Editorial Type:
Article
Article Type:
Research Article

The Sensitivity of Rainfall Estimation to Error Assumptions in a Bayesian Passive Microwave Retrieval Algorithm

Gregory S. Elsaesser Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado

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Christian D. Kummerow Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado

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Print Publication:
01 Feb 2015
DOI:
https://doi.org/10.1175/JAMC-D-14-0105.1
Page(s):
408–422
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Abstract

The Goddard profiling algorithm (GPROF) uses Bayesian probability theory to retrieve rainfall over the global oceans. A critical component of GPROF and most Bayes theorem–based retrieval frameworks is the specification of uncertainty in the observations being utilized to retrieve the parameter of interest. In the case of GPROF, for any sensor, uncertainties in microwave brightness temperatures (Tbs) arise from radiative transfer model errors, satellite sensor noise and/or degradation, and nonlinear, scene-dependent Tb offsets added during sensor intercalibration procedures. All mentioned sources impact sensors in a varying fashion, in part because of sensor-dependent fields of view. It is found that small errors in assumed Tb uncertainty (ranging from 0.57 K at 10 GHz to 2.29 K at 85 GHz) lead to a 3.6% change in the retrieved global-average oceanic rainfall rate, and 10%–20% (20%–40%) shifts in the pixel-level (monthly) frequency distributions for given rainfall bins. A mathematical expression describing the sensitivity of retrieved rainfall to uncertainty is developed here. The strong global sensitivity is linked to rainfall variance scaling systematically as Tb varies. For ocean scenes, the same emission-dominated rainfall–Tb physics used in passive microwave rainfall retrieval is also responsible for the substantial underestimation (overestimation) of global rainfall if uncertainty is overestimated (underestimated). Proper uncertainties are required to quantify variability in surface rainfall, assess long-term trends, and provide robust rainfall benchmarks for general circulation model evaluations. The implications for assessing global and regional biases in active versus passive microwave rainfall products, and for achieving rainfall product agreement among a constellation of orbiting microwave radiometers [employed in the Global Precipitation Measurement (GPM) mission], are also discussed.

Corresponding author address: Gregory Elsaesser, Dept. of Atmospheric Science, Colorado State University, 1371 Campus Delivery, Fort Collins, CO 80523-1371. E-mail: elsaesser@atmos.colostate.edu

Abstract

The Goddard profiling algorithm (GPROF) uses Bayesian probability theory to retrieve rainfall over the global oceans. A critical component of GPROF and most Bayes theorem–based retrieval frameworks is the specification of uncertainty in the observations being utilized to retrieve the parameter of interest. In the case of GPROF, for any sensor, uncertainties in microwave brightness temperatures (Tbs) arise from radiative transfer model errors, satellite sensor noise and/or degradation, and nonlinear, scene-dependent Tb offsets added during sensor intercalibration procedures. All mentioned sources impact sensors in a varying fashion, in part because of sensor-dependent fields of view. It is found that small errors in assumed Tb uncertainty (ranging from 0.57 K at 10 GHz to 2.29 K at 85 GHz) lead to a 3.6% change in the retrieved global-average oceanic rainfall rate, and 10%–20% (20%–40%) shifts in the pixel-level (monthly) frequency distributions for given rainfall bins. A mathematical expression describing the sensitivity of retrieved rainfall to uncertainty is developed here. The strong global sensitivity is linked to rainfall variance scaling systematically as Tb varies. For ocean scenes, the same emission-dominated rainfall–Tb physics used in passive microwave rainfall retrieval is also responsible for the substantial underestimation (overestimation) of global rainfall if uncertainty is overestimated (underestimated). Proper uncertainties are required to quantify variability in surface rainfall, assess long-term trends, and provide robust rainfall benchmarks for general circulation model evaluations. The implications for assessing global and regional biases in active versus passive microwave rainfall products, and for achieving rainfall product agreement among a constellation of orbiting microwave radiometers [employed in the Global Precipitation Measurement (GPM) mission], are also discussed.

Corresponding author address: Gregory Elsaesser, Dept. of Atmospheric Science, Colorado State University, 1371 Campus Delivery, Fort Collins, CO 80523-1371. E-mail: elsaesser@atmos.colostate.edu
Keywords: Climate records; Microwave observations; Remote sensing; Satellite observations; Sensitivity studies; Bayesian methods

1. Introduction

Quantifying global precipitation at a 3-hourly temporal resolution is a goal of the Global Precipitation Measurement (GPM; Hou et al. 2014) mission. To achieve such a resolution, GPM will utilize the observations from all existing spaceborne microwave (window channel and sounder) radiometers in conjunction with data from the new GPM Core Observatory satellite carrying both a dual-frequency precipitation radar (DPR) and a multichannel microwave imager (GMI). These new measurements will advance our understanding of the global water and energy budgets, serve to constrain numerical weather prediction models, and enable improved quantification of the global precipitation distribution (including snowfall, drought frequency, and flood-producing rainfall intensity/frequency).

The precipitation distribution, in addition to the mean precipitation state, may systematically change as the climate warms (Allan and Soden 2007; Allan et al. 2013; Lau et al. 2013; Scoccimarro et al. 2013). Such changes can have large societal impacts (e.g., effects on crop growth, natural hazard planning, and water management). Quantifying trends or systematic shifts in the distribution has proven difficult since intersatellite product differences have often been found to exceed the differences expected to be associated with climate change (e.g., John et al. 2009). Such rainfall product discrepancies arise when each satellite sensor’s characteristics (e.g., active vs passive observations, potential drift with time, calibration differences, field-of-view differences, channel differences) impact a retrieval algorithm to such an extent that the final retrieved states differ. From the passive microwave (PMW) instrument perspective, such issues have motivated extensive intercalibration efforts (Ruf 2000; Yan and Weng 2008; Biswas et al. 2010; Yang et al. 2011; Berg et al. 2013; Sapiano et al. 2013; Wilheit 2013) and improvements to algorithms (Kummerow et al. 2011) with the expectation that once discrepancies in microwave-based precipitation records can be understood and reconciled, robust climate data records can be created.

The state-of-the-art PMW rainfall retrieval algorithms, particularly ones applied to oceanic scenes, typically incorporate Bayesian inversion methods [e.g., the ocean algorithm component of the Goddard Profiling algorithm (GPROF; Kummerow et al. (2001)]. A key component of a Bayesian inversion framework is the specification of uncertainty in the observations (and radiative transfer model, if so used). At present, it is not known how sensitive any given PMW sensor rainfall record is to assumed uncertainty, let alone accounting for varying sensor-to-sensor uncertainty and the implications for multisensor rainfall product agreement. An understanding of the sensitivity of PMW retrieved rainfall to the specified uncertainty is even more important with the launch of GPM, since the overland rainfall retrieval algorithm is transitioning from an inversion technique based on rainfall–brightness temperature scattering relationships (Ferraro and Marks 1995; Ferraro 1997) to a Bayesian framework consistent with the over-ocean algorithm.

