## 1. Introduction

Observations of clouds and precipitation processes in the microwave (MW) domain from space have been performed since the late 1980s (Spencer et al. 1989). The launch of the Tropical Rainfall Measurement Mission (TRMM) satellite in 1997, carrying a Precipitation Radar (PR) along with the passive TRMM Microwave Imager (TMI), allowed an unprecedented amount of collocated MW multispectral atmospheric signatures and radar-derived vertical profiles of hydrometeor type and density (Kummerow et al. 1998). This enabled the development of algorithms for the estimation of the surface (or near surface) precipitation rate from passive MW observations. Among these algorithms are the Goddard profiling algorithm (GPROF) and GSMaP, developed by NASA and JAXA, respectively (Kummerow et al. 2001; Aonashi et al. 2009). Other passive MW imagers similar to TMI have been sent into orbit since then: SSMI/S on board the DMSP platforms, AMSR on board Advanced Earth Observing Satellite-II (ADEOS-II), AMSR-E on board *Aqua*, Microwave Analysis and Detection of Rain and Atmospheric Structures (MADRAS) on board *Megha-Tropiques*, AMSR-2 on board *Global Change Observation Mission–Water* (*GCOM-W1*), etc. In 2014, as a follow-up to TRMM, the Global Precipitation Measurement (GPM) *Core Observatory* satellite carrying the GPM Microwave Imager (GMI) passive MW imager and the Dual-Frequency Precipitation Radar (DPR) was launched, making available a new set of collocated passive MW and radar observations of the atmosphere (Hou et al. 2014).

Several studies have evaluated the performance of satellite passive MW estimates of instantaneous precipitation rates by comparison with collocated radar or rain gauge data (e.g., Petty and Li 2013; Tang et al. 2014; Oliveira et al. 2015; Ebtehaj et al. 2016). These studies generally evaluate the ability of MW estimates to reproduce the statistical distribution of rain rates. They also quantify the similarity of MW estimates with the reference dataset through metrics such as mean squared difference (MSD) or the linear correlation coefficient.

Another important criterion one may consider in assessing the performance of the MW retrievals is their “effective resolution,” that is, the finest spatial scale at which they can accurately reproduce desired properties of precipitation fields. Determining the effective resolution of the surface rain rate estimated through the inversion of a multispectral signature is in fact far from trivial. Indeed, for a given spaceborne MW imager, the size of the footprint on the ground varies with the channel (Fig. 1). The scanning geometry and sampling rate combined with the satellite displacement during the measurement may lead to overlapping instantaneous fields of view for the low-frequency channels and to undersampling of the observed scene for the high-frequency channels (Kummerow et al. 1998; Petty and Bennartz 2017). The relative importance of each channel, and thus the effective resolution of the retrieval, may vary depending on parameters such as the surface type (Ferraro et al. 2013). Beyond the issue of the nominal resolution of each channel of the MW imagers, because of the indirect relation between observed radiances and the surface precipitation, the inversion is always ambiguous, and the MW signal may not contain sufficient information to resolve the finescale variability. Thus, the computation method used to deduce the surface precipitation rate from the measured MW radiances may induce a filtering effect or generate “computational noise” affecting the effective resolution of the estimate.

The above issues limit the classical point-to-point or pixel-to-pixel comparison with a reference dataset and calls for developing validation methods and metrics that compare the multiscale spatial structure and geometrical properties of the retrieved precipitation fields. Object-based approaches have been proposed to handle this issue (e.g., Nesbitt et al. 2006; Aghakouchak et al. 2011; Demaria et al. 2011). Objects are defined as contiguous areas with precipitation intensity above a given threshold. One can consider the number of objects, their size, their convexity or compactness, etc., and evaluate the ability of MW estimates to preserve these characteristics. To comprehensively assess the ability of the estimates to reproduce precipitation patterns, the objects’ statistics must be computed with multiple thresholds, covering the whole range of possible precipitation intensities. These methods may be computationally intensive and lead to a large amount of computed parameters to be interpreted.

