A Near-Global Survey of the Horizontal Variability of Rainfall

Transcription

1 A Near-Global Survey of the Horizontal Variability of Rainfall Atul K. Varma and Guosheng Liu Department of Meteorology, Florida State University Tallahassee, FL 32306, USA Corresponding Author Address: A. K. Varma Department of Meteorology 404 Love Bldg. Florida State University Tallahassee, FL (850) (850) (fax) (Submitted to Monthly Weather Review revised September 2004)

2 Abstract The characteristics of the horizontal variability of rain are presented in a global perspective through two important attributes, namely, fractional rain cover (FRC) and conditional probability distribution function (PDF) of instantaneous rain rates, at a spatial scale of ~30 km x 30 km, comparable to the footprint size of microwave satellite pixels and the grid size of numerical weather prediction models. The global variability of these attributes, separately for convective and stratiform rain, is investigated within 37 S to 37 N latitudes using 3 years of the precipitation radar data from Tropical Rainfall Measuring Mission satellite. The results show that the dominant variability for FRC and PDF of rain rates arises from the difference in rain types, i.e., convective versus stratiform rain. Given the same area averaged rain rate, convective rain type corresponds to a broader conditional PDF and a smaller FRC. The second order of the variability results from the difference in surface type, i.e., land versus ocean, for convective rain, but latitudinal locations for stratiform rain. Rain rates over land (in high latitudes) have a broader PDF than those over ocean (in lower latitudes) for convective (stratiform) rain. A third order of latitudinal variability in convective rain and land-ocean contrast in stratiform rain is also found. The possible causes for the land-ocean difference of convective rain and latitudinal difference of stratiform rain are discussed. The conditional PDFs of rain rates are found to resemble a lognormal distribution, which is modeled as a sum of two Gaussian functions in this study; and the coefficients in these functions are parameterized in terms of the area averaged rain rate. The satellite data-derived results are also compared with ground-based radar observations. 1

3 1. Introduction Precipitation is a meteorological parameter having various scales of spatial variability ranging from a few meters to hundreds of kilometers (McCollum and Krajewski, 1998; Tustison et al., 2003). The large-scale spatial variability of precipitation and cloudiness has been documented by several investigators (e.g., Rao et al., 1976; Gray and Jacobson, 1977; Short and Wallace, 1980; Varma et al., 2001; Berg et al., 2002; Samir et al., 2003). In this paper, we investigate the distribution of instantaneous rain rates within a small 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 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 rain rate variability, which was not possible until the space-borne precipitation radar data were available. The study intends to provide observational inputs for improving precipitation retrieval from satellite microwave measurements and better parameterizing clouds and precipitation in numerical weather prediction models, besides having potential applications in hydrology (e.g., Eltahir and Bras, 1993; Arora et al., 2001) and radio communication engineering (e.g., Paulson, 2002). The importance of cumulus cloud parameterization in atmospheric general circulation models (GCMs) was earlier described by Arakawa (1975). GCMs often do not simulate the cloudiness correctly due to inadequate representation of various cloud processes in them (e.g., Cess et al., 1996; Wears et al., 1996; Arakawa; 2004). Cess et al. (1990, 1996) studied the effect of cloud parameterization on the behavior of GCMs in simulating climate change. They studied whether clouds potentially amplify or dampen a possible climate change induced by a changed 2