The goal of this investigation is to perform such a sensitivity study through use of a slightly modified (details in section 2) version of GPROF. The investigation is performed using Tropical Rainfall Measuring Mission (TRMM; Kummerow et al. 1998) Microwave Imager (TMI) brightness temperature observations. While the specific results are shown for TMI, the results are fundamental to many Bayes theorem–based retrievals utilizing PMW imager observations, because of the nature of the physical relationship between rainfall and brightness temperature (occasionally referred to as Tb) at microwave frequencies. While accurate and consistent rainfall climate data records require intercalibrated satellite sensors, we argue that an improved accounting of uncertainty associated with each sensor’s brightness temperature product is necessary for rainfall product agreement among all constellation sensors and, more broadly, for accurate global rainfall estimation and closure of regional/global water budgets.

Substantial prior (and ongoing) research has also been focused on regional and synoptic-scale differences between PMW (e.g., TMI) and active microwave [e.g., TRMM Precipitation Radar (PR; Iguchi et al. (2000, 2009)] rainfall products, often within the context of understanding differing microphysics and rainfall structure assumptions in each sensor’s retrieval algorithm (Masunaga et al. 2002; Ikai and Nakamura 2003; Nesbitt et al. 2004; Berg et al. 2006; Sanderson et al. 2006; Fisher and Wolff 2011; Zagrodnik and Jiang 2013; Wang et al. 2014). Despite making no changes to algorithm microphysics/structure assumptions, we find that the regional spatial patterns/magnitudes of TMI–PR biases can be substantially altered by small changes in the observational/model uncertainty used in the passive microwave retrieval. To this end, we emphasize that understanding and reconciling the differences in active versus passive retrieval (Bayesian based) products, both at the mesoscale and larger (global; interannual) scales, requires a more robust accounting of observation/model uncertainty in addition to understanding the differences in active/passive algorithm microphysics assumptions.

2. Algorithm, data, and methodology

a. Summary description of the GPROF over-ocean retrieval algorithm

Inversion algorithms, incorporating Bayes’s theorem and applied to the passive microwave rainfall retrieval problem, have a well-documented history (Marzano et al. 1999; Kummerow et al. 2001; Bauer et al. 2002; Di Michele et al. 2005; Chiu and Petty 2006; Olson et al. 2006; Tassa et al. 2006; Petty and Li 2013). Derivations in these studies all lead to a rainfall retrieval equation of the form
e1
where E(rr) is the expected rainfall rate (i.e., the retrieval); P(rr) is the a priori probability distribution of rainfall; Tbs and Tbo denote the simulated and observed vectors of brightness temperatures, respectively; and is the (observations + model) error covariance matrix. The discretized form of Eq. (1) is used in the ocean algorithm component of GPROF and is expanded to global scenes with the launch of GPM, as mentioned.

For a given sensor, algorithms incorporating Eq. (1) exhibit varying skill according to how the following terms of Eq. (1) are derived: P(rr), Tbs, and . In the current version of GPROF (hereinafter, often referred to as GPROF 2010), P(rr) and Tbs are derived using observations from the TMI and TRMM PR sensors such that the computed rainfall distribution (and associated radar reflectivities) and simulated Tbs agree with both PR and TMI multichannel observations within an allotted range [see Kummerow et al. (2011) for a detailed description].

For TMI [a nine-channel (10V,1 10H, 19V, 19H, 21V, 37V, 37H, 85V, and 85H) radiometer], the sensor utilized for this study, is shown in Fig. 1a. The diagonal elements of are then computed as the root-mean-square (RMS) differences between the TMI and simulated Tbs. Off-diagonal elements are the covariances between the observed and simulated brightness temperatures of interest. The corresponding correlation matrix is shown in Fig. 1b. Mostly positive covariances and correlations indicate a tendency for the TMI channels to respond in a similar statistical manner as rainfall increases; that is, as the amount of liquid/rainfall increases, over the TMI field of view, Tbs typically increase. Thus, emission dominates over scattering, which is not surprising since nonraining and lightly raining (non-ice) scenes compose the majority population of the rainfall distribution. As currently formulated, off-diagonal elements of are neglected in GPROF 2010 (for all sensors, in fact). In the case of TMI, then, is forced to be a 9 × 9 diagonal matrix, with the nine RMS magnitudes equivalent to the diagonal elements (bottom left to top right) of the matrix shown in Fig. 1a, and all off-diagonal elements set to 0. These RMS estimates are assumed constant for all global scenes/pixels [also discussed in Kummerow et al. (2011)].

Fig. 1.
Fig. 1.

(a) TMI GPROF 2010 database (Kummerow et al. 2011) all-channel simulated Tb covariance matrix (40°N–40°S, over ocean): H = horizontal polarization, and V = vertical polarization. (b) Corresponding correlation matrix.

Citation: Journal of Applied Meteorology and Climatology 54, 2; 10.1175/JAMC-D-14-0105.1

Passive microwave rainfall retrieval is underconstrained in that different rainfall profiles often yield similar Tb vectors. Therefore, during the inversion process, additional information is often required in order to “retrieve” the best rainfall for cases where multiple answers are possible given the same Tb. A partial remedy to this issue has been achieved by using ancillary sea surface temperature (SST; Reynolds et al. 2007) and total precipitable water (TPW; Elsaesser and Kummerow 2008) fields to subset the entire a priori database. The assumption is that rainfall profiles associated with cold surfaces and drier atmospheres are likely to be different from those found in environments with high SST and substantial atmospheric moisture; thus, the use of these environmental parameters aids in the selection of the best rainfall estimate. Therefore, in GPROF 2010, for a given satellite pixel, all profiles associated with the input TPW and SST are initially selected, with E(rr) then being computed through the use of Eq. (1) applied only to the initially selected profiles.

b. Modification to GPROF and methodology chosen for rainfall sensitivity investigation

The only modification made to GPROF 2010 throughout this study pertains to changing the elements of . Specifically, the following modifications to are made: 1) the case where nondiagonal elements of the matrix are retained and not forced to zero; 2) the case where a diagonal matrix is still assumed, as in GPROF 2010, but instead, diagonal elements are systematically increased or decreased through the use of a scalar multiplier applied to the Tb uncertainties (which we term the uncertainty scale-factor approach); and 3) the case where a diagonal matrix is assumed, but whose elements are allowed to vary as a function of SST and TPW. The modifications to are performed independently of each other, and the resulting tropical (40°N–40°S) retrieved rainfall states are compared with GPROF 2010 rainfall for the year spanning May 1999–June 2000. The same TMI Tb data (version-7 1B11 product; V7) and ancillary SST/TPW (section 2a) are used in each case.

