Small-Scale Horizontal Rainrate Variability Observed by Satellite

Small-Scale Horizontal Rainrate Variability Observed by Satellite Atul K. Varma and Guosheng Liu Department of Meteorology, Florida State University Tallahassee, FL 32306, USA Corresponding Author Address: Atul. K. Varma Meteorology and Oceanography Group Space Applications Centre Indian Space Research Organization Ahmedabad 380 015, India 91-79-2691 6050 91-2717-23 5431 avarma@sac.isro.gov.in (Submitted to Monthly Weather Review Note, September 2005)

Abstract The horizontal distribution of rainrates within an area comparable to the pixel size of satellite microwave radiometers and the grid size of numerical weather prediction models has been studied over the global tropics using three years of Tropical Rainfall Measuring Mission satellite precipitation radar (PR) data. The global distribution of rainrate standard deviation derived from the PR data suggests that the horizontal variability of rainrates is largely influenced by two factors: surface type (land or ocean) and latitudinal location (tropical or extratropical). Except for light stratiform rain, the land-ocean contrast seems to be the dominant feature for the differences in conditional probability density functions (PDF) of rainrate. That is, oceanic rainrate distribution is narrower when rainrate is low, but becomes broader when rainrate is high. For light stratiform rain, there is no clear difference among the rainrate PDFs for rain events over land and ocean. The latitudinal variation of rainrate PDFs seems to be greater for heavy rain than for light rain. In particular, there is no measurable difference in over-land convective rainrate PDFs between tropics and extratropics. Based on three years of observational data, two attributes, fractional rain cover and conditional rainrate PDF, are parameterized as a function of 0.25 x 0.25 areal rainrate. These parameterizations are particularly useful in satellite microwave rainfall retrieval and assimilation of satellite microwave radiance data in numerical weather prediction models. 1

1. Introduction Precipitation spatial variability ranges from a few meters to hundreds of kilometers (McCollum and Krajewski, 1998; Tustison et al., 2003). In this paper, we investigate the distribution of instantaneous rainrates within an area comparable to the footprint size of microwave satellite pixels and the grid size of numerical weather prediction models, as well as the behavior of this distribution over the global area between 40 N and 40 S. Data from the Precipitation Radar (PR) onboard the Tropical Rainfall Measuring Mission (TRMM) satellite are utilized. The unique feature of this study is that it provides the global distribution (within the domain of TRMM) of the attributes describing the small-scale rainrate variability; this was not possible until the space-borne precipitation radar data were available. While it provides a better understanding of the natural variability of rain fields, the current study particularly aims at deriving observational rainrate statistics for improving rainfall retrieval from satellite microwave measurements. Knowledge of the sub-pixel scale rainrate variability is imperative in dealing with beam-filling problems associated with rain retrieval from satellite microwave radiometry data. The beam-filling problem results from an inhomogeneous distribution of rainrate over the large satellite radiometric field-of-view (e.g., Wilheit et al., 1977; Spencer et al., 1983; Chiu et al., 1990; Harris et al. 2003; Kummerow et al., 2004). Kummerow et al. (2004) discussed the effects of beam-filling problem on climate scale rainfall estimation from passive microwave sensors. Following Chiu et al. (1990) and taking into consideration the slant path radiative transfer calculation, they estimated the average beam-filling problem biases the retrieval by a factor of 1.26. 2