We propose here to use a spatial multiscale decomposition of the retrieved precipitation fields to assess the ability of MW-based methods to reproduce the spatial patterns of precipitation at various scales. A two-dimensional discrete orthogonal Haar wavelet decomposition is used to perform the analysis. It allows characterizing specifically each scale of variation of the precipitation fields in a nonredundant way. Wavelets being differential operators result in wavelet coefficients representing the local spatial gradients (or fluctuations) of precipitation fields at various scales. The orthogonal wavelet decomposition erases the spatial correlation of the analyzed signal, that is, the wavelet coefficient fields are spatially uncorrelated even if the original signal is spatially correlated. Wavelet filtering also removes possible nonstationarities in the original fields (in the case of the discrete Haar wavelet, it removes piecewise constant-level trends), rendering statistical interpretations more meaningful. Wavelets have been used successfully to characterize the spatial structure of the precipitation fields and the dependence of local precipitation rates on their surroundings (e.g., Kumar and Foufoula-Georgiou 1994, 1997). Using a discrete orthogonal wavelet basis, which is a reconstructive basis, leads to a nonparametric reversible decomposition by which all the information contained in the original signal is contained in the wavelet coefficients. In this paper, MW estimates from TMI and GMI based on two different versions of the GPROF algorithm are compared with radar data in the wavelet domain. A wavelet spectral analysis is also performed by computing second-order statistics of the wavelet coefficients, to assess how the energy (spatial variance) of the signal is distributed across scales.

The main objective of this study is to assess the improvement in retrieval of instantaneous precipitation rates allowed by the last version of GPROF with the 13 channels of GMI compared to the TRMM-era version of the algorithm with the 9 channels of TMI. The smaller footprint of GMI compared with TMI (Fig. 1) is expected to allow resolving of precipitation structures at finer scales. However, the effective resolution as defined herein is a result of not only the nominal resolution of the instrument, but also of the information content of each MW channel and the ability of the retrieval algorithm to accurately interpret this information to reproduce the precipitation spatial variability and structure at fine scales. The MW estimated precipitation rates from TMI and GMI are compared to the precipitation rates measured by the PR on board TRMM and the DPR on board GPM, respectively. The radar-derived precipitation rates are considered an accurate enough reference product for the evaluation of the MW estimates, meaning that the radar retrieval error is considered negligible relative to the MW error. Besides, the regional discrepancies revealed by our analysis can always be assessed in the context of possible errors stemming from regional variations in the accuracy of the radar estimates. The potential inaccuracies in the radar estimates are discussed in the last section of this paper.

The article is organized as follows. The passive MW and radar datasets are presented in section 1. Section 2 is dedicated to the evaluation methodology relying upon the wavelet transform. In section 3, results are presented, first with an illustrative case study and then at a global scale with multiyear statistics. Section 4 discusses the results.

## 2. Data

### a. TRMM era: 6 years of TMI estimates collocated with PR observations

The PR on board the TRMM satellite produced measurements of the instantaneous surface rain rate under the track of the satellite from 1997 to 2015, at 13.8 GHz (Ku band), with a 5-km horizontal resolution. The swath width of the radar is 245 km. The wider swath (880 km) of the TRMM Microwave Imager (TMI) encompasses the swath of the PR, allowing collocated observations. Here, 6 years (2002, 2005, 2008, 2011, 2012, and 2013) of retrieved surface rain rates derived from TMI through GPROF-2010 (product 2A12v7; TRMM 2011a) are compared with the surface rain rates from the PR (product 2A25v7; TRMM 2011b). For the comparison, TMI-GPROF estimates are projected on the 5-km grid of the 2A25v7 product using a nearest neighbor interpolation. It is to be noted that the computation schemes of GPROF-2010 are different over land and oceans. Over oceans, a Bayesian inversion scheme is used to retrieve the most probable hydrometeor profiles from a database made of 65 million coincident TMI and PR observations. Over land, stratiform and convective precipitation areas are first delimited using a scattering index. For each type of precipitation, an empirical relationship between precipitation rate and 85-GHz brightness temperatures is used (Kummerow et al. 2015; Petković and Kummerow 2017).