The modifications to are chosen based on an investigation of the GPROF 2010 database (Kummerow et al. 2011). The first variation simply allows for use of the full covariance matrix (as described in section 2a) in the retrieval [Eq. (1)]. The latter two changes to are partly motivated by the results depicted in Fig. 2. Depending on the database surface rainfall rate or the ancillary SST and TPW being used, substantial variation in RMS uncertainty is observed in the database (shown for the vertical-polarized TMI channels only). Considering the 19V channel, while the tropical RMS uncertainty is ~1.8 K for TMI (Fig. 1a; this is the fixed value used in GPROF 2010), magnitudes vary from 1.5 to 8 K as rainfall increases to 3 mm h−1. Similar variation is found over the spectrum of TPW (right column in Fig. 2). Variations are larger for horizontal-polarized channels (not shown). Therefore, the scale-factor approach developed here allows uncertainties to simply be scaled (up or down) as a function of SST and TPW and, thus, we can attempt to account for the fact that RMS uncertainty is not constant as assumed in GPROF 2010. Physically, such systematic variations in RMS uncertainty imply that it is more difficult to simulate brightness temperatures in some environments (perhaps characterized by SST or TPW) versus others.

Fig. 2.
Fig. 2.

(left) TMI (vertical polarized channels only) over-ocean total (model + observation) RMS uncertainty in brightness temperature as a function of the TRMM V7 PR surface rainfall rate (averaged to the 19-GHz TMI FOV). (right) Uncertainty as a function of SST and TPW.

Citation: Journal of Applied Meteorology and Climatology 54, 2; 10.1175/JAMC-D-14-0105.1

Additionally, because the RMS uncertainties in are computed explicitly through a sum-of-the-squared-errors method combining model error and sensor noise (here, sensor noise is representative of TMI), rainfall retrieval sensitivity experiments involving small changes in scale factors represent experiments for understanding changes in the rainfall retrieval as one transitions from one PMW imager (e.g., TMI) to another (e.g., GMI). Model error differences between two such sensors are generally small (due to both sensors having similar Tb frequencies). However, changes in sensor characteristics (e.g., sensor fields of view or calibration characteristics) may lead to changes in sensor noise estimates (occasionally referred to as NEDT) beyond published values. Therefore, we perform an analysis of the extent to which the rainfall distribution changes given small changes in the scale factor used for . Factors leading to changes in sensor noise along with expected magnitudes are discussed in the following section.

c. Uncertainty scale factors in relation to current PMW sensor Tb uncertainties

The brightness temperature uncertainty discussed in sections 2a and 2b is most applicable to TMI. This is due to TMI Tbs serving as the observational benchmarks against which cloud/rainfall profiles are adjusted to yield simulated Tbs that most closely match the observed Tbs (Kummerow et al. 2011). To perform a retrieval using another PMW sensor, the TMI uncertainties must be scaled up or down to match those that would be characteristic of another sensor (e.g., GMI). A simple way to do this would be to remove the squared error term associated with TMI sensor noise, and then add the new squared error term associated with the new sensor’s observed brightness temperatures (i.e., new NEDT).

There are a number of reasons why sensor Tb noise estimates may differ among sensors or differ from published values (and even change with time). From the sensor mechanics side, there are errors in the relative calibration between sensors as a function of brightness temperature, including nonlinearities from the typically two-point calibration used by almost all previous and current imagers besides GMI (it has a four-point onboard calibration). There are also time-dependent calibration errors due to sensor degradation and other factors. Examples include changes in solar heating, and warm/cold load intrusions due to orbit decay of the Special Sensor Microwave Imager/Sounder (SSMIS) sensors on board Defense Meteorological Satellite Program satellites F-16, F-17, and F-18. Since current calibration efforts, while aiming to address systematic biases in brightness temperatures, are not suited for addressing variances found during sensor Tb intercomparisons, there is an unknown amount of uncertainty added to most sensor Tb channels during the intercalibration process. The ability to quantify sensor-dependent uncertainty is confounded further since each sensor has its own channel fields of view (FOVs). Since radiative transfer errors can be masked or amplified depending on the FOV over which brightness temperatures are convolved, it is not clear that model uncertainties computed using a TMI–PR database are applicable to sensors with different FOVs (a problem exacerbated further if slightly different channel or incidence angles exist).

The overall result is that the quantifying uncertainty, including small trends and regime-dependent variability, is difficult. In short, for all passive microwave sensors, the statement that “uncertainty exists in specified retrieval uncertainty” is a universal one. The extent to which this is an issue to be addressed depends on the degree to which changes in uncertainty (on the order of a few kelvins) impact passive microwave rainfall retrieval. From a climate-trend perspective, knowing how large the sensitivity is in relation to expected/observed changes in rainfall due to interannual variability or systematic climate change is important [estimates vary considerably, ranging from no change up to a 7% increase in global rainfall per kelvin of surface warming: Lau and Wu (2007), Gu et al. (2007), Wentz et al. (2007), Zhou et al. (2011), Allan et al. (2013), Chou et al. (2013), and Lau et al. (2013)]. From the perspective of multisensor rainfall product generation, mischaracterization of retrieval uncertainties can lead to artificial jumps or trends in rainfall as current satellite sensors drop out of a multidecadal record, or as new ones are introduced. Therefore, our analyses of changes in the retrieved rainfall distribution for small scale-factor changes are meant to illustrate situations for which changes in total Tb uncertainties are driven by the differences in sensor NEDT as one transitions from one sensor to another.

3. Results

a. Varying uncertainty and implications for PMW rainfall product agreement and detectability of rainfall trends

As discussed, scale factors denote the multiplicative adjustments that are made to the Tb uncertainties used in the TMI GPROF 2010 algorithm. For TMI, the diagonal entries of are equal to the squares of the boldface numbers corresponding to a scale factor of 1.0 in Table 1. For GPROF 2010, RMS uncertainty ranges from roughly 1.4 K at the 10V channel to 5.5 K at the 85H channel (Table 1). A scale factor change to 2.0 implies total net increases in RMS uncertainty of only 0.74 K at 19V and 2.29 K at 85H (Table 1). From the perspective of an average brightness temperature magnitude (e.g., 250 K), this amounts to less than a 1% increase in the error on the simulated Tb.

Table 1.

For the scale factors discussed in the text, the associated uncertainties in TMI Tbs (K) are shown. A scale factor of 1.0 corresponds to the operational rainfall retrieval algorithm (in boldface; GPROF 2010). The June 1999–May 2000 global-average oceanic rainfall rate (mm day−1) retrieved utilizing the specified uncertainties is shown in the last column.