In rain retrievals, the brightness temperature (T B ) measured from a satellite microwave radiometer is the integrated radiation from all raining and no-rain areas within a pixel. It may be expressed by the following equation (Varma et al., 2004): T = [ 1 FRC( Rav )] TB (0) + FRC( Rav ) TB ( x) PDF( x, Rav dx (1) B ) where: x=ln(r); T B (0) is the brightness temperature at no-rain area; and T B (x) is the brightness temperature at rainrate R. FRC(R av ) and PDF(R av ) are, respectively, the fractional rain cover (FRC) and sub-pixel rainrate probability density function (PDF) at the pixel-averaged rainrate, R av. The retrieval problem is to estimate R av from the observed T B s at several frequencies. Clearly, the characteristics of the PDF of sub-pixel rainrates as well as the FRC are needed information in this inversion problem. A similar argument can be made for direct assimilation of satellite brightness temperature data into numerical weather prediction models (e.g., Aonashi and Liu, 1999) that provide rainrate (or rain water content) averaged over a model grid while the satellite observes the pixel averaged brightness temperature; to connect model and observation variables, it is necessary to know the sub-pixel (or subgrid) distribution of rainrates (Arakawa, 2005; Nykanen et al., 1997). Varma et al. (2004) used surface radar data collected near Japan and in the tropical Pacific warm pool to determine the conditional PDF of rainrate and FRC, and parameterized these two attributes in terms of R av. By applying the parameterization to (1), they obtained a closer agreement between satellite-observed and radiative transfer model simulated brightness temperatures than by assuming homogeneous rainrate within a pixel, which reduces the systematic retrieval error caused by the beam-filling problem. However, since the surface radar data only cover a limited area, the applicability of their R 3

rainrate distribution model to global rainfall retrieval is unwarranted. In this study, we take advantage of the satellite radar measurements to study the horizontal variability of rainrate in a global perspective. By analyzing the rainrate variability over the global tropics, we attempt to answer the following questions: What are the main factors that influence the horizontal variability of rainrate? How can we parameterize the horizontal variability? 2. Data Data used in this study are from the TRMM PR, which is a Ku band (13.8 GHz) cross-scanning precipitation radar. The low orbiting (at 350 km) TRMM satellite provides coverage over the tropical region between about 37 S to 37 N (Kummerow et al., 1998). The PR onboard TRMM scans ±17 from the nadir with 49 positions, resulting in a 220 km swath with a spatial resolution of 4.3 km. Our analysis utilizes the recently-released version 6 of the standard data product, 2A25 orbital data, that contains instantaneous rainrates from surface to 20 km altitude with a 250 m vertical and a 4.3 km horizontal resolution and a minimum detectable rainrate of ~0.7 mm h -1. The version 6 dataset offers an improved rain classification scheme described by Awaka et al. (2004). A hybrid of the Hitschfeld-Bordan method and the surface reference method (Iguchi and Meneghini, 1994) are used to estimate the vertical profile of the attenuation-corrected effective radar reflectivity factor (Z e ). The vertical rainrate profile is then calculated from the estimated Z e profile by using an appropriate Z e -R relationship. One major difference of the PR algorithm from Iguchi and Meneghini (1994) is that, in order to deal with the uncertainties in measurements of the scattering cross section of the surface as well as the 4

rain echoes, a probabilistic method is used. Since radar rain echo from near the surface is hidden in the strong surface echo, the rain estimate at the lowest point in the clutter-free region is given as the near-surface rainrate for each angle bin. The time period covered by the study is 3 years from 1 December 1997 to 30 November 2000. During this period the PR provided about 7.7 billion observations over its coverage area. The TRMM datasets are available from the NASA Data Active Archive Center. The dataset also contains the information on rain type, such as stratiform or convective. There are two methods for classifying rain types: the vertical profile method (V-method) and the horizontal pattern method (H-method). Both methods classify rain into three categories: stratiform, convective, and other (Awaka, 1997). The V-method is based on detection of bright band. When bright band is present, the rain is classified as stratiform. It detects the convective rain that is characterized by strong radar echo. When rain type is neither convective nor stratiform, it is defined as type other. The H- method, orginally based on Steiner et al. (1995), classifies rain type based on the horizontal pattern of radar reflectivity. It detects convective rain by determining the convective core and rain adjacent to it. When rain is not convective, it is generally classified as stratiform. But if the echo is very weak, it assigns the rain type as other. Both methods are used to classify rain type, but a merged type is provided in the PR 2A25 standard dataset. The merging rules are defined by Awaka (1997) as follows. (1) When it is stratiform (convective) by one method and is stratiform (convective) or other by the other method, the merged rain type is stratiform (convective). (2) When it is convective by V-method but is stratiform by H-method, the merged rain type is convective. (3) When it is statiform by V-method (i.e., bright band is detected) but is 5

