUS (Fig. 4). All three parameters in both TMPA and NLDAS-2 exhibit higher values in the eastern United States and the Pacific coastal mountains than in the western United States. This should be expected as a result of the higher amounts of precipitation in these parts of the country (see Fig. 1). TMPA tends to overestimate and and underestimate relative to NLDAS-2, except in the Pacific coastal mountains and the Rocky Mountains. Differences in and in the western United States are lower in magnitude, though the relative differences are approximately uniform except for over mountains. Isolated or small clusters of seemingly anomalous parameter values can be seen in TMPA but not in NLDAS-2. Visual inspection shows that these are collocated with water bodies such as lakes and reservoirs that are known to influence PMW-based precipitation estimates (Tian and Peters-Lidard 2007).
POP cannot be evaluated directly from Fig. 4. Over the CONUS, POP for TMPA is more uniform and significantly lower than in NLDAS-2, suggesting that the precipitation detection limits imposed by the satellite sensors or processing algorithms exert strong controls (Fig. 5). The TRMM sensor package was designed to detect moderate to heavy rainfall and thus tends to underestimate light precipitation and mixed phase/falling snow. GPM can see a much broader spectrum of precipitation. As with the parameter estimates in Fig. 4, anomalous isolated POP values are collocated with water bodies. We do not explore this issue further in this study, but Maggioni et al. (2014) suggest that a minimum detection threshold of 0.25 mm day−1 may be a reasonable approximation in TMPA, and their PUSH error model utilizes this threshold to distinguish between precipitation and nonprecipitation. The linear and nonlinear conditional CSGD models described in section 3b do allow for nonzero true precipitation even when , and thus the CSGD approach need not explicitly consider detection thresholds.
b. Error modeling using the conditional CSGD framework
Before showing CONUS-wide error modeling and correction results using the CSGD framework, we provide a more detailed illustration of the linear and nonlinear models and comparison with the PUSH model from Maggioni et al. (2014) for the 0.25° grid cell nearest to Charlotte, North Carolina (Fig. 6). The models and data, including the 1998–2013 training period and 2014 validation period, are shown on both linear (left panels) and logarithmic scales (right panels). For both Charlotte and other locations across the CONUS, TMPA tends to overestimate at higher precipitation rates. This overestimation is consistent with previous studies (AghaKouchak et al. 2011; Tian et al. 2009) and may be due to the joint effect of TMPA’s monthly bias correction and poor light precipitation detection, which would tend to introduce a high bias in precipitation magnitude (Tian et al. 2009; Wright et al. 2017). However, since NLDAS-2 does not account for gauge undercatch, it almost certainly underestimates true heavy precipitation to an unknown degree. Thus, the extent to which TMPA overestimates true precipitation for large events is difficult to assess without a more detailed reference dataset.
The linear and nonlinear versions of the CSGD-based error model provide good fits to the data for both the training and validation periods, and the nonlinear variant better captures the nonlinearity in conditional bias that is evident in high precipitation. PUSH greatly overestimates conditional bias for high precipitation, and no points fall outside of the lower bound of that model’s 95% spread, which is unrealistic given the relatively large sample size. In contrast, approximately 5% of points fall outside of the 95% quantile spread for the CSGD model (note that not all data points are clearly visible in Fig. 6, particularly those that fall very close to either axis).
We evaluate a range of conditional CSGD error model complexities, specifically, models using different versions of Eqs. (3), (6), and (7) to estimate . CONUS-wide evaluation using root-mean-square error (RMSE) from two versions, the linear model without covariates and the nonlinear model with MERRA-2 precipitation, is shown in Fig. 7. Here and in subsequent calculations using CSGD error models, RMSE and other error metrics are computed between NLDAS-2 and the conditional CSGD median. As noted in section 3b, the conditional CSGD mean is always nonzero and greater than the median, which for low precipitation rates can be equal to zero. This means that neither the conditional mean nor median are ideal measures of the central tendency, but investigation of a more appropriate summary statistic is beyond the scope of this study. The linear model improves upon the TMPA dataset (i.e., reduces RMSE) except in the Rockies and Pacific coastal mountains, where performance is poor. The nonlinear model with MERRA-2 precipitation offers further improvement, including in these mountainous areas. Reductions in RMSE are greatest in the northern part of the country (particularly in the nonlinear model with MERRA-2 precipitation) and in the high-altitude but lower-relief portions of the Intermountain West such as the upper Rio Grande in southern Colorado and northern New Mexico and the Snake River Plain in southern Idaho.