### b. GPM era: 3 years of GMI estimates collocated with Ku-PR observations

The equivalent of the 2A12 product for the GPM era is the experimental 2A-GPROF-GMI product, obtained by the inversion of the GMI spectral signatures with the new GPROF-2014 algorithm (GPM Science Team 2016a). For GPROF-2014, the same Bayesian scheme is used over land and oceans (Kummerow et al. 2015). Over oceans, the same database as in GPROF-2010 is used. Over land a database of coincident ground radar observations in North America and simulated radiances (from SSMI/S observations) is used. The GMI-GPROF estimates are collocated with the surface precipitation rates from the Ku-PR (lowest frequency of the DPR; GPM Science Team 2016b). The Ku PR on board GPM has the same characteristics as the TRMM PR. The measurements in the Ka band are not used in this study to ensure consistency of the reference datasets for the TRMM and GPM eras and because of the reduced swath width of the Ka PR (120 km only). We consider here 3 years of data, from March 2014 to March 2017. The GMI-GPROF estimates are projected on the Ku-PR 5-km grid for the comparison.

## 3. Methodology

### a. Motivation

*R*

_{MW}would be to compare it to

*R*

_{R}pixel by pixel and compute scoring metrics such as the linear correlation coefficient and the mean squared difference. Doing so with the rain rates in the two images shown in Fig. 2, we obtain a correlation coefficient of 0.58 and a root-mean-square difference of 7 mm h

^{−1}. This approach is adapted when one seeks to characterize a retrieval error that is stationary, independent of the reference rain rate

Passive MW retrieval methods such as GPROF rely on the inversion of a multispectral MW signature to infer the surface precipitation rate. These methods are generally designed to minimize a quadratic cost function—for example, the mean squared error (MSE)—or to provide the most likely rain rate given the observed parameters. Minimum MSE methods, Bayesian or empirical Bayesian methods such as the maximum likelihood, maximum a posteriori, best linear unbiased estimator, etc., tend to produce smooth estimates, discarding the extreme values (DeGroot 2004; Carroll et al. 2006; Buzas et al. 2014; Foufoula-Georgiou et al. 2014). In that case, the variance of the estimates is always smaller than the variance of the reference rain rates, which is inconsistent with the additive random error model [Eq. (2)]. These smooth estimates are in fact affected by a conditional systematic bias (Ciach et al. 2000; Kirstetter et al. 2013a), that is, the error necessarily depends on the reference rain rate. This smoothing of the rain rates naturally leads to a smoothing of the spatial gradients. The smoothing effect on the gradients is complex as gradients at different scales may be affected differently. Therefore, the effect on the effective resolution is not trivial. As an illustration, Fig. 3 shows the distribution of local gradients computed as the differences between two adjacent pixels (whose centers are 5 km apart), for the fields *R*_{MW}(*x*, *y*) and *R*_{R}(*x*, *y*) displayed on Fig. 2. The gradients of *R*_{R} have a wider distribution with more frequent extreme values compared to *R*_{MW}. Moreover, a very low correlation, 0.085, is found between the MW and radar gradients (to be compared with the 0.58 correlation coefficient for the precipitation rates themselves), showing that the finescale spatial features of the radar field are not reproduced by the MW estimate.

The correlation coefficient and MSD between the precipitation rates *R*_{MW}(*x*, *y*) and *R*_{R}(*x*, *y*), as pointwise or pixelwise metrics fail to reveal information about how well the spatial organization of the precipitation fields is captured in the retrieval and also do not provide information on the spatial structure of the retrieval error. Instead of quantifying absolute rain intensity, the wavelet coefficients represent the local differences of intensity at various scales. Thus, they quantify the spatial structure of precipitation fields, and therefore, correlation coefficients and MSD computed on the wavelet coefficients at each scale provide more specific information than the correlation coefficients and MSD of rain rates. In addition, because the wavelet coefficients are spatially uncorrelated, there is no ambiguity in the interpretation of their variance or covariance.

### b. Implementation

The two-dimensional discrete Haar wavelet transform used in this study is an orthogonal decomposition, decomposing a field into series of wavelet coefficients encoding the spatial variations (i.e., gradients) of the field at various scales (Mallat 2008; Kumar and Foufoula-Georgiou 1993). While any orthogonal wavelet basis may theoretically be used, the Haar wavelet offers simplicity in interpretation, as the Haar wavelet coefficients can be interpreted in terms of simple finite differences and the smoothing function associated with the Haar wavelet as simple window averaging at different scales.