convective by H-method, the merged rain type is convective or stratiform depending upon the level of confidence in the bright band detection. (4) When it is other by both methods, the merged rain type is other. The statistics of the horizontal variability of rainrates were examined within an area of 0.25 (latitude) x 0.25 (longitude) (~30 x 30 km 2 ). This was carried out by moving a window of 0.25 x 0.25 by window-size area in east-west and north-south direction over each PR pass. The size of the window is similar to the footprint of past and presently available 19 GHz channels of space-borne microwave radiometers, which provide the emission signal for rain measurement. In the following discussions, we refer the 0.25 x 0.25 area as a window. Each window allows approximately 40 PR pixels in it. The windows are also recognized by their surface type (land or ocean) and rain type (convective or stratiform). The rain type of a window is considered convective if more than a certain fraction of raining PR pixels therein is determined to be convective in the 2A25 product. Otherwise, the rain type of the window is considered to be stratiform. Thus a threshold value defined by the convective rain fraction (CRF) within a window is used to determine the rain type of the window. In the present study, we have considered two threshold values of CRF, 0.1 and 0.33, to classify the rain type of a window and analyzed the impact of the threshold value of CRF on our results. The threshold value of 0.33 divides the total convective and stratiform windows in a ratio of about 1:3 that is close to the ratio suggested by Schumacher and Houze (2003). To study the impact of a possible over-classification of the stratiform rain by PR, as apprehended by Schumacher and Houze (2003), on our results, a smaller value, 0.1, of threshold is also considered. The three-year PR data provide approximately 11.16 and 8.17 million convective, and 6

11.00 and 14.02 million stratiform windows with 0.10 and 0.33 threshold values, respectively. The windows found to be other type had window averaged rain always < 1 mm h -1, and were fewer in number (0.018 and 0.033 million over land and ocean, respectively). The other type windows are discarded from the analysis. Figure 1 shows the global distribution of the number of rainy (R av >0) windows separately for convective and stratiform rain that are defined using CRF threshold value of 0.33. The plots generated with CRF threshold value of 0.1 for rain type classification show the different number densities but similar variability patterns and hence are not presented here. These maps are proxies of frequency distribution of rain occurrence. While convective windows are most common near the equator, stratiform windows have a more frequent occurrence in the higher latitudes. An exception is the Atlantic Gulf Stream region east of the United States where both convective and stratiform windows have a frequent occurrence. The number of rainy windows is low over several climatologically low-rain areas, e.g., west coasts of South America and California; Australia; and northern Africa. 3. Results While analyzing the data, we found that the features of rainrate variability deviate with rainfall intensity. To better capture this information, we carried out data analysis by dividing data into three rainrate intensity categories: light (R av <2.5 mm h -1 ), moderate (R av =2.5-10 mm h -1 ) and heavy (R av >10 mm h -1 ). Figures 2 and 3 show the global distribution of the rainrate standard deviation within 0.25 x 0.25 windows averaged over the period of three years with the CRF value for rain classification as 0.1 and 0.33, 7

respectively. In these figures, we only show the standard deviation in the areas where there are enough observations to ensure a 90% confidence level. Both Figures 2 and 3 show similar patterns except for the difference in the absolute values of averaged standard deviation. Both figures indicate that two major features appear to exist in the global distributions of standard distributions. One is the latitudinal variation with greater variability of rainrates in the tropical rain belts, particularly in the oceanic regions. The other feature is the land-ocean contrast, which is dominant in the light convective rain category. The standard deviation is greater over land than over ocean for light rain, but it is smaller for the moderate and heavy rain categories. It is hard to identify a latitudinal trend of the standard deviation over land but its latitudinal variation is evident over oceanic areas. The land-ocean difference in the convective systems has been described by many investigators in the past (e.g., Liu and Fu, 2001; Schumacher and Houze, 2003). But this difference has not been studied as a function of rainrate by any of the previous researchers. Here we found that the land-ocean difference in moderate/heavy rain categories is small and has opposite sign compared to that in the light rain category. One plausible explanation is that the moderate/heavy rain accumulated over a 0.25 x 0.25 area arises from organized large convective systems, the rainrate distribution of which is largely governed by large-scale atmospheric conditions. Therefore, the rainrate field in this case is relatively less sensitive to surface forcing. On the other hand, a large part of the light convective rain results from isolated convective systems that often produce intense rain over small areas while the averaged rainrate over window size of 0.25 x 0.25 is low. Such small rain systems can be greatly influenced by heating from the 8