4 composition of the atmosphere. They defined the cloud feedback parameter as the change of the effects clouds exert on the radiative fluxes at the top of the atmosphere, divided by an assumed direct radiative forcing due to an increase in the concentration of atmospheric CO 2, and found that the value of cloud feedback parameter was largely inconsistent among different models. This highlighted the large uncertainty in the representation of clouds in climate models and the need for their improvement, which is possibly due to the poor understanding of many processes that act on scales smaller than those can be resolved by the large-scale models. Arakawa (2004) asserted that, It is important to keep it in mind that the need for parameterizing physical processes is not limited to numerical models. Formulating the statistical behavior of small-scale processes is needed for understanding large-scale phenomena regardless of whether we are using numerical, theoretical, or conceptual models. Using the Fifth Generation Mesoscale Model (MM5), Zhang and Foufoula-Georgiou (1997) studied the effects of subgrid scale variability of rainfall in a land-atmosphere system. Their results indicated that the inclusion of subgrid scale spatial variability of rainfall could significantly affect the distribution of surface temperature and short-term (<24 h) prediction of rainfall intensity. Nykanen and Foufoula-Georgiou (2001) found the inclusion of subgrid scale rainfall variability affects the spatial organization of the storm, as well as surface temperature, soil moisture and sensible and latent heat fluxes. These effects were found to occur at spatial scales much larger than the scale at which rainfall variability was introduced. Thus understanding the characteristics of small-scale variability of rain rates has important implications for developing and verifying numerical models. The knowledge of the small-scale rain rate variability is imperative in dealing with beamfilling problem associated with precipitation measurement from satellite microwave radiometers. The beam-filling problem results from an inhomogeneous distribution of precipitation over the 3

5 large satellite radiometric instantaneous-field-of-view (e.g., Wilheit, 1977; Spencer et al., 1983; Chiu et al., 1990; Kummerow and Giglio, 1994; Kummerow, 1998; Turk et al., 1998). Chiu et al. (1990) utilized the rain rate data collected during the Global Atmospheric Research Program (GARP) Atlantic Tropical Experiment (GATE) and rain rate versus brightness temperature relationship derived from the model results of Wilheit et al. (1977) to show that the bias in rain rate estimation due to the beam-filling problem, to the first order, could be expressed as a product of a function of the rain rate versus brightness temperature relation and the rain rate variance. They pointed that the magnitude of the underestimation is about 25% (30%) for a footprint size of 8 km and increases to about 40% (45%) for a footprint size of 40 km for phase I (II) of GATE. Short and North (1990) found that the underestimation of rainfall for tropical areas due to the beam-filling problem could be of the order of 2. Kummerow et al. (2004) discussed the effects of beam-filling problem on climate scale rainfall estimation from passive microwave sensors. They highlighted the importance of TRMM PR for estimation of sub-pixel scale rainfall variability for the correction of beam-filling problem. Following Chiu et al. (1990) and taking into consideration the slant path radiative transfer calculations they reported an average beamfilling bias of The unaccounted sub-pixel scale variability of precipitation could result in biases in precipitation retrievals when using brightness temperature measurements from TRMM radiometers (Harris et al, 2003). Varma et al. (2004) offered a remedy for the beam-filling problem for precipitation estimation. Their method consists of convolving the radiative transfer calculations of brightness temperature associated to an underlying sub-pixel rain rate (R) with a probability function representing the likelihood of R given the field of view averaged rain rate (R av ). They used surface radar data to determine conditional probability distribution function of the rain rate and fractional rain cover, and parameterized these two attributes of sub-pixel rain 4

6 rates in terms of averaged rain rate within the field of view. Their results showed a closer agreement between satellite-observed and radiative transfer model simulated brightness temperatures. Like Kummerow et al. (2004), they also emphasized the need for studying radiometric sub-pixel scale rain variability with TRMM PR observations to address the beamfilling problem in a global scale. Several studies have been published in the literature on the horizontal variability of rain rates derived using surface-based observations. López (1976) analyzed radar data from northwestern Atlantic tropical disturbances and found that the frequency distributions obtained for the maximum attained echo area and height are lognormal. Similar distributions were reported by Biondini (1976) who analyzed radar data over Florida. López (1977) attempted to explain the characteristic growth pattern of height, horizontal size, and duration of cloud and radar echo population on the basis of the law of proportionate effects that sets the genesis of lognormal distribution. He proposed two hypotheses of cloud development one is based on growth by turbulent diffusion, and the other is based on merger of smaller cloud elements. The growth by diffusion is adopted by Kedem (1994) to propose a probability distribution model for rain rate. LeMone and Zipser (1980) found that in the cumulonimbus clouds in GATE area, the distribution of the average vertical velocity, the maximum vertical velocity, diameter and mass fluxes for drafts and cores at five different altitudes closely follow a lognormal form. Austin and Houze (1972) proposed a doubling rule for the intensity of rainfall with decreasing spatial scale. As the area decreases by the factor of 10, the rain rates roughly doubles. Tustison et al. (2001) highlighted the importance of multiscale variability of rain when observations at two different scales are compared; and Tustison et al. (2003) proposed a scale-recursive estimation framework to resolve this issue. 5

