1. Introduction
The number of precipitation-relevant observation platforms and algorithmic developments has increased in recent decades, yielding a large corpus of satellite quantitative precipitation estimation (QPE) products over the tropics. The range of applications of the products includes climatology (Biasutti and Yuter 2013; Roca et al. 2014), hydrological modeling (Bitew and Gebremichael 2011; Cassé et al. 2015), vegetation monitoring (Pierre et al. 2011), and infectious disease risk management (Guilloteau et al. 2014). Many validation studies of these products have been undertaken based on comparison with ground data (Sapiano and Arkin 2009; Gebremichael et al. 2014) at resolutions down to 0.25° and 3 h, and occasionally finer (Hong et al. 2007; Behrangi et al. 2012; Habib et al. 2012). Few of these validation studies have focused on West Africa (Nicholson et al. 2003; Roca et al. 2010; Gosset et al. 2013), where rain gauge networks are very sparse (Lorenz and Kunstmann 2012). These validation studies over Africa have been performed at a coarse spatiotemporal resolution (i.e., 1° and 1 day or coarser). Some products provide much more finely gridded data (down to 2.8-km instantaneous estimates) but remain unevaluated at their full resolution in West Africa.
Because of the intermittent nature of rain, QPE can be thought of as a double exercise: 1) identification of rainy areas and 2) estimation of rain intensity. The rain/no rain discrimination from passive satelliteborne sensors is far from trivial. Microwave and infrared brightness temperatures measured by the sensors cannot be unambiguously associated with a unique hydrometeor profile (Stephens and Kummerow 2007). Spatiotemporal variability of accumulated rain depth is partially driven by the rain/no rain variability. For a given period and over a given area, the cumulated rain depth is the product of precipitation fraction (i.e., fraction of space and time that is precipitating) and the mean rain intensity. The relative importance of each term in explaining rainfall variability depends on the considered resolution and the type of rainfall regime. Over the tropical continents, where a few hours of rain per year can produce most of the annual rain depth, the variability of the precipitation fraction is a key determinant (Morrissey et al. 1994; D’Amato and Lebel 1998; Kebe et al. 2005). In West Africa, most rainfall is provided by organized mesoscale convective systems (Houze 2004) of spatial expansion ranging from 102 to 106 km2 and propagating from east to west over thousands of kilometers. These systems may live up to a few days, but they produce rainfall for only a few hours over a given point. These physical processes give rise to a specific finescale signature in the rain fields.
Hossain and Huffman (2008) recommended to systematically analyze the dependence of error metrics to scale when assessing satellite rainfall data. To this end, Turk et al. (2009) and Sohn et al. (2010) aggregated satellite rain fields at various spatiotemporal resolutions to compare them with ground data. In this paper, the rain/no rain discrimination ability of a suite of high-resolution products derived from spaceborne passive sensors is evaluated in West Africa. Products considered are the Tropical Amount of Precipitation with an Estimate of Errors (TAPEER) intermediate data rain mask, Climate Prediction Center morphing technique (CMORPH), Global Satellite Mapping of Precipitation (GSMaP), and Precipitation Estimation from Remotely Sensed Information Using Artificial Neural Networks–Cloud Classification System (PERSIANN-CCS). The last three products are also evaluated as rain masks. A multiscale approach based on discrete wavelet decomposition is used to investigate the scale dependence of the masks’ performance. Satellite-derived rain masks are compared with ground radar–derived rain masks over Burkina Faso and with TRMM Precipitation Radar (PR)-derived rain masks over the whole West Africa. The wavelet coefficients resulting from the decomposition of the masks and characterizing the variations of these masks at various scales are compared. At each scale, the variances and the covariance of the coefficients are computed. This is equivalent to analyzing the masks through various bandpass filters and comparable to what is done in Turk et al. (2009) and Sohn et al. (2010), where the aggregation can be seen as a low-pass filtering.
The objective is to determine the relative contribution of each scale to the masks’ variance and covariance and to the variance of their difference. The scales for which the variance of the difference of the coefficients is greater than the variance of the radar-derived coefficients can be considered as noninformative. At these scales the information provided by the satellites degrades the estimation of the precipitation fraction.