surface. In this study, we found that rainrates over land have larger horizontal variability than those over ocean for light convective rain category. The difference of rainrate variability over the tropics and extratropics is possibly due to the different mechanisms involved in the formation of clouds. Several studies have indicated different cloud microphysics of convective cloud systems over tropics and extratropics (e.g., Bringi et al., 2003). Similarly, latitudinal difference is also expected for stratiform rain. While extratropical stratiform rain is mainly associated with large-scale lifting near cyclones and fronts, in tropics stratiform rain occurs in anvils or regions of dissipating convections (Houze, 1997). To better extract rainrate variability characteristics in different regimes, further analysis is conducted by dividing data into the following: two latitudinal regions, i.e., tropics (20 N - 20 S) and extratropics (20-40 S or 20-40 N), and two surface types (land and ocean), and two rain types (convective and stratiform). a. Fractional Rain Cover The FRC in each 0.25 x 0.25 window is computed separately for convective and stratiform rain, and for ocean and land regions. For a clear representation of the relationship between FRC and window averaged rain rate R av, we have averaged values of R av in each 2.5% FRC bins. Figure 4 shows the FRC versus R av plot averaged for the entire period of study with a CRF threshold of 0.33 and a 90% confidence interval for R av in each FRC bin. The 90% confidence intervals for R av are very small in all FRC bins as compared to mean R av values in corresponding bins. The corresponding plots for CRF threshold of 0.10 show a similar pattern and hence they have not been shown. Although the difference of the FRC-R av relations caused by surface type and latitudinal location is 9

small, there is a clear distinction between convective and stratiform rain, i.e., a steeper slope for stratiform than for convective rain. In other words, given the same areal rainrate, a much larger (smaller) fraction of the area is probably filled with rainfall if the rain type is stratiform (convective). This is logical given the fact that, while stratiform rain covers a large area uniformly, convective rain comes from individual cells. The FRC shown in Fig.4 can be fitted with the following function: FRC = 100 [1 exp( arav )], (2) where FRC is in %, R av is in mm h -1, and a (0.2701 for convective and 0.9578 for stratiform rain) is a regression coefficient. When using (2) with the above values for a to evaluate FRC, the correlation coefficient and root-mean-square (rms) difference between estimated and observed FRCs are 0.99 and ~3.5%, respectively. b. Conditional Probability Distribution Function Figure 5 shows the averaged conditional PDFs of rainrates separated by: rain type (determined by a CRF threshold of 0.33), surface type, latitudinal location, and rainrate category. The 90% confidence intervals for frequencies calculated for these figures are less than 0.01 in all ln(r) intervals. The corresponding plots for a CRF threshold of 0.10 are very similar and hence they have not been shown. For both convective and stratiform rain, PDFs of rainrates for light rain are positively skewed (i.e., higher than modal rainrates are more populated), whereas PDFs for heavy rain are negatively skewed. PDFs for moderate rain are generally symmetrical about the mode. It is also observed that stratiform rain has a narrower distribution than convective rain, which reflects its relative 10