7 In this study, we take the advantage of the satellite radar measurements to characterize the horizontal variability of rain rate in a global perspective. By analyzing the characteristics of the fractional rain cover and the conditional probability distribution function of rain rate over the global tropics, we attempt to answer the following three questions in this study: How are the attributes representing the horizontal variability distributed globally? What are the main factors that determine the rain rate horizontal variability? How can we classify and parameterize the horizontal variability? In section 2, we describe the datasets and analysis method used for the study. Results on horizontal variability of rain rates are described in section 3. A mathematical parameterization of the probability distribution function of rain rates is presented in section 4. Finally, the conclusions are given in section Data Description and Analysis Method Data used in this study are mainly from TRMM PR, which is a Ku band (13.8 GHz) cross-scanning precipitation radar. The low orbiting (at 350 km from the surface of the earth) TRMM satellite provides the coverage over the tropical region. Kummerow et al. (1998) have described the complete sensors package of TRMM. The PR onboard TRMM scans ±17 from the nadir with 49 positions resulting in a 220 km swath with a horizontal resolution of 4.3 km covering area between about 37 S to 37 N with about 16 orbital passes per day in the tropics. TRMM s orbit allows each Earth location to be covered at a different local time each day, enabling the analysis of the diurnal cycle of precipitation. Our analysis utilizes the standard data product, 2A25 orbital data (radar rainfall rate and profile) that contain instantaneous rain rates from surface to 20 km altitude with a 250 m vertical and a 4.3 km horizontal resolution and a 6

8 minimum detectable rain rate of ~0.7 mm h -1. As described in the TRMM PR algorithm instruction manual (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 attenuationcorrected effective radar reflectivity factor (Z e ). The vertical rain rate profile is then calculated from the estimated Z e profile by using an appropriate Z e -R relationship. One major difference of PR algorithm from Iguchi and Meneghini (1994) is that in order to deal with the uncertainties in measurements of the scattering cross section of surface as well as the 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 rain rate for each angle bin. The time period covered by the study is 3 years from December 1997 to November During this period PR provided about 7.7 billion observations over its coverage area. The TRMM datasets are available from the NASA Data Active Archive Center (website: 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, viz., stratiform, convective, and others (Awaka, 1997). The V-method is based on detection of bright band. When bright band is present, the precipitation is classified as startiform. It detects the convective precipitation that is characterized by strong radar echo. When precipitation type is neither convective nor startiform it is defined as other type. The H-method, orginally based on Steiner et al. (1995), classifies rain type based on horizontal pattern of radar reflectivity. It detects a convective precipitation by determining the convective core and precipitation adjucent to it. 7

9 When a precipitation is not convective, it is generally classified as startiform. 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 PR 2A25 standard data set. The merging rules are defined by Awaka (1997) as follows. (1) When it is stratiform (convective) by one method and is startiform (convective) or other by the other method, the merged precipitation type is startiform (convective). (2) When it is convective by V-method but is startiform by H-method, the merged precipitation type is convective. (3) When it is statiform by V-method (i.e., bright band is detected), but is convective by H-method, the merged precipitation type is convective or stratiform depends upon level of confidence in the bright band detection. (4) When it is other by both methods, the merged precipitation type is other. A ground-based radar dataset is also used in the study for the purpose of validation. The standard data products used are referred as 2A53 and 2A54, which provide surface rain rate and rain type (convective/stratiform) over an area of 300x300 km 2. The dataset comprises of data from 5 validation radars located at Darwin, Guam, Houston, Melbourne (Florida) and Kwajalein. The period of data used for the study depends upon the data availability that varies for different radars. Table 1 shows the geographical locations of the radars and the period of the data used in the present study. The details about TRMM ground radar are available at the following website: The following two parameters representing the horizontal variability of rain rate are examined: the fractional rain cover (FRC) and the conditional probability distribution function (PDF) of rain rates within an area of 0.25 x 0.25 (~30 x 30 km 2 ), comparable to the field of view of a satellite microwave radiometer and the grid size of numerical weather prediction models. The size is similar to the footprint of past and presently available 19 GHz channels of 8