The substantial improvements provided by the nonlinear model with MERRA-2 covariates in the northeastern and northwestern parts of the country are likely attributable to the relatively higher proportion of stratiform precipitation in those regions, which is generally better estimated by atmospheric models than by satellite sensors. The more complex model also improves upon simpler versions in most of the Rockies and West Coast mountains. Visual inspection of results for a range of models reveals that most of this improvement stems from inclusion of MERRA-2, rather than from the nonlinear model structure (results not shown). Error reductions are associated with the identification and removal of systematic errors and, in the case of models that include MERRA-2 covariates, some further reduction of random errors.
We compute the RMSE and mean absolute error (MAE) normalized by the mean daily precipitation (henceforth referred to as NRMSE and NMAE, respectively) for each 0.25° grid cell across the CONUS for a range of CSGD model configurations. This allows us to compare the relative reduction in errors achieved in various precipitation hydroclimates. Results are then summarized by computing the CONUS-wide median and interquartile range (IQR) of NRMSE and NMAE (Table 1). These nonparametric summary statistics were chosen rather than the mean and standard deviation because in arid parts of the country, normalizing by a daily mean precipitation close to zero can produce spurious results.
The NRMSE and NMAE for the uncorrected TMPA dataset show slightly increased accuracy for the validation period, relative to the training period, possibly associated with improvements in the number and quality of satellite sensors over the lifetime of TMPA. In contrast, the error statistics for the CSGD models tend be unchanged or slightly worse for the 2014 validation period, though in all cases the validation performance is within 7% of the reference period in terms of RMSE and within 5% in terms of MAE, suggesting relatively robust model performance.
The linear (nonlinear) model improved median NRMSE by 20% (22%) and median NMAE by 17% (19%) for the training period, with similar performance in the validation period. MERRA-2 covariates improved upon this “baseline” CSGD model performance. The inclusion of MERRA-2 precipitation offers robust improvements to both NRMSE and NMAE (32% and 33%, respectively, in the case of the nonlinear version). Inclusion of MERRA-2 TPW alone (i.e., without MERRA-2 precipitation) offers very little improvement in both the linear and nonlinear models. When both MERRA-2 TPW and precipitation are included, neither linear nor nonlinear models show much improvement over when only the precipitation covariate is included. This implies that precipitation from MERRA-2 is a much stronger predictor of true precipitation than TPW. It also suggests that MERRA-2 precipitation and TPW are highly correlated, which is unsurprising.
A linear CSGD error model was tested in which the size of the TMPA and NLDAS-2 samples at each grid cell were expanded by concatenating the data from the eight adjacent grid cells for model fitting. Referred to in Table 1 as “linear with spatial pooling,” this model produced similar results to the linear model fitted only to data from individual grid cells (“linear” in Table 1). This has several implications. In complex terrain or near water bodies, precipitation can vary over relatively short distances. In such cases, spatial pooling may create an enlarged sample that does not properly represent precipitation statistics in the grid cell in question. Visual inspection of RMSE maps shows similar performance between pooled and unpooled linear CSGD models in the eastern portion of the country and lower performance using pooling in the Intermountain West, consistent with this intuition (results not shown). In addition, the value added through spatial pooling is inherently limited if there is substantial spatial correlation in the precipitation estimates and errors between adjacent grid cells. The similar performance between pooled and unpooled models in less varied terrain also implies that the model-fitting procedure is relatively robust to small samples.
We evaluate the relationships between errors in TMPA, as a function of correlation between TMPA and NLDAS-2, before and after applying a nonlinear CSGD model with MERRA-2 precipitation (Fig. 8). The influence of land surface elevation, as a proxy for topographic relief, is also evaluated, since this impact is somewhat difficult to assess in Fig. 7. Both the absolute values and the variability in NRMSE and NMAE are relatively low for locations with high correlation, while the variability (though not the central tendency) in these statistics increases for locations with lower correlation, and there is a relatively weak inverse relationship between error magnitude and correlation between the SMPP and reference. Neither correlation nor elevation appears to be the primary controls on NRMSE or NMAE, even though correlation values for higher-elevation locations tend to be relatively low. It also appears from Fig. 8 that similar reductions in NRMSE and NMAE can be achieved regardless of correlation or land surface elevation. Qualitatively similar results were produced with the simpler linear model (not shown).