uniformity of rain field. This is quantitatively described in Table 1 that provides the value of skewness and kurtosis for all the plots shown in Fig. 5. For convective rain over land, the PDFs of rainrate do not show a difference between the tropics and extratropics regardless of the intensity of rain. However, the land-ocean difference in convective rainrate PDFs is particularly evident for the light and heavy rain categories. For stratiform rain in the light rain category, the four PDFs of rainrate do not show much difference among them. Although the difference among them increases as rainfall intensity increases, the land-ocean difference continues to dominate the variability of the PDFs. The influence of latitudinal location on the pattern of rainrate PDFs is greater for heavy rain than for light rain. As described by (1), the PDFs of rainrate are needed in the effort to reduce systematic error caused by the beam-filling problem in microwave rainrate retrievals and data assimilation of microwave radiances. In the following, we fit the PDFs of rainrate by a mathematical function. To examine the quality of fit using the Kolmogorov-Smirnov test, we use 2 years of PR data (Dec. 1997 - Nov. 1999) for developing the function, and one year of data (Dec. 1999 Nov. 2000) for testing it. We only examine those PDFs that have 90% confidence intervals for frequencies less than 0.1. We refer to the data used to develop the function as analysis-data and the data used for testing the function as testdata. The study is carried out by first fitting a large number of possible functions to the conditional PDFs of rainrate from the analysis-data and then comparing the correlation coefficient and the rms error of the fittings. While fitting the functions, we also analyze the standard error, t-statistics and P-value associated with each of the coefficients. The standard error associated with a coefficient is the uncertainties in the estimates of the 11

regression coefficients. The t-value tests the null hypothesis that the coefficient of the independent variable is zero, that is, the hypothesis that the independent variable does not contribute to predicting the dependent variable. The P-value is the probability of being wrong in concluding that there is an association between dependent and independent variables (i.e., the probability of falsely rejecting the null hypothesis). We ensured that standard error is at least an order smaller than the value of the coefficient, the t-value is large and P-value is small. After many attempts, we found that a double Gaussian function [in ln(r av ) scale] fits to all PDFs with reasonable accuracy. The form of the function is as follows: 2 2 2 x x = i PDF (%) = exp 0.5, (3) i 1 bi bi where x = ln(r). The PDF is the conditional probability distribution of ln(r) in percentage for a given interval x+dx and for a given window s averaged rainrate, R av. We fit this function to the observed PDFs derived from the analysis-data. The two unknown parameters, x i and b i, are related to the window averaged rainrate, R av. The mathematical form of their relationship and the values of coefficients for PDFs (classified over tropical and extratropical land and oceans and by rain type using a CRF threshold of 0.33) are summarized in Table 2. The PDFs similarly classified with a CRF threshold of 0.1 are not found to be much different and hence values of coefficients are not separately estimated for them. 12

Figure 6 shows an example of the comparison of actual PDFs at different R av from both observed and estimated values from (3) and Table 2 of convective rain over extratropical oceans with test-data. Figure 7 shows the comparison of the estimated values of PDFs from (3) using coefficients in Table 2 against observed values for all PDFs at different R av from the test-data. The correlation coefficients are ~0.99 and rms differences are less than 0.4 for all the diagrams in Fig. 7. We have further compared observed and estimated PDFs from test data with Kolmogorov-Smirnov test and examined the D value at the 5% level of significance. The test indicates that computed D values are an order smaller than the critical value of the D at the 5% level of significance. Thus, it is concluded that the probability calculated by the parameterized function (3) agrees well with the observed one for all realistic values of convective and stratiform rainrates. 4. Conclusions The horizontal distribution of rainrates within an area of 0.25 x 0.25 has been studied over the global tropics using three years (Dec. 1997 Nov. 2000) of TRMM PR data. In the data analysis, we categorized the 0.25 x 0.25 rainy windows by rain type (convective or stratiform), rain intensity (light, moderate or heavy), surface type (land or ocean) and latitudinal location (tropics or extratropics). The rain type is defined by CRF threshold values of 0.1 and 0.33. It is noticed that at these two threshold values our results are not much different. We use two attributes to characterize the small-scale rainrate variability: fractional rain cover, or FRC, and conditional rainrate probability density function, or PDF. It is found that FRC is closely related to areal averaged rainrate, 13