Table 1.

For the year of June 1999–May 2000, the effects of varying uncertainty on the ocean-only rainfall retrieval distribution are shown in Fig. 3a. For each investigated scale-factor retrieval experiment, all orbital-level pixel (resolution of TMI 19-GHz field of view, or ~25 km) rainfall estimates are gathered, and a frequency distribution is derived (in %). For the scale-factor change discussed above (1.0–2.0; Table 1), the higher end (1–10 mm h−1) of the rainfall probability distribution decreases by 1%–4%, while the probability decreases by over 15% for a rainfall rate of 0.3 mm h−1. These differences are computed by taking the distribution derived from the retrieval with a given covariance matrix scale factor, and subtracting the frequency distribution derived using GPROF 2010. Pixel-level estimates are also averaged to a monthly time scale and 1° latitude–longitude grid, and frequency distributions are computed as they are for the pixel-level data. The sensitivity of the monthly rainfall distributions to varying scale factors is shown in Fig. 3b. Even larger differences are found in the monthly frequency distributions as the scale factor increases to 2.0 (10%–40% decreases in the occurrences for rainfall rates over 5 mm h−1, and 10%–20% increases in the occurrences of lighter rainfall rates). A change in the monthly rainfall distribution implies regional changes in rainfall; a map of such changes is shown in Fig. 3c. These small increases in global uncertainty lead to substantial changes in the rainfall spatial distribution [a 6%–8% decrease in rainfall in the southeastern Pacific Ocean, a 1%–4% decrease in the tropical rainfall belt, and a positive (2%–3%) increase over the offshore regions adjacent to South America, Africa, and the Baja Peninsula/southwestern United States]. Such increases in rainfall span the climatologically important low-cloud (stratocumulus) regions of the subtropics, where many satellite rainfall products typically underestimate rainfall (Rapp et al. 2013).

Fig. 3.
Fig. 3.

(a) Retrieved TMI surface rainfall-rate PDF percent difference (relative to the GPROF 2010–retrieved PDF) as a function of the covariance matrix scale-factor parameter (see text for description of factor). (b) As in (a), but for pixel-level rainfall converted to monthly estimates. (c) Change in regional TMI rainfall estimates for a scale factor of 2. (d) Retrieved tropical ocean (40°N–40°S) average rainfall difference (relative to the average rainfall estimated using the official Tb uncertainties specified in the GPROF 2010 retrieval) as a function of varying Tb uncertainties. In (d), vertical lines are shown for visual reference only and horizontal lines denote −5%, 0%, and 5% changes in the global average relative to the official estimate.

Citation: Journal of Applied Meteorology and Climatology 54, 2; 10.1175/JAMC-D-14-0105.1

Even more important, such small uniform increases in uncertainty lead to a >3.6% decrease in the tropical average ocean-only rainfall (black line in Fig. 3d as the scale factor increases from 1 to 2). This extreme sensitivity is not unique to GPROF as currently formulated. A retrieval that incorporates additional information (through the use of the full covariance matrix; red line in Fig. 3d) or one that uses the 19V channel only (straight emission-dominated retrieval; blue line in Fig. 3d) yields similar sensitivity. As discussed, for TMI, uncertainty varies as a function of the environment and can vary with time according to sensor instrument mechanics. A similar argument can be made for any other sensor composing the GPM constellation. The strong sensitivity of rainfall to uncertainty, both in the global average and from a distribution/regional perspective, implies that global rainfall retrieval and intersensor agreement is dependent on the accurate quantification of uncertainty.

b. Varying uncertainty and implications for passive and active microwave rainfall retrieval comparison studies

There are expected regional differences between the TMI database and PR surface rainfall fields. During the TMI passive microwave database construction, drop-size distributions (DSDs), ice microphysics, and rainfall are all adjusted to ensure agreement between simulated and observed PR reflectivity profiles, and simulated and observed TMI brightness temperatures (Kummerow et al. 2011). At this stage, the microphysics is consistent between the radar and radiometer (within the observational capabilities of each sensor). After iterating to achieve such agreement, the final surface rainfall map difference that emerges is shown in Fig. 4a. The comparison is illustrated from the perspective of percent differences [(TMI − PR)/PR × 100] in Fig. 4b. Coherent regions of positive and negative rainfall biases are evident (globally, the bias is negligible). It is important to emphasize that these regional biases (Figs. 4a,b) are expected. If DSD changes as a function of region impact the PR retrieval but have little impact on TMI Tbs, then the database TMI rainfall rate is expected to be different from the PR rainfall estimate. From the inversion point of view, with unbiased simulated Tbs and proper uncertainties utilized, a rainfall retrieval utilizing TMI observations should reproduce the TMI database rainfall (or, stated differently, the TMI-retrieved rainfall minus the TMI database rainfall should be zero, and the TMI–PR-expected rainfall differences should be replicated).

Fig. 4.
Fig. 4.

(a) Regional map of the GPROF 2010 TMI database surface rainfall rates minus the TRMM V7 PR surface rainfall rates. Regions 1–5 are discussed in the text. (c) As in (a), except that official GPROF 2010–retrieved rainfall rates are used in the comparison. (e) As in (c), except that a modified GPROF retrieval is performed whereby global uncertainty estimates are substituted for estimates computed as a function of TPW and SST. (g) As in (c), except that the assumption of a diagonal covariance matrix is removed (i.e., correlations in errors among different TMI channels are taken into account in the retrieval). (b),(d),(f),(h) The same experiments as in (a),(c),(e), and (g), except that TMI − PR differences are depicted as percent differences, defined as (TMI − PR)/PR × 100.

Citation: Journal of Applied Meteorology and Climatology 54, 2; 10.1175/JAMC-D-14-0105.1

The expected differences, however, are not reproduced in the retrieval. For this same year, the difference between GPROF 2010 and the PR surface rainfall is shown in Figs. 4c,d. Here, the operational algorithm uses the tropical–global uncertainties (scale factor = 1 in Fig. 3 and Table 1) derived in the database construction process. While global Tb biases are removed, if the regional (or rainfall/environmental) Tb biases are small, and if the global uncertainties used are representative of all areas, then the difference maps in Figs. 4a,b should be faithfully reproduced in the retrieval. On a pixel-by-pixel scale, of course, the retrieved rainfall rate minus the database rainfall rate would be different (substantial noise). However, the differing biases shown in Figs. 4c,d (relative to Figs. 4a,b) indicate an overall inaccurate selection of uncertainty. Regions 1, the southeastern half of region 2, region 3, and the southern half of region 5 all show a much larger difference (TMI minus PR, hereinafter TMI − PR) than expected. The opposite is observed in the northwestern part of region 2, all of region 4, and the equatorward portion of region 5. Here, the retrieval yields a very negative bias relative to the database TMI rainfall, thus implying that the database TMI − PR difference is much smaller (and even slightly positive).