The subdaily variations of the precipitation fraction are partially driven by the diurnal cycle that is prominent in the tropics (Nesbitt and Zipser 2003; Roca et al. 2010). The satellite-derived mean diurnal cycle of rain occurrence is evaluated here against gauges in Benin and Niger.
This paper is organized as follows. Section 2 presents the datasets used in this study. In section 3, the method for multiscale qualification through wavelet transform is explained. Results of the decomposition, multiscale skill scores, and mean diurnal cycle are presented in section 4, with emphasis on both local and regional scales.
2. Data
a. High-resolution multisatellite rainfall products
The products evaluated are time series of high-resolution (i.e., finer than 0.1°) mapped estimates. Each estimate is an instantaneous snapshot of the surface rain rate at a given time. All the products have a sampling period shorter than 1 h. For all the products, microwave (MW) radiances measured from satellites forming the GPM constellation (Hou et al. 2014) and infrared (IR) images from geostationary satellites are used as primary or auxiliary sources of information. Spatial resolution, sampling period, and other main characteristics of the products are summarized in Table 1.
Main characteristics and references of the satellite-based products. All products are instantaneous rainfall estimation. For the third column from the left, the products PERSIANN-CCS, GSMaP, CMORPH, and IMERG are mapped on a grid with a constant increment in latitude and longitude (0.04°, 0.1°, 0.073°, and 0.1°, respectively). TAPEER rain mask is defined on the original irregular grid of SEVIRI images. Values are equivalent resolutions at the ground in the Xport radar coverage area.
TAPEER is a 1°, 1-day quantitative rain estimation algorithm based on the Universally Adjusted Global Precipitation Index (UAGPI) technique (Xu et al. 1999; Chambon et al. 2013a). TAPEER was developed under the French–Indian Megha-Tropiques mission framework (Roca et al. 2015). TAPEER combines thermal infrared (10.8 μm) brightness temperatures 
For each 1° × 1° × 1 day estimation volume, a 3° × 3° × 1 day neighborhood or training volume is defined.
The
Histograms of collocated data are computed and a rain/no rain threshold
A binary indicator field, the rain mask
- Finally,
where
The final estimation 
PERSIANN-CCS is produced by the Center for Hydrometeorology and Remote Sensing of the University of California, Irvine (Hong et al. 2004). The algorithm relies on the identification of rainy clouds using features such as cloud height, areal extent, and texture from IR images. PERSIANN-CCS algorithm also uses MW observations to statistically adjust IR-based estimates.
GSMaP is a JAXA product (Ushio et al. 2009). MW radiances are used to estimate rain rates through a radiative transfer model. To compensate for the sparse MW observations, rain fields are advected in space and time through a simple motion model called moving vector. Each new MW observation is assimilated using Kalman filtering. IR geostationary images are used for the computation of motion vectors. The near-real-time version of the product, which does not integrate gauges, is used.
CMORPH (Joyce et al. 2004) uses MW-based rainfall estimates as the primary source of information. From every MW observation, two “predictions” are made, forward and backward in time, using advection vectors computed from IR images. Estimated rain rates between two MW observations result from the merging of the two predictions (morphing). In this work, the gauge-free version 1.0 of CMORPH is used.
Integrated Multisatellite Retrievals for GPM (IMERG) recently developed by the U.S. Global Precipitation Mission team is a synthesis of PERSIANN-CCS and CMORPH algorithms (Huffman et al. 2015). The IMERG algorithm also inherited the correction procedure from gauge data of the TRMM Multisatellite Precipitation Analysis (TMPA) algorithm (Huffman et al. 2007). The PERSIANN-CCS algorithm is first run independently and PERSIANN-CCS rain fields are then optimally integrated into a CMORPH interpolation scheme using a Kalman filter. The data used here are the uncalibrated precipitation fields, which do not include correction from gauges. As this article is being written, IMERG has not been processed for 2012 and 2013 yet, so only the 2014 rainy season is considered. A limited assessment of IMERG detection capabilities focused on the energy spectrum is presented in section 4b.