R av, and the greatest cause in varying the FRC R av relation is rain type. Given the same R av, FRC for stratiform rain is larger than FRC for convective rain. FRC increases with R av to 100% at R av ~3 mm h -1 for stratiform rain and ~8 mm h -1 for convective rain. By studying the global distribution of rainrate standard deviation, we found that the horizontal variability of rainrates is largely influenced by two factors: surface type and latitudinal location. Except for light stratiform rain, the land-ocean contrast seems to be the dominant feature for the differences in conditional PDF of rainrate. Oceanic rainrate distribution is narrower when rainrate is low but becomes broader when rainrate is high. For light stratiform rain, there is no clear difference among the rainrate PDFs for rain events over land and ocean. The latitudinal variation of rainrate PDFs seems to be greater for heavy rain than for light rain. In particular, there is no measurable difference in overland convective rainrate PDFs between tropics and extratropics. On a logarithmic scale, the conditional PDFs of rainrate show similar patterns to a normal distribution for the moderate rain category. But the distribution is skewed to the right (larger rainrate) for light rain and to the left for heavy rain categories. Additionally, the conditional rainrate PDFs are broader for convective rain than for stratiform rain. Using the available data, FRC and conditional rainrate PDF are parameterized as a function of the 0.25 x 0.25 R av with divisions of convective versus stratiform for FRC and divisions according to surface type and latitudinal location for conditional rainrate PDF. These parameterizations are believed particularly useful in satellite microwave rainfall retrieval and assimilation of satellite microwave radiance data in numerical weather prediction models. However, the usefulness of the results depends upon classification of rain type from brightness temperature measurements. Though some efforts have been made by 14

several researchers to classify rain as convective/stratiform [e.g., Hong et al. (1999)] with satellite measured microwave brightness temperatures, we are also advancing the development of a reliable method to classify the rain type that will be reported in the near future. Acknowledgement. TRMM data were provided by NASA Goddard Space Flight Center DAAC. Comments from four anonymous reviewers were very helpful. This research has been supported by NASA grants NNG04GB04G and NNG05GJ17G. 15

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Table Captions: Table 1: Kurtosis and Skewness measured from the distributions presented in Fig 5 Table 2: Parametric equations of the coefficients in (3) for different regions and rain types. 19

Figure Captions: Figure 1: Distribution of the number of window samples based on CRF threshold of 0.33 for (a) convective rain and (b) stratiform rain. Figure 2: Global distribution of standard deviation of the rainrates within 0.25 x 0.25 windows for light rain (R 2.5 mm h -1 ), moderate rain (2.5 mm h -1 < R 10 mm h -1 ) and heavy rain (R > 10 mm h -1 ) with CRF threshold of 0.10. Figure 3: Same as Fig. 2 but with CRF threshold of 0.33. Figure 4: Fractional rain cover (FRC) versus window average rainrate (R av ) with CRF threshold of 0.33. Figure 5: The averaged conditional PDFs of rainrates separated by surface type (land and ocean), rain type (convective and stratiform as defined by CRF threshold of 0.33), latitudinal location (tropical and extratropical) and rainrate category (light, moderate and heavy). Figure 6: Observed conditional PDFs of extratropical convective rain rates over oceans (circles) and their corresponding estimated values from (3) and Table 2 (triangles). Figure 7: Comparison between estimated values of the rainrate PDFs from (3) and Table 2 versus observed values for the test-data. The rain types were determined by a CRF threshold of 0.33. 20