Another attempt at explaining these results could include a discussion related to the sole use of passive microwave information in the retrieval versus the combined active and passive framework utilized in the database construction. In other words, the database might add/subtract an amount of rainfall that does not equate to an amount that lends agreement with the observed TMI Tbs. If this were the dominant physical issue, then regional/regime-dependent biases in Tbs would likely be substantial. However, Kummerow et al. (2011) show that after the database is fully constructed, residual Tb biases (simulated − observed) are slightly negative in regions of higher TPW/greater rainfall. The opposite is the case in lower TPW/SST (less rainfall) regimes. Such biases, while small in magnitude, are in the wrong direction for reproducing the expected TMI–PR rainfall differences spanning the wetter portions of the tropics [e.g., the intertropical convergence zone (ITCZ) east of the date line]. Accounting for such biases in the retrieval may actually introduce discrepancies in a number of other regions [e.g., the tropical west Pacific, South Pacific convergence zone (SPCZ)], where the expected differences are better replicated.

There are other ways of representing Tb uncertainty in the rainfall retrieval. Figures 4e–h show retrieval results for the cases where uncertainties are characterized according to the environment, and where correlations among sensor errors are taken into account during the retrieval process (these cases are discussed in section 2). In Figs. 4e,f, the uncertainty selected is now a function of environmental SST/TPW. Despite the change, there is virtually no difference in the pronounced climatological biases computed using global uncertainties. When global covariance information is utilized (Figs. 4g,h), there is also negligible improvement toward reproducing the expected differences in Figs. 4a,b. Thus far, differing ways of indexing uncertainty, or using covariance information, offer little insight. There is no well-explored framework for determining how uncertainty should be selected in a Bayes theorem–based retrieval applied, globally, to all sensors (as GPROF is). Uncertainty is not a unique function of SST/TPW, let alone any one environmental parameter. Uncertainty as a function of rainfall could be a robust stratifying method; however, that would require prior knowledge of rainfall. Since we are retrieving rainfall, the circular nature of this latter argument is self-evident.

Despite a method for appropriately characterizing uncertainty not being available, the expected differences are known for the entire tropical belt (for the database year only). For this entire year (40°N–40°S, over ocean), the TMI retrieval is run 18 times with scale factors ranging from 0.05 to 10, and the “ideal” scale factor that best reproduces the results in Figs. 4a,b is sought (i.e., Monte Carlo search). The converged-upon scale factors are shown in Fig. 5b. Scale factors > 1 (< 1) highlight regions where the uncertainty had to be increased (decreased). The corresponding rainfall from TRMM PR is shown in Fig. 5a. In a broad sense, the spatial patterns associated with increased uncertainty follow rainfall (or even rainfall gradients). Notably, though, the correlation is much less than unity, and therefore a rainfall stratifying approach does not facilitate reproduction of all expected biases (despite offering marked improvement). Regions that stand out include the northern Atlantic Ocean and, interestingly, the entire eastern Pacific/Atlantic basins where rainfall/TPW is much less substantial (scale factors here are about 2.0–2.5, but their patterns are very large scale and coherent). These, among others, are regions where uncertainty/covariance indexing may not be best served through consideration of the climatological rainfall. Another issue at hand is the assumption that uncertainty can simply be scaled for all channels through the assumed scale-factor idea developed here. The Monte Carlo search, as currently designed, does not take this situation into account (only a general scale-factor search is performed). As a result, uncertainty could be drastically increased in some areas to compensate for the “bad” assumption that uncertainty can be scaled by simply applying a multiplicative factor to the array of variances associated with all nine TMI Tbs.

Fig. 5.
Fig. 5.

(a) Map of the TRMM V7 PR surface rainfall rates. (b) The scale factors (which multiply the variances of the global covariance matrix) needed for the TMI retrieval to produce estimated rainfall rates that replicate the expected TMI − PR rainfall rates computed in Fig. 4a.

Citation: Journal of Applied Meteorology and Climatology 54, 2; 10.1175/JAMC-D-14-0105.1

c. Understanding the sensitivity of global rainfall retrieval to uncertainty

For the range of plausible uncertainties (i.e., scale factors less than 10 in Fig. 3d), the absolute rate of change of the retrieved ocean-average rainfall rate increases as the assumed uncertainty decreases. The rate of change approaches zero as the uncertainty approaches infinity; in this case, the database mean rainfall is retrieved everywhere (and no weight is placed on observed Tbs). Recall that Fig. 3d also shows a similar response when the retrieval is performed using the 19V channel only. Conveniently then, a mathematical expression governing the sensitivity of the retrieved rainfall to uncertainty can be more simply derived using this channel only. Consider the one-channel formulation of Eq. (1), where is now a scalar value denoting the uncertainty (expressed as variance) in 19V brightness temperatures (σ2). Equation (1) can be rewritten as
e2
where and . Subscripts o and s denote observed and simulated Tbs, respectively. Taking the derivative of Eq. (2) with respect to σ yields
e3
Factoring out common terms and using Eq. (2) again gives
e4
Substituting into Eq. (4) yields
e5
where the weighted average is now represented as
e6
In the absence of weighting {i.e., [P(rr)W] = constant}, note that Eq. (5) would be equivalent to the covariance of rr and ΔTb2 (divided by σ3). In this case though, Eq. (5) can be rewritten as a weighted covariance, or
e7
where the subscript w denotes the weighting scheme in Eq. (6). Equation (7) can also be written exactly as
e8
where corrw(⋅⋅⋅) and σw(⋅⋅⋅) denote the weighted correlation and standard deviation operators, respectively.
An approximate expression for ΔTb2 can be developed here that will facilitate further understanding of Eqs. (7) and (8). Short and North (1990) modeled the 19-GHz over-ocean brightness temperature as
e9
where A, B, and C are constants (FOV/sensor dependent). A Taylor series expansion for Tbs,19v at the observed rainfall rate, retaining only the first-order term, yields
e10
where Δrr = (rrs – rro) and Tbs,19v(rro) is the simulated brightness temperature at the observed rainfall rate. The weighting scheme chosen [Eq. (6)] ensures that simulated Tbs close (in a Euclidean-distance sense) to Tbs,19v(rro) are most heavily weighted; these are also scenes for which Δrr is small, on average. This is the reason that the first-order term of the Taylor series expansion for ΔTb2 is the only one needed. Let Tbo,19v = [Tbs,19v(rro) + δ], where δ represents observational noise, sensor calibration uncertainty, and/or other parameter-driven changes in Tbs (assumed uncorrelated with rainfall). Rearranging the expression for the observed brightness temperature, substituting it into Eq. (10), and solving for ΔTb2 yields
e11
where f = BC exp(−C × rro). With suitable constants (B and C) and a measure of δ, Eq. (11) could be substituted into Eqs. (7) and (8) in place of ΔTb2, and a central set of equations could be developed that predicts the sensitivity of retrieved rainfall to assumed uncertainty on a sensor-by-sensor basis.
Equation (7) can be integrated over latitude and longitude to give an expression for the expected global change in retrieved rainfall as the uncertainty assumption changes. However, for reasons that will become clear, the same result can be achieved by integrating Eq. (7) over the observed 19V brightness temperature distribution. Doing so gives the following expression for the global average derivative:
e12
where the overbar on E(rr) denotes the tropical-wide average. When Tbo is “cool” (much less than the median Tb), as is the case in nonraining or lightly raining regimes, ΔTb2 scales with rainfall [or, using Eq. (11), Δrr2 would positively covary with rainfall]. Therefore, rainfall in these regimes increases on average. The opposite occurs when Tbo is warm (i.e., ΔTb2 increases with decreasing rainfall). Equation (12) is a mathematical expression for the statement that as uncertainty increases, the lighter rainfall rates become heavier and the more intense rainfall rates become lighter. This is essentially the behavior seen in Figs. 3a–c.