Like NRMSE and NMAE, correlation between the uncorrected TMPA and NLDAS-2 is slightly higher in the validation period than the training period, again likely associated with improvements in the quality and number of sensors. Interestingly, linear and nonparametric correlations between corrected SMPP time series and NLDAS-2 reduce somewhat when TMPA is fed through a linear CSGD model without covariates and remain relatively unchanged when a nonlinear model is used instead (Table 2). This may be because of the limitations of using either the CSGD mean or median and because of the implicit bias adjustment in the CSGD framework. When MERRA-2 precipitation is included as a covariate, however, correlation between the corrected SMPP time series and NLDAS-2 increases. This highlights the ability of MERRA-2 covariates (particularly precipitation) to reduce random errors in TMPA.
We also examined the real-time version (TMPA-RT) with several CSGD models (Table 3). NRMSE and NMAE in the original TMPA-RT dataset are 14% larger in terms of NRMSE and 8% larger in terms of NMAE than the research version analyzed previously. Results are qualitatively similar to Table 1, with all CSGD models showing improvement over the uncorrected TMPA-RT dataset, and with the largest improvements coming from the nonlinear model with MERRA-2 precipitation. Likewise, error statistics are generally comparable for the 2014 validation period, showing minimal loss of performance as compared to the training period. The degree of error reduction achieved by the CSGD models is greater using TMPA-RT than TMPA. For example, relative to the uncorrected TMPA-RT, the linear CSGD model reduced NRMSE (NMAE) by 25% (20%), while the same model reduced error for the research version by 20% (17%). Reduction in NRMSE (NMAE) relative to the uncorrected TMPA-RT was as high as 39% (37%) for the nonlinear CSGD with MERRA-2 precipitation. These results are consistent with the notion that error models identify and remove systematic biases, since Maggioni et al. (2016) reported higher systematic errors in TMPA-RT than the research version.
c. Parameter sensitivity
The results for the validation period shown in Tables 1 and 2 provide an initial indication that the CSGD framework can be applied outside of the training period. To investigate this issue further, we re-estimate the CSGD parameters for NLDAS-2 and TMPA, as well as the regression parameters for the linear version of the conditional CSGD model for each year individually from 1998 to 2013 and for successively longer time periods (i.e., 1998–99, 1998–2000, etc.) for the grid cell nearest to Charlotte, North Carolina (Fig. 9). While parameters vary somewhat from year to year, estimates using longer time periods converge to relatively stable values after several years. Exceptions are the slight downward trend in and upward trend in . It is well known that the spatial and temporal statistical consistency of precipitation datasets vary according to input data availability, such as the number of rain gauges (Hamlet and Lettenmaier 2005) or the quality and type of satellite sensor (Cho and Chun 2008). The trends in and are consistent with improvement in precipitation estimation in TMPA (i.e., reduction in the weight given to the regression intercept and increase in weight given to ). Parameters for the nonlinear model and for other locations are similarly stable over time (results not shown).
These results suggest that the continuous ranked probability score-based parameter estimation procedure for the climatological CSGD and the conditional CSGD regression framework is relatively efficient with respect to data requirements and that several years of coincident reference data may be sufficient. It would be worthwhile to evaluate this issue using error metrics such as RMSE or MAE. We leave this as a topic of future work, though it is worth noting that Scheuerer and Hamill (2015) found relatively poor conditional CSGD performance with a 1-yr training sample but good performance with modest increases in training record length.
5. Summary and discussion
Using the censored, shifted gamma distribution (CSGD), we characterize the climatology of daily precipitation over the CONUS of TMPA, a satellite multisensor precipitation product (SMPP) and NLDAS-2, a reference (i.e., rain gauge based) dataset. We also use a conditional CSGD error modeling framework to quantify and reduce errors in TMPA. The CSGD describes both precipitation occurrence and magnitude and reveals significant differences between TMPA and NLDAS-2, including poor satellite-based estimation over inland water bodies and mountainous regions. The CSGD-based error modeling framework considers errors both in the detection and magnitude of precipitation and can model systematic bias either as a linear or nonlinear function of precipitation rate. Both versions perform better than an existing error model from Maggioni et al. (2014) over a wide range of precipitation magnitudes for daily precipitation.