Figure 4 illustrates the discrete orthogonal decomposition of the two-dimensional MW precipitation field *k* = {1, 2, 3}, and one series of low-pass (LP) coefficients denoted as *k* = {1, 2, 3} and *k* ={1, 2, 3} and *λ* is the scale index. The three sets of wavelet coefficients encode the variations of the field in three directions of the two-dimensional space and are necessary to comprehensively represent the spatial variability [see Eqs. (1)–(4) in Perica and Foufoula-Georgiou (1996) for interpretation of wavelet coefficients in terms of discrete approximations of spatial gradients]. In the present study, the three directions, imposed by the radars’ scanning geometry and the satellite trajectory, are along track, cross track, and diagonal. Because the smoothing function associated with the Haar wavelet is a simple averaging operator,

A necessary condition for the decomposition to be orthogonal and reconstructive is that the scale decomposition is dyadic, that is, the length scale *λ* = {1, 2, …, 5}. Finally, all the information contained in the original field *λ* = {1, 2, …, 5}. In the following, *R*_{R}(*x*, *y*) to allow the multiscale comparison of the two fields.

*λ*. From Eq. (3) we can see that the wavelet decomposition allows us to quantify the contribution of each scale to the total energy of a field. Similarly, by computing the squared difference and the covariance between the wavelet coefficients

*E*[ ] denotes the expected value. Note that, at any scale, the expected value of the wavelet coefficients

*λ*is effectively resolved if the NS efficiency of

Although one could study independently the three directional wavelet components

## 4. Results

### a. Analysis of a case study

The purpose of this case study is to introduce and explain the significance of the metrics used for comparison. Here we compare the radar and MW fields shown in Fig. 2, corresponding to collocated GMI and Ku-PR observations over the Pacific on 15 November 2015. The wavelet and LP coefficients obtained from the decompositions of *R*_{MW} and *R*_{R} are displayed in Fig. 4. The corresponding wavelet energy spectra are displayed in Fig. 5a. The larger the spectral energy, the larger the spatial variability of the precipitation field at the corresponding scale. Comparing the *R*_{MW} and *R*_{R} wavelet spectra, it can be seen that the energy of the wavelet coefficients is lower for *R*_{MW} than for *R*_{R} at all scales between 5 and 80 km. This deficit of energy is particularly strong at the scales of 5, 10, and 20 km, illustrating the fact that the finescale spatial gradients are smoother in the *R*_{MW} field than in the *R*_{R} field. The radar sees sharper transitions, as it can detect small cells of very high rain intensity or identify small isolated rainy areas, while these are smoothed out in the MW estimates. The deficit of variability at 5 km, which was already revealed in Fig. 3, is not surprising, considering that the *R*_{MW} original nominal resolution is 12.5 km, spatially interpolated to the 5 km grid of *R*_{R}. Nevertheless, the variability at scales of 10–80 km is also underrepresented in the MW field, demonstrating that the MW retrieval methodology is not able to reproduce the finescale spatial variability of the precipitation signal. Also, the fact that the variance of *R*′_{MW,λ} is consistently lower than the variance of the reference *R*′_{R,λ} at all scales confirms that the retrieval error cannot be described as an additive zero-mean random noise [Eqs. (1) and (2)], suggesting a systematic and scale-dependent conditional bias in the retrieved spatial gradients. On the contrary, for the LP component (which captures precipitation variability at scales coarser than 80 km), as

As a consequence of the underrepresentation of finescale variability in the MW estimates, the energy of the error (*R*′_{MW,λ} − *R*′_{R,λ}) increases and the correlation between *R*′_{MW,λ} and *R*′_{R,λ} decreases with finer scales (Fig. 5b). Specifically, at scales of 5–20 km, the energy of the error is of the same order as the energy of the reference signal *R*′_{R,λ}, leading to low NS efficiency, implying that the MW retrieval provides little information on the spatial variability of precipitation at those scales. From the NS efficiency of the wavelet coefficients, the effective resolution of *R*_{MW} is found to be between 20 and 40 km in this case study.