There is, however, significant asymmetry in this rainfall adjustment, as the negative-weighted covariances far exceed the positive-weighted covariances when integrated over equal population sizes (from Tbmin to median Tb, and from median Tb to Tbmax). The weighted covariance and correlation as a function of 19V Tbo are illustrated in Figs. 6c,d (vertical dashed lines denote the median Tb). As predicted, for the lower (upper) half of the Tb population, correlations are mostly positive (negative). However, despite similar correlation magnitudes (of opposite signs), the positive weighted covariances hover near zero (unlike the case for the top half of the Tb distribution). Recalling Eq. (8), this can be understood by viewing the graph of the weighted variance of rainfall against Tb (Fig. 6e). For scenes with Tbo less than the median, there is little variation in rainfall. Variations in brightness temperatures here are increasingly driven by parameters not associated with rainfall. Using Eq. (11), ΔTb2 is largely driven by the non-rainfall-correlated δ term, and therefore, the weighted covariances are nearly zero. Nonprecipitating parameters play a lesser role as rainfall increases, and thus variations in rainfall largely drive variations in Tbs. As a result, the weighted variance of rainfall drastically increases beyond the median Tb. An increase in the scatter of rainfall at higher Tbs is also attributed to the saturation of Tb as rainfall increases (since Tb increases logarithmically with rainfall). Combined, these physics govern the net sensitivity.

Fig. 6.
Fig. 6.

(a) Histogram of TMI-observed 19V Tb. (b) GPROF TMI database rainfall rate as a function of observed Tb. (c) Weighted covariance of as a function of observed Tb, where the o and s subscripts to Tb denote observed and simulated Tbs, respectively. (d) Weighted correlation of as a function of observed Tb. (e) Weighted variance in PR surface rainfall (averaged to the TMI 19GHz FOV) as a function of observed Tb. (f) Weighted variance in as a function of observed Tb. The weighting scheme is described in the text. The vertical dashed line in all panels denotes the median 19V-observed TMI Tb.

Citation: Journal of Applied Meteorology and Climatology 54, 2; 10.1175/JAMC-D-14-0105.1

In short, for ocean scenes, the nature of the rainfall–brightness temperature physical relationship (nonlinear, emission based) ensures that an overall under- (over-) estimation of global-ocean PMW-retrieved rainfall will occur if the assumed uncertainty is too large (small). Importantly though, such sensitivity can be understood and approximately reproduced even without fitting a nonlinear curve to Tb and rainfall. This can also be understood from Eq. (11). Here, f dictates the degree of nonlinearity; in the case of a linear relationship between rr and Tb, f is a constant. However, even in such a situation, at cooler brightness temperatures, Δrr is still negligible, and in return, so is the fΔrr component of ΔTb2. At warmer brightness temperatures, Δrr would still be negative, and thus fΔrr would systematically vary with observed Tb (i.e., rain-rate variance would still scale with Tb in the case of a linear fit). Either way, the result is that for equal distributions of Tbs (below and above the median Tb), the average negative covariance far outweighs the slight positive covariance, and thus the ocean-average rainfall decreases quite substantially as uncertainty increases. This global retrieved rainfall tendency was depicted in Fig. 3d.

4. Conclusions

Physical rainfall algorithms are typically formulated within a Bayesian framework, where the specification of uncertainty is a central component. Superposed upon model uncertainty (which itself is sensor dependent, due in part to differences in sensor FOVs) is sensor noise (NEDT). Given that model uncertainty is strongly related to rainfall, model uncertainty can slowly trend if there are subtle shifts in the rainfall spectrum over time. Increases in published sensor NEDT values arise over time because of changing sensor characteristics (possible degradation) and intercalibration efforts (where Tb offsets may be added as a function of scene, in a nonlinear fashion, thus affecting overall variances). Such issues imply that there is an element of error associated with an assumed array of Tb uncertainties used in Bayes theorem–based retrievals. With systematically varying FOVs across sensors, determining whether to increase or decrease uncertainties (say, those corresponding to a database tuned to one satellite) for a different sensor requires a thorough investigation.

If small changes in assumed uncertainty have negligible impact on retrieved rainfall, then the discussion becomes one limited to reconciling small details. However, the retrieved rainfall rates, both regionally and globally, exhibit substantial sensitivity to specified uncertainty. Specifically, increases in uncertainty, ranging from 0.57 K at 10V GHz to 2.24 K at 85H GHz, lead to a 3.6% decrease in the global average, and 10%–20% (20%–40%) shifts in the pixel-level (monthly) probability distribution computations for given rainfall bins. Such changes are similar to GCM and current-observed variations/trends in rainfall mentioned in section 2c. A mathematical expression has been derived in order to explain the sensitivity of retrieved global rainfall to specified uncertainty. The smaller the initial specified uncertainty is (e.g., the case of small NEDT or model error), the larger the retrieval sensitivity is [partly resulting from the σ−3 dependence in Eq. (7), and further illustrated in Fig. 3d]. It has been shown that emission-dominated rainfall–Tb physics leads to a global underestimation (overestimation) of rainfall if the assumed uncertainty is too large (small). Such sensitivity depends on rainfall variance scaling systematically as a function of Tb. With the expansion of the Bayesian algorithm to land scenes, if such variance scaling is common over land, a sensitivity of global average overland rainfall to uncertainty will also be present (the sign/magnitude of which would depend on the dominant brightness temperature–rainfall relationship).