b. Xport polarimetric radar data in Burkina Faso
Xport is an X-band (9.4 GHz) dual-polarization Doppler precipitation radar (Koffi et al. 2014). It operated in Ouagadougou, Burkina Faso (12.4°N, 1.5°W, Sahelian climate) during the 2012 rainy season (i.e., May–October) as a part of the Megha-Tropiques ground validation campaign. Its ground coverage is a 120-km-radius disk (Fig. 1). Its radial resolution is 200 m and its angular resolution is 1° (equivalent to 2-km width at the maximum range). The radar performs a complete scan of the surrounding area every 6 min. Rain rates used here are derived from differential phase shift between horizontal and vertical signals (Matrosov et al. 2002; Koffi et al. 2014). The intercomparison of several Xport rain fields derived from various independent variables shows a very good consistency in terms of rain detection at the resolutions considered in this study (i.e., 2.8 km and larger; Kacou 2014). Radar rain fields have also been validated against gauge data (Kacou 2014). Xport rain detection fields are considered as a reference dataset for the evaluation of TAPEER rain mask and the other satellite products presented above. Because of its high spatiotemporal resolution, the radar is an ideal tool to evaluate both temporal and spatial finescale rain variability. On the other hand, spatial scales larger than the radar coverage cannot be evaluated with a single radar, and conclusions obtained from local data cannot be extrapolated to a larger regional scale without additional information.
Map of West Africa with the areas studied. The red circle indicates the Xport radar coverage area in Burkina Faso, the green squares indicate the AMMA-CATCH gauge networks in Niger (Sahelian climate) and in Benin (Sudanese climate), and the blue rectangle indicates the extended region where regional spectra are computed and 
Citation: Journal of Hydrometeorology 17, 7; 10.1175/JHM-D-15-0148.1
c. TRMM PR
The TRMM satellite carried a Ku-band radar from November 1997 to October 2014 to estimate the rain intensity from reflectivity profiles. The data used here are the TRMM 2A25, version 7, near surface rain (Iguchi et al. 2000) provided with a 5-km spatial resolution. The radar swath width at the ground is 247 km, not significantly greater than Xport’s range, but TRMM PR provides coverage of the whole West Africa region. More than 3000 orbit sections over West Africa during the 2012–14 rainy seasons are considered here (Fig. 1). The data are publicly available online (http://mirador.gsfc.nasa.gov/cgi-bin/mirador/presentNavigation.pl?tree=project&dataset=2A25%20%28Version%20007%29:%20Radar%20Rainfall%20Rate%20and%20Profile%20%28PR%29&project=TRMM&dataGroup=Orbital&version=007).
d. AMMA-CATCH gauge networks in Benin and Niger
Two dense gauge networks setup in Benin (Sudanese climate) and in Niger (Sahelian climate) since the early 1990s have been operated as an element of the Couplage de l’Atmosphère Tropicale et Cycle Hydrologique (CATCH) observatory of the African Monsoon Multidisciplinary Analysis (AMMA) program (Lebel et al. 2010). Both networks are made of 40–45 rain gauges covering a square area of 1° (Fig. 1). Gauges are automatic tipping buckets with a tip every 0.5 mm cumulated depth, leading to a delay of 30 min between two tips for a 1 mm h−1 rain rate. Because of sparse spatial sampling and time-integrated measurements, rain gauges are weak in representing instantaneous rain fields at high resolution (Ciach 2003). Here, rain gauges are not used for a direct comparison with satellite instantaneous estimates, but only to infer statistical properties such as the multiyear mean diurnal cycle of rain occurrence.
e. Binary indicators generation
The comparison of two datasets with different resolutions is performed at the coarsest resolution: Xport data are aggregated to 2.8, 4.4, 8, and 11 km for the comparison with TAPEER rain mask, PERSIANN-CCS, CMORPH, and GSMaP. TAPEER rain mask is aggregated to 5-km resolution for the comparison with TRMM PR.