### b. Global results during the TRMM era

Collocated

In all cells, the energy of the wavelet coefficients *λ* and summed over all TRMM overpasses. The energy of the wavelet coefficients at all scales from 5 to 80 km, shown as percentage of the total energy of

Figure 8 shows the map of the correlation of wavelet coefficients

Figures 9a–c displays the global wavelet spectra of

Over land, the error appears to be much more complex, as the spatial variability is overestimated by the MW retrieval at scales of 40 and 80 km and underestimated at scales of 5 and 10 km. At the 20-km scale,

Over the Himalayas, at all scales from 10 to 80 km, the energy of

From the wavelet decomposition the *R*_{MW} field can be reconstructed as linear combinations of the LP fields *R*_{MW} and *R*_{R} are linear combinations of the correlations and MSD of the wavelet and LP coefficients. The last row of Fig. 8 shows the map of correlation and NS efficiency of precipitation rates *R*_{MW} against *R*_{R}. Figure 10 shows the effective resolution of the GMI-GPROF estimates, computed from 6 years of data, for each 3° × 3° cell. Over oceans, the effective resolution is generally between 20 and 40 km. Even if the effective resolution is also between 20 and 40 km in some continental areas, over land, the effective resolution is more often found to be coarser than 40 km and even coarser than 80 km in some regions.

### c. Global results during the GPM era

The same diagnostics used to analyze six years of collocated PR and TMI observations are applied on the 2014–17 collocated GMI and Ku-PR observations (more than 16 000 GPM orbits). Rain rates are derived from GMI measured radiances through the GPROF-2014 algorithm. As for the TRMM era, the large-scale global patterns of the MW-derived mean intensity are very consistent with the radar estimates (Fig. 11). The overestimation over the Himalayas that characterized the TRMM-era retrievals (Fig. 6) does not appear anymore.

Figure 12 is the equivalent of Fig. 8 for the GPM era and shows the distribution of energy across scales for the MW and radar fields. Note that because of the reduced number of observations per 3° × 3° grid cell compared to the TRMM era (700 on average from the GPM era against 3000 for the TRMM era), the results show less spatial coherency, the “noisy” appearance of the maps being an effect of sample size. The orbit of GPM allows the observation of the regions between 35° and 65° of latitude, which was not observed by TRMM. As for TMI during the TRMM era, the GMI fields show less energy than the radar fields in the scales of 5, 10, and 20 km over oceans. Note that this deficit of energy is also observed over land while it was not the case during the TRMM era. This may result from the adoption of Bayesian retrieval scheme over land for the 2014 version of the GPROF algorithm, leading to smooth estimates both over land and oceans. The energy in scales of 40 and 80 km and in the LP term is consistent between the MW and radar fields, over land and oceans. Note that the LP term strongly contributes to energy of the MW and radar fields over medium- and high-latitude oceans. This reveals that over medium- and high-latitude oceans, precipitating systems tend to be dominated by large-scale features and be correlated over long distances, with relatively small finescale variability.

Figure 13 is the equivalent of Fig. 8 for the GPM era. As previously observed, globally, the finer the scale, the lower the correlation and NS efficiency. At all scales, the correlations between

Over oceans, the MW retrieval smooths spatial variations at scales of 5–40 km. For the 80-km scale and the LP term, the error is very low. The relative variances of

Over land the error appears to be of the same type as over oceans: very little spatial variability is resolved for scales finer than 80 km (smooth spatial gradients), and unbiased estimates (random errors) are found for scales equal to or larger than 80 km. The relative variance of the LP error is higher over land than over oceans. Although the variance of

The computed local effective resolution of 2014–17 GMI-GPROF-2014 estimates all over the globe is displayed in Fig. 15. The GMI estimates have an effective resolution between 10 and 20 km over most of the oceanic areas, which actually corresponds to the nominal resolution of the product (12.5 km). Although significant improvements in error reduction at the coarse scale are observed over land compared to the TRMM-era estimates, the effective resolution is still found to be coarser than 40 km over most of the continental areas for GMI estimates.