At present, approximately 64 million profiles go into the GPROF 2010 database. Future databases are likely to be larger (e.g., GPM is sampling globally). It is impractical for operational algorithms to search all profiles at every observational time step. Database reduction methods, often achieved through database clustering, become necessary. The results of this sensitivity study imply that any reduction methods that modify the rainfall variance per observed Tb interval will systematically bias the sensitivity of the retrieved rainfall to the assumed uncertainty. For example, algorithms aiming to minimize within-cluster variances [i.e., grouping a diverse number of rainfall–Tb profiles into one cluster (e.g., k-means clustering techniques)] would be ill-suited for database reduction, as the rainfall variance at warm brightness temperatures would be substantially decreased. A reduction algorithm that retains the natural variation of rainfall as Tb changes is most desirable. A bottom-up hierarchical clustering approach in which any rainfall–Tb profiles are merged only if Tbs are indistinguishable (i.e., the same within-sensor NEDT) is one such promising approach for retaining all database information. This clustering technique would approximately reproduce the natural scatter that exists in the rr–Tb relationship, and would lead to a replication of Fig. 6b (with an apparent filtering of points essentially applied). This database reduction approach is currently planned for GPM constellation PMW retrievals.

A robust, physical algorithm applied to the constellation of orbiting radiometers composing the GPM mission is tasked with producing consistent precipitation products so that our confidence in quantifying rainfall, extreme rainfall rates, and snowfall—globally—is increased. For the global scenes where rainfall variance changes systematically with Tb, it is clear that confidently quantifying variations/trends in rainfall distributions (including extremes) and achieving satellite–sensor rainfall product agreement is tied to appropriate accounting of the sensor uncertainty. Results suggest that efforts aimed at understanding sensor uncertainty added during calibration efforts and understanding potential trends in uncertainty due to changing rainfall/environment would be particularly useful. Furthermore, the results of this study suggest that the incorporation of covariance matrix information into the retrieval may not aid in relaxing the sensitivity of the retrieval to uncertainty, nor fundamentally affect the regional TMI/PR climate biases relative to what is achieved through the sole use of the diagonal entries of the brightness temperature covariance matrix. From the regional bias perspective, given that uncertainty seems most related to rainfall itself, the investigation of particular brightness temperature combinations serving as an “index” for rainfall (before actually retrieving it), thus informing which multichannel array of uncertainties should be used, could have more impact.

Acknowledgments

The comments on the manuscript by Wesley K. Berg and three anonymous reviewers have been useful in improving the manuscript. This research was supported by NASA Grants NNX13AG31G and NNX13AN44G.

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1

Here, V = vertical polarization; H = horizontal polarization.

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Journal of Applied Meteorology and Climatology

  • Allan, R. P., and B. J. Soden, 2007: Large discrepancy between observed and simulated precipitation trends in the ascending and descending branches of the tropical circulation. Geophys. Res. Lett., 34, L18705, doi:10.1029/2007GL031460.

    • Search Google Scholar
    • Export Citation

  • Allan, R. P., C. Liu, M. Zahn, D. Lavers, E. Koukouvagias, and A. Bodas-Salcedo, 2013: Physically consistent responses of the global atmospheric hydrological cycle in models and observations. Surv. Geophys., 35, 533–552, doi:10.1007/s10712-012-9213-z.

    • Search Google Scholar
    • Export Citation

  • Bauer, P., J.-F. Mahfouf, W. S. Olson, F. S. Marzano, S. Di Michele, A. Tassa, and A. Mugnai, 2002: Error analysis of TMI rainfall estimates over ocean for variational data assimilation. Quart. J. Roy. Meteor. Soc., 128, 2129–2144, doi:10.1256/003590002320603575.

    • Search Google Scholar
    • Export Citation

  • Berg, W., T. L’Ecuyer, and C. Kummerow, 2006: Rainfall climate regimes: The relationship of regional TRMM rainfall biases to the environment. J. Appl. Meteor. Climatol., 45, 434–454, doi:10.1175/JAM2331.1.

    • Search Google Scholar
    • Export Citation

  • Berg, W., M. Sapiano, J. Horsman, and C. Kummerow, 2013: Improved geolocation and Earth incidence angle information for a fundamental climate data record of the SSM/I sensors. IEEE Trans. Geosci. Remote Sens., 51, 1504–1513, doi:10.1109/TGRS.2012.2199761.

    • Search Google Scholar
    • Export Citation

  • Biswas, S. K., K. Gopalan, W. L. Jones, and S. Bilanow, 2010: Correction of time-varying radiometric errors in TRMM Microwave Imager calibrated brightness temperature products. IEEE Geosci. Remote Sens. Lett., 7, 851–855, doi:10.1109/LGRS.2010.2050135.

    • Search Google Scholar
    • Export Citation

  • Chiu, J. C., and G. W. Petty, 2006: Bayesian retrieval of complete posterior PDFs of oceanic rain rate from microwave observations. J. Appl. Meteor. Climatol., 45, 1073–1095, doi:10.1175/JAM2392.1.

    • Search Google Scholar
    • Export Citation

  • Chou, C., J. C. H. Chiang, C.-W. Lan, C.-H. Chung, Y.-C. Liao, and C.-J. Lee, 2013: Increase in the range between wet and dry season precipitation. Nat. Geosci., 6, 263–267, doi:10.1038/ngeo1744.

    • Search Google Scholar
    • Export Citation

  • Di Michele, S., A. Tassa, A. Mugnai, F. Marzano, P. Bauer, and J. Baptista, 2005: Bayesian algorithm for microwave-based precipitation retrieval: Description and application to TMI measurements over ocean. IEEE Trans. Geosci. Remote Sens., 43, 778–791, doi:10.1109/TGRS.2005.844726.

    • Search Google Scholar
    • Export Citation

  • Elsaesser, G. S., and C. D. Kummerow, 2008: Toward a fully parametric retrieval of the nonraining parameters over the global oceans. J. Appl. Meteor. Climatol., 47, 1599–1618, doi:10.1175/2007JAMC1712.1.

    • Search Google Scholar
    • Export Citation

  • Ferraro, R. R., 1997: Special Sensor Microwave Imager-derived global rainfall estimates for climatological applications. J. Geophys. Res., 102, 16 715–16 735, doi:10.1029/97JD01210.