Thresholds for the rain intensity fields are defined to generate rain detection masks. When comparing several rain masks, the differences in rain detection sensitivity associated with various passive and active sensors and various detection methods must be accounted for. Figure 2 shows the probability of exceedance for rain rates between 0 and 5 mm h−1 computed on Xport data and on collocated CMORPH, GSMaP, and PERSIANN-CCS data for the 2012 rainy season. The various datasets have different statistical distributions for low rain rates. All three products overestimate the occurrence of rain compared to Xport data. For the same period and area, the rate of rainy pixels in TAPEER rain mask is 12%. To generate the indicators 
Probability of exceedance as a function of rain rate. Black, dark gray, gray, and light gray curves represent the Xport radar data at 2.8-, 4.4-, 8.0-, and 11-km resolution, respectively. The red curve represents GSMaP (11 km), the green curve represents CMORPH (8 km), the purple curve represents PERSIANN-CCS (4.4 km), and the dashed blue curve represents the TAPEER rain mask (2.8 km). The computations are carried out over the Xport coverage domain for the period May–October 2012.
Citation: Journal of Hydrometeorology 17, 7; 10.1175/JHM-D-15-0148.1
When aggregated to a resolution coarser than its original resolution, the indicator I can be interpreted as a precipitation fraction ∈ [0, 1] (where the brackets indicate a closed interval). The precipitation fraction is a surface ratio and is therefore expressed in square meters per square meters. At 1° and 1-day resolution, the correlation between the precipitation fraction derived from 
3. Methodology
As stated in section 2e, the satellite detection fields 
The pixel-to-pixel comparison of 
Several methods are well suited to describe the spatiotemporal structure and the covariation of two variables. Among them are geostatistics (Grimes and Pardo‐Igúzquiza 2010) and multiscale analysis through fractal theory (Lovejoy and Mandelbrot 1985) or wavelet transform (Kumar and Foufoula-Georgiou 1997; Venugopal et al. 2006). We chose the last one for its simplicity and its computational efficiency when dealing with massive data. Wavelet transform presents many similarities with the well-known Fourier transform (Flandrin 1998). In the Fourier domain, the wavelet decomposition is equivalent to a filter bank decomposition (Vetterli and Herley 1992). Wavelet spectra and Fourier spectra can be interpreted in a similar way. Wavelet transform is a lossless (reversible) operation. It does not rely on any approximation. It does not require any specific property such as data stationarity, which is questionable when dealing with precipitation data (Over and Gupta 1996). Wavelet coefficients are localized in space and time identically to the original data. The correlation between two series of wavelet coefficients can be interpreted in the space–time domain.
The use of the wavelet transform for the comparison of observed or modeled fields has been proposed by Briggs and Levine (1997). Kumar and Foufoula-Georgiou (1993, 1997), Venugopal and Foufoula-Georgiou (1996), Turner et al. (2004), Casati et al. (2004), and Johnson et al. (2014) showed the applicability of this method specifically for the analysis of rain fields. In this study, the Haar wavelet is used because it is well suited for representing binary fields (Kumar and Foufoula-Georgiou 1997; Domingues et al. 2005). The Haar scaling function is a simple averaging operator, and the wavelet coefficients are computed as finite differences (see appendix B). The Haar wavelet has been used by Casati et al. (2004) and Saux Picart et al. (2012) to analyze binary images, as it is done in the present study.
a. The discrete wavelet transform
For the largest scale m = M, wavelet coefficients should be interpreted differently from the case when m < M. The coefficient series associated with the index M can be seen as the residual of an uncompleted decomposition (because this decomposition has a finite number of levels). The term 
Spatial wavelet transform applied to 
Citation: Journal of Hydrometeorology 17, 7; 10.1175/JHM-D-15-0148.1
Each instantaneous rain mask image is first decomposed by spatial scale. The depth M of the decomposition is limited by the Xport radar coverage, which is around 200 km. Variable M also depends on the products’ original resolution. For the TAPEER rain mask, as
Each of the M + 1 time series is then decomposed through a temporal wavelet transform of depth N.