## 5. Conclusions

The ability of passive MW precipitation retrievals to reproduce the spatial variability of precipitation fields at various scales is evaluated during the TRMM era and during the GPM era, seeking to quantify the improvement allowed by the enhanced instrumental capabilities of GPM and algorithmic developments. Instead of comparing the MW estimates with a reference dataset in terms of rain rates, our evaluation relies on comparing the wavelet coefficients resulting from a two-dimensional dyadic orthogonal wavelet decomposition of MW and radar precipitation fields. The wavelet methodology offers advantages as it filters out possible trends and nonstationarities in the original fields; it captures “edges,” that is, intensity contrasts due to local precipitation features of different scales, and it renders the wavelet coefficient fields spatially and across-scale independent for more meaningful statistical analysis. Using the radar precipitation fields as the reference, our analysis allowed us to determine which scales of variability can be effectively retrieved through passive MW. The main findings are as follows:

- Over oceans, rainfall variability down to spatial scales of 20–40 km is accurately retrieved from TRMM, while improvements down to scales of 10–20 km have been realized from GPM.
- Over land, in most areas, only scales around 40–80 km or coarser are resolved for both the TRMM era and the GPM era. The adoption of the Bayesian retrieval over land for the GPM era led to smoother estimates. The magnitude of the finescale noise over land is reduced compared to the TRMM era, but the finescale spatial variability is still unresolved over land during the GPM era.
- The spatial variability of precipitation rates at a given scale may not be resolved by the MW retrieval for two different reasons. First, they may be filtered out during the retrieval. In that case, the variance of the wavelet coefficients is low, and the information at the corresponding scale is missing. Second, they may be altered by noise. In that case, the wavelet coefficients have a high variance; they contain information, but this information appears to be unrelated to the precipitation process. These two cases may coexist at different scales and can hardly be represented by a unique error model.
- The low-pass term representing the coarse scale (coarser than 80 km) spatial variations of the fields is well retrieved everywhere except over mountainous areas during both the TRMM and GPM eras.

The results help us better understand the scale dependence of the retrieval error. As expected, the finescale variability is generally less accurately retrieved than the large-scale variability. The exception is mountainous areas for which the retrieval of coarse-scale variability is also problematic. In most areas, the larger part of the error in MW rain rates comes from the underrepresentation of the finescale spatial variability (at scales finer than 40 km). A much smaller part comes from the error affecting the coarse-scale fields; this coarse-scale error has the characteristics of a random noise.

It must be noted that while GPROF-2014 retrieval over oceans has been developed in the continuity of GPROF-2010, over complex terrains it marks a deviation from previous versions, as GPROF-2014 is the first version of GPROF implementing the Bayesian retrieval over land. As expected, the Bayesian retrieval produces smoother fields than the empirical relation used for the TRMM-era overland retrieval. The passive MW retrieval over land remains more uncertain than over oceans because of the heterogeneity of land surface emissivity. If the spatial variability at scales finer than 80 km is still hardly retrieved over land, the coarse-scale variability is well reproduced: coarse-scale correlations with radar are high everywhere, and except for mountainous areas, no large-scale systematic bias is noted. The next step in the ongoing development of the GPROF algorithm is the integration of coincident GMI and DPR observations in the database used for the retrieval.

In this paper, the radar precipitation fields from the PR on board TRMM and the DPR on board GPM are used as a reference. It is assumed that the radars are able to resolve precipitation spatial variability at scales down to 5 km. The radar products have been validated and calibrated regionally against ground radar data (Islam et al. 2012; Kirstetter et al. 2013b; Chen and Chandrasekar 2016). Nevertheless, part of the discrepancies between the passive MW and radar estimates may come from errors of the radar estimates. Frozen precipitation is known to be challenging for both radar and passive MW retrieval, which may explain part of the discrepancies over mountainous areas and high-latitude regions. Moreover, the effective resolution of the radar estimates is also eventually questionable, and an approach similar to the one presented here may be used to determine it.

## Acknowledgments

This work was supported by the NASA Global Precipitation Measurement Program under Grant NNX16AO56G. The authors thank Dr. A. M. Ebtehaj, who performed the TMI-PR collocation for the 2002–13 period. The wavelet decomposition procedure was implemented in R distributed under GNU General Public License, using the Waveslim 1.7.5 package (Whitcher 2015).

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