Table 1: Skewness and Kurtosis of the probability distributions presented in Fig. 5. Convective Rain Light Rain Moderate rain Heavy rain Stratiform Rain Light Rain Moderate Rain Heavy Rain Extratropical Land Extratropical Ocean Tropical Land Tropical Ocean Extratropical Land Extratropical Ocean Tropical Land Tropical Ocean Extratropical Land Extratropical Ocean Tropical Land Tropical Ocean Extratropical Land Extratropical Ocean Tropical Land Tropical Ocean Extratropical Land Extratropical Ocean Tropical Land Tropical Ocean Extratropical Land Extratropical Ocean Tropical Land Tropical Ocean Kurtosis 2.08 2.82 2.10 2.77 2.16 2.33 2.20 2.30 3.70 3.10 3.60 2.75 3.01 2.97 2.92 2.99 3.22 3.26 3.22 3.15 5.27 3.75 4.90 3.28 Skewness.15.48.12.41 -.29 -.18 -.31 -.07 -.92 -.58 -.86 -.49.48.42.50.46 -.40 -.28 -.43 -.17 -.53 -.49 -.44 -.43 21

Table-2: Parametric equations of the coefficients in (3) for different regions and rain types. Convective Rain Extratropical Land Tropical Land Extratropical Ocean Tropical Ocean Stratiform Rain Extratropical Land Tropical Land Extratropical Ocean Tropical Ocean b = c + c R + c exp (c R ) i i0 i1 av i2 i3 av x = d + d d 2 exp( d 3 R ) i i0 i1 Rav + i i av c 1j c 2j d 1j d 2j j= 0 2.0997 0.4591 3.2647 1.7372 1-0.0301 0 0 0.0446 2-1.4118 0.5946-4.3285-1.0461 3-0.0773-0.1640-0.0546-0.4671 j= 0 2.1067 0.496 2.9445 3.1126 1-0.0284 0 0 0 2-1.4359 0.5083-3.9739-1.9926 3-0.0745-0.1968-0.0620-0.0753 j= 0 1.4439 0.8404 2.4035 1.3857 1-0.0153-0.0099 0 0.0662 2-1.2423 0.5237-3.2502-2.5694 3 0.2642 0.8354-0.0641-0.8574 j= 0 1.3267 0.9779 2.9585 3.4931 1-0.0108-0.0146 0 0 2-1.0693 0-3.6990-2.9681 3 0.2936 0-0.0404-0.0785 j= 0 1.08-0.7865 2.2598 1.0855 1 0 0.0495 0 0.0701 2 0 1.6014-3.4017-3.7168 3 0-0.0929-0.1486-0.5125 j= 0-1.7831 1.2125 1.0404 2.2577 1 0.0782 0 0.0717 0 2 2.4741-0.5494-5.5615-2.9843 3-0.0589-0.0808-0.6955-0.1239 j= 0 0.3580 1.4155 1.0247 1.2413 1 0.0094-0.0134 0.0768 0.0258 2 0.4025-0.8934-2.2306-3.5373 3-0.2669-0.3268-0.3575-0.3199 j= 0 0.65 1.5357 0.7712 1.8358 1 0-0.0298 0.0974 0 2 0-1.1011-4.3446-2.3037 3 0-0.3226-0.6274-0.1119 22

Figure 1: Distribution of window samples based on CRF threshold of 0.33 (a) for convective rain and (b) stratiform rain. 23

Figure 2: Global distribution of standard deviation of the rainrates within 0.25 x 0.25 windows for light rain (R 2.5 mm h -1 ), moderate rain (2.5 mm h -1 < R 10 mm h -1 ) and heavy rain (R > 10 mm h -1 ) with CRF threshold of 0.10. 24

Figure 3: Same as Fig. 2 but with CRF threshold of 0.33. 25

Figure 4: Fractional rain cover (FRC) versus window average rainrate (R av ) with CRF threshold of 0.33. 26

Figure 5: The averaged conditional PDFs of rainrates separated by surface type (land and ocean), rain type (convective and stratiform as defined by CRF threshold of 0.33), latitudinal location (tropical and extratropical) and rainrate category (light, moderate and heavy). 27

Figure 6: Observed conditional PDFs of extratropical convective rain rates over oceans (circles) and their corresponding estimated values from (3) and Table 2 (triangles). 28

Figure 7: Comparison between estimated values of the rainrate PDFs from (3) and Table 2 versus observed values for the test-data. The rain types were determined by a CRF threshold of 0.33. 29