Finally, (M + 1) × (N + 1) time series of wavelet coefficients result from the two successive decompositions. Each of these series represents the signal’s variation at a given spatiotemporal scale and is designated by spatial and temporal length scales
b. Discrete wavelet energy spectrum and cospectrum
4. Results
a. Local comparison of satellite rain masks with Xport rain masks
1) Coarse spatial scale:
As stated in section 3, the largest spatial-scale coefficient 
Temporal evolution of 180 km × 180 km instantaneous 
Citation: Journal of Hydrometeorology 17, 7; 10.1175/JHM-D-15-0148.1
Performance of satellite rain masks against Xport rain mask, considering only the coarse-spatial-scale coefficients 
Figure 5 shows the spectra resulting from the wavelet temporal decompositions of 
Temporal spectra of coarse-spatial-scale components 
Citation: Journal of Hydrometeorology 17, 7; 10.1175/JHM-D-15-0148.1
2) Fine spatial scales:
Figure 5 shows the spectra of 
(top) Spatiotemporal energy spectrum [(m2 m−2)2] of 
Citation: Journal of Hydrometeorology 17, 7; 10.1175/JHM-D-15-0148.1
Spatiotemporal normalized energy spectra of (top) 
Citation: Journal of Hydrometeorology 17, 7; 10.1175/JHM-D-15-0148.1
Figure 6 (bottom) shows the 
Correlation of wavelet coefficients 
Citation: Journal of Hydrometeorology 17, 7; 10.1175/JHM-D-15-0148.1
3) Multiyear mean diurnal cycle of rain occurrence
The spatiotemporal spectral analysis (Fig. 6) shows that 90% of 
Mean diurnal cycle of rain occurrence in (top) Benin and (bottom) Niger for the period July–September 2012–14. Black is AMMA-CATCH rain gauges (with sampling variance), blue is TAPEER rain mask, green is CMORPH, red is GSMaP, and purple is PERSIANN-CCS.
Citation: Journal of Hydrometeorology 17, 7; 10.1175/JHM-D-15-0148.1
This demonstrates that despite the low correlations shown in Fig. 8, finescale temporal variations are not purely random and the correlations are still significantly positive. Averaging several realizations of satellite detection series enables us to suppress random variability and reveals relevant finescale temporal variations. We may also expect temporal averaging to exhibit coherent small-scale spatial patterns. This cannot be verified with Xport radar data or AMMA-CATCH gauges, as covered regions do not exhibit a significant climatological gradient of rain occurrence, and long-term averages are spatially homogeneous.
b. Energy spectra at regional scale, comparison of 
The energy spectra are computed on a larger area. TAPEER rain mask is compared with TRMM PR to check if local results obtained with Xport data are still relevant on the West Africa regional scale. Figure 10 shows the energy spectra from spatial wavelet decomposition of the four satellite rain detection fields 
Normalized spatial wavelet energy spectra, that is, variance of wavelet coefficients 
Citation: Journal of Hydrometeorology 17, 7; 10.1175/JHM-D-15-0148.1
TRMM PR is used to evaluate the spatial patterns in TAPEER rain mask, with a regional perspective. A threshold with 
Spatial wavelet energy spectra [(m2 m−2)2]. Black is 
Citation: Journal of Hydrometeorology 17, 7; 10.1175/JHM-D-15-0148.1
In addition to the local study of the diurnal cycle in Niger and Benin, the 3-yr mean diurnal cycle of rain occurrence was also mapped over the West African region for TAPEER rain mask, CMORPH, GSMaP, and PERSIANN-CDR. Figure 12 shows maps of mean diurnal rain occurrences for 3-h windows. The diurnal cycles from the four products are in remarkably good agreement, showing again that averaging along one dimension (here time) tends to remove the random part of satellite fields. The salient patterns are orographic effects, with rainy systems forming over elevated terrains (i.e., Marrah Mountains in Sudan, Mandara Mountains between Cameroon and Nigeria, and Aïr Mountains in Niger) around 1800 local solar time and then propagating westward.