The framework suffers most in areas of high topographic relief (though not necessarily in areas of high elevation). Error reduction at a specific location depends on the relative balance of systematic and random error in the SMPP at that location. Preliminary analyses demonstrate that parameter estimation of both the CSGD and the CSGD-based error framework are relatively insensitive to record length for periods of record longer than several years.
In addition, we show that errors in TMPA can be reduced by incorporating covariates from the MERRA-2 atmospheric reanalysis, despite its relatively coarse resolution. This is the first study that we are aware of in which the potential benefits of merging numerical weather prediction and SMPP is explored quantitatively. Precipitation from MERRA-2 offers robust increases in performance, particularly in mountainous areas, while MERRA-2 precipitable water provides little improvement. The improvements offered by MERRA-2 appear to be due to the better performance of numerical models relative to satellite-based instruments in resolving stratiform precipitation. Other numerical weather models that have higher resolution or that assimilate more independent observations would likely provide additional improvement.
It should be emphasized that precipitation error models can only isolate and thus remove systematic errors. The errors remaining after the removal of systematic bias, that is, the random errors, can be described statistically but not reduced or eliminated. The variability in these residuals can only be explained via the inclusion of additional information. Except for models that include MERRA-2 covariates, therefore, the error reductions shown throughout section 4b stem solely from the identification and removal of systematic errors. MERRA-2 covariates can explain some amount of residual (i.e., random) error, as evidenced by the further reductions in errors and increased correlations.
The error reduction achieved in this study is generally consistent with the levels of systematic error found over the eastern United States at the same spatial and temporal resolution by Maggioni et al. (2016), though more work is needed to reconcile discrepancies between the degree of systematic bias shown here and shown by those authors in the Intermountain West. Also consistent with Maggioni et al. (2016), improvements in TMPA-RT were relatively greater than for the gauge-corrected non-real-time version, suggesting that the CSGD approach has particular advantages for near-real-time applications. The CSGD approach, coupled with real-time numerical weather prediction estimates such as those generated using NASA’s GEOS-5 (Rienecker et al. 2008), offer a pathway to improve the accuracy of near-real-time SMPP and for parameterizing remaining random errors.
Certain relevant issues were not explored in this study. Maggioni et al. (2014) concluded that seasonally varying model parameters offered no major advantage in their error model, and our initial investigations into seasonality, which are omitted here in the interest of brevity, confirm this. Errors in the NLDAS-2 reference data, including those due to rain gauge undercatch, were not considered and can be significant, particularly in the cold season and in steep terrain.
Many applications, such as hydrologic modeling, can require subdaily precipitation inputs. SMPP errors in magnitude grow with increasing resolution. The autocorrelation of daily precipitation is relatively low, but increases as temporal resolution becomes finer. Thus, generating a realistic high-resolution time series of precipitation using the CSGD approach or other error models requires consideration of this autocorrelation. The same is true for generating spatially correlated precipitation fields.
One key challenge with the CSGD framework, and precipitation error modeling more generally, is transferability to regions that lack reference data. This issue requires significant further effort, but several previous studies have shown promise (Gebregiorgis and Hossain 2014, 2013). The CSGD framework would be a strong candidate for such efforts because of the relatively simple structure, robust performance, and the ability to include relevant atmospheric variables from numerical weather prediction, which may potentially be even more useful in data-limited settings. Resolving such issues would constitute a major step toward quantifying and reducing errors in satellite precipitation estimates and helping users to better understand the implications of remaining irreducible random errors.
This work was supported by NASA’s Precipitation Measurement Mission Grant Number NNX16AH72G and the Wisconsin Alumni Research Foundation and used computing resources and assistance from the UW-Madison Center for High Throughput Computing (CHTC). CHTC is supported by UW-Madison, the Advanced Computing Initiative, the Wisconsin Alumni Research Foundation, the Wisconsin Institutes for Discovery, and the National Science Foundation and is an active member of the Open Science Grid, which is supported by the National Science Foundation and the U.S. Department of Energy. We thank Dr. Michael Scheuerer at the University of Colorado Cooperative Institute for Research in Environmental Sciences and NOAA/Earth System Research Laboratory for providing advice and his original CSGD code. We also thank Dr. Viviana Maggioni at George Mason University for direction regarding PUSH.