    • Search Google Scholar
    • Export Citation

  • Ferraro, R. R., and G. F. Marks, 1995: The development of SSM/I rain-rate retrieval algorithms using ground-based radar measurements. J. Atmos. Oceanic Technol., 12, 755–770, doi:10.1175/1520-0426(1995)012<0755:TDOSRR>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Fisher, B., and D. B. Wolff, 2011: Satellite sampling and retrieval errors in regional monthly rain estimates from TMI, AMSR-E, SSM/I, AMSU-B, and the TRMM PR. J. Appl. Meteor. Climatol., 50, 994–1023, doi:10.1175/2010JAMC2487.1.

    • Search Google Scholar
    • Export Citation

  • Gu, G., R. F. Adler, G. J. Huffman, and S. Curtis, 2007: Tropical rainfall variability on interannual-to-interdecadal and longer time scales derived from the GPCP monthly product. J. Climate, 20, 4033–4046, doi:10.1175/JCLI4227.1.

    • Search Google Scholar
    • Export Citation

  • Hou, A. Y., and Coauthors, 2014: The Global Precipitation Measurement Mission. Bull. Amer. Meteor. Soc., 95, 701–722, doi:10.1175/BAMS-D-13-00164.1.

    • Search Google Scholar
    • Export Citation

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  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    (a) TMI GPROF 2010 database (Kummerow et al. 2011) all-channel simulated Tb covariance matrix (40°N–40°S, over ocean): H = horizontal polarization, and V = vertical polarization. (b) Corresponding correlation matrix.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    (left) TMI (vertical polarized channels only) over-ocean total (model + observation) RMS uncertainty in brightness temperature as a function of the TRMM V7 PR surface rainfall rate (averaged to the 19-GHz TMI FOV). (right) Uncertainty as a function of SST and TPW.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    (a) Retrieved TMI surface rainfall-rate PDF percent difference (relative to the GPROF 2010–retrieved PDF) as a function of the covariance matrix scale-factor parameter (see text for description of factor). (b) As in (a), but for pixel-level rainfall converted to monthly estimates. (c) Change in regional TMI rainfall estimates for a scale factor of 2. (d) Retrieved tropical ocean (40°N–40°S) average rainfall difference (relative to the average rainfall estimated using the official Tb uncertainties specified in the GPROF 2010 retrieval) as a function of varying Tb uncertainties. In (d), vertical lines are shown for visual reference only and horizontal lines denote −5%, 0%, and 5% changes in the global average relative to the official estimate.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    (a) Regional map of the GPROF 2010 TMI database surface rainfall rates minus the TRMM V7 PR surface rainfall rates. Regions 1–5 are discussed in the text. (c) As in (a), except that official GPROF 2010–retrieved rainfall rates are used in the comparison. (e) As in (c), except that a modified GPROF retrieval is performed whereby global uncertainty estimates are substituted for estimates computed as a function of TPW and SST. (g) As in (c), except that the assumption of a diagonal covariance matrix is removed (i.e., correlations in errors among different TMI channels are taken into account in the retrieval). (b),(d),(f),(h) The same experiments as in (a),(c),(e), and (g), except that TMI − PR differences are depicted as percent differences, defined as (TMI − PR)/PR × 100.

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    (a) Map of the TRMM V7 PR surface rainfall rates. (b) The scale factors (which multiply the variances of the global covariance matrix) needed for the TMI retrieval to produce estimated rainfall rates that replicate the expected TMI − PR rainfall rates computed in Fig. 4a.

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    (a) Histogram of TMI-observed 19V Tb. (b) GPROF TMI database rainfall rate as a function of observed Tb. (c) Weighted covariance of as a function of observed Tb, where the o and s subscripts to Tb denote observed and simulated Tbs, respectively. (d) Weighted correlation of as a function of observed Tb. (e) Weighted variance in PR surface rainfall (averaged to the TMI 19GHz FOV) as a function of observed Tb. (f) Weighted variance in as a function of observed Tb. The weighting scheme is described in the text. The vertical dashed line in all panels denotes the median 19V-observed TMI Tb.

Fig. 1.

(a) TMI GPROF 2010 database (Kummerow et al. 2011) all-channel simulated Tb covariance matrix (40°N–40°S, over ocean): H = horizontal polarization, and V = vertical polarization. (b) Corresponding correlation matrix.
Fig. 2.

(left) TMI (vertical polarized channels only) over-ocean total (model + observation) RMS uncertainty in brightness temperature as a function of the TRMM V7 PR surface rainfall rate (averaged to the 19-GHz TMI FOV). (right) Uncertainty as a function of SST and TPW.
Fig. 3.

(a) Retrieved TMI surface rainfall-rate PDF percent difference (relative to the GPROF 2010–retrieved PDF) as a function of the covariance matrix scale-factor parameter (see text for description of factor). (b) As in (a), but for pixel-level rainfall converted to monthly estimates. (c) Change in regional TMI rainfall estimates for a scale factor of 2. (d) Retrieved tropical ocean (40°N–40°S) average rainfall difference (relative to the average rainfall estimated using the official Tb uncertainties specified in the GPROF 2010 retrieval) as a function of varying Tb uncertainties. In (d), vertical lines are shown for visual reference only and horizontal lines denote −5%, 0%, and 5% changes in the global average relative to the official estimate.
Fig. 4.

(a) Regional map of the GPROF 2010 TMI database surface rainfall rates minus the TRMM V7 PR surface rainfall rates. Regions 1–5 are discussed in the text. (c) As in (a), except that official GPROF 2010–retrieved rainfall rates are used in the comparison. (e) As in (c), except that a modified GPROF retrieval is performed whereby global uncertainty estimates are substituted for estimates computed as a function of TPW and SST. (g) As in (c), except that the assumption of a diagonal covariance matrix is removed (i.e., correlations in errors among different TMI channels are taken into account in the retrieval). (b),(d),(f),(h) The same experiments as in (a),(c),(e), and (g), except that TMI − PR differences are depicted as percent differences, defined as (TMI − PR)/PR × 100.
Fig. 5.

(a) Map of the TRMM V7 PR surface rainfall rates. (b) The scale factors (which multiply the variances of the global covariance matrix) needed for the TMI retrieval to produce estimated rainfall rates that replicate the expected TMI − PR rainfall rates computed in Fig. 4a.
Fig. 6.

(a) Histogram of TMI-observed 19V Tb. (b) GPROF TMI database rainfall rate as a function of observed Tb. (c) Weighted covariance of as a function of observed Tb, where the o and s subscripts to Tb denote observed and simulated Tbs, respectively. (d) Weighted correlation of as a function of observed Tb. (e) Weighted variance in PR surface rainfall (averaged to the TMI 19GHz FOV) as a function of observed Tb. (f) Weighted variance in as a function of observed Tb. The weighting scheme is described in the text. The vertical dashed line in all panels denotes the median 19V-observed TMI Tb.
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