Normalized mean diurnal cycles of rain occurrence of (from left to right) TAPEER rain mask, CMORPH, GSMaP, and PERSIANN-CCS for the period July–September 2012–14. The color scale expresses the contribution of each time interval to total daily mean precipitation fraction. Time intervals are 3 h long, centered at 0000, 0300, 0600, 0900, 1200, 1500, 1800, and 2100 UTC. The diurnal cycles are only displayed for areas that are rainy during more than 1% of the time steps.
Citation: Journal of Hydrometeorology 17, 7; 10.1175/JHM-D-15-0148.1
5. Conclusions
The wavelet transform highlights the contribution of each spatial and temporal scale to signals’ variances and covariances. All four satellite-based, high-resolution rain masks considered exhibit energy spectra that are consistent with each other and with the ground radar spectrum. In particular, despite a small deficit of variance at scales 
The multiscale method highlights the fact that standard pointwise scores such as correlation and mean quadratic difference are a combination of different scale-specific performances. A few specific temporal and spatial scales dominate the satellite product’s overall performance as revealed by the concentration of spectral energy.
All satellite-based products show a similar evolution of performance regarding spatial and temporal scales. The correlation between satellite detection fields and radar detection fields is essentially explained by their good agreement at coarse spatial and temporal scales. The relation between surface rain and 
All products considered have skills in reproducing the mean diurnal cycle of rain occurrence at 1-h temporal scale in Niger and Benin. This demonstrates that averaging along one dimension enables us to separate meaningful small-scale information from random variations. Similar analysis with radar data from a region with a strong local climatic gradient, such as a coastal or mountainous area, would be interesting to assess the effect of temporal averaging on finescale spatial patterns.
The results on the local scale were confirmed using TRMM PR data over the whole West African region and over a 3-yr period.
The results presented focused on rainfall detection above a fixed rain intensity threshold. They are relevant for the question of quantitative estimation of rainfall amounts because precipitation fraction and rainfall cumulated depth are related. How much of the rainfall variability can be captured by the rain mask alone depends on the scales considered, on the precipitation regime, and on the value of the selected intensity threshold. The method presented could be applied to evaluate rain amounts (rather than a single rain mask) by repeating the multiscale binary indicators analysis with varying values of the intensity threshold (Casati et al. 2004).
In the region studied, the spectral contents of satellite rain masks are not optimal regarding the mean-squared difference between satellite and radar data. The satellite-based rainfall estimate cannot be optimal in terms of mean-squared error and at the same time preserve statistical properties of actual rain. An adapted filtering of the finescale variations would reduce the mean-squared difference but would decrease the consistency between the radar and satellite power spectra. It would also degrade the mean diurnal cycle dynamics, which is currently satisfactory.
These results suggest that deterministic approaches to rain detection from passive sensors at high resolution are intrinsically limited by the nature of the precipitation process and by the inherent ambiguity of the relation between surface rain and cloud-related variables measured by spaceborne passive sensors (Stephens and Kummerow 2007). The development of a probabilistic approach for high-resolution rain detection deserves further investigation. Ensemble generation by wavelet transform associated with stochastic methods (Perica and Foufoula-Georgiou 1996), with rain mask as constraining data, will be investigated for this purpose.
Acknowledgments
The authors would like to acknowledge the support from CNRS, IRD, and Université de Toulouse III Paul Sabatier. Part of this work was supported by the CNES. The authors thank the AMMA-CATCH team for providing gauges data, and Dr. K. Hsu and Mr. D. Braithwaite from the University of California, Irvine, CHRS for providing PERSIANN-CCS full resolution data. Nicolas Taburet and Estelle Lorant are thanked for providing the TAPEER rain mask data. The authors also thank the NOAA/Climate Prediction Center, the JAXA Precipitation Measurement Mission science team, and the U.S. Global Precipitation Mission team for respectively making available CMORPH, GSMaP, and IMERG data. Audine Laurian is thanked for her proofreading and correction work.
APPENDIX A
Standard Metrics
Uncentered correlation is preferably used when both series contain many zeroes. The classical correlation would be artificially enhanced by the large number of correctly detected zeros 
APPENDIX B
Discrete Wavelet Transform, Haar Wavelet
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