10 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. The analysis is carried out first by calculating FRC and conditional PDF of surface rain rate from TRMM PR passes for each 0.25 x 0.25 window. The window is then shifted by 0.05 at a time over the entire PR pass in both north-south and west-east directions. The window size of 0.25 x 0.25 allows approximately 40 PR pixels of size 4.3 km in it. This process is repeated for all PR passes from December 1997 through November The windows are also recognized by their surface type (land, ocean or coast) and rain type (convective or stratiform). A window is considered over coast if the PR pixels therein are located partly over both land and ocean. Rain type of a window is considered convective if more than 35% of raining PR pixels therein are determined to be convective in the 2A25 product. Otherwise, the rain type of the window is considered to be stratiform. The sliding of the window provides approximately 21, 70 and 5 million convective, and 77, 350 and 15 million stratiform windows over land, ocean and coast, respectively. Figure 1 shows the number of window samples as a function of window averaged rain rate (R av ) accumulated over the entire TRMM covered area during the three years. At R av > 30 mm h -1, the number of windows for stratiform rain becomes nearly ~1000, and much smaller than that for convective rain. In the following analysis, we limited ourselves only to windows with R av < 30 mm h -1 for stratiform rain, and R av < 70 mm h -1 for convective rain for obtaining stable statistics. Figure 2 shows the global distribution of the number of rainy (R av >0) windows, separately for convective and stratiform rain. These maps are proxies of frequency distribution of rain occurrence. Generally speaking, there are much more stratiform windows than convective windows. While convective windows are mostly populated near the equator, stratiform windows have a greater occurrence in the higher latitudes. An exception is the Atlantic 9

11 Gulf Stream region east of United States, where both convective and stratiform windows have a large 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. We find that there are often not enough data for generating stable statistics for these areas, particularly for moderate to high rain rates. 3. Results a. Classification of the Window Rain Type The rain type of a window is classified into convective or stratiform. Although the classification of a window to convective or stratiform presented herein is somewhat subjective, its conception is based on the analysis of the percentage of convective pixels out of total number of raining pixels (referred as fraction of convection) and the standard deviation of rain rates in a 0.25 x0.25 window. The present definition of convective or stratiform window is based on the premises that stratiform clouds are more homogeneously structured over land and ocean (e.g., Liu and Fu, 2001; Schumacher and Houze, 2003). On the other hand, because convection over ocean are more sustainable than over land (Schumacher and Houze, 2003), the standard deviation of rain rates in convective rain systems over ocean is expected to be smaller than that over land. Figure 3 (a) and (b) show the standard deviation of PR determined rain rates within a window as a function of fraction of convection for 3 years of PR observations, averaged in each 0.5% bin and separated for ocean and land. Figure 3 (c) shows the difference of the standard deviation between over land and over ocean versus the fraction of convection. It is known that convective precipitation is characterized by small, intense and horizontally inhomogeneous radar echoes whereas stratiform precipitation areas are identified with widespread and horizontally 10

12 homogeneous radar echoes (Houze, 1993; Houze, 1997; Schumacher and Houze, 2003). Thus a window filled with more convective rain rates (thus with higher fraction of convection) would have higher value of standard deviation of rain rates. It is noted from Fig. 3 (a) and (b) that standard deviation increases steadily with fraction of convection. From Fig. 3 (c), it is seen that standard deviations over land and ocean remain nearly the same when the fraction of convection is small. There, however, appears to be a point of inflection somewhere between 30% and 45% of fraction of convection. The large difference between standard deviations over land and ocean for windows with higher fraction of convection reflects the land-ocean difference in convective rain. Based on the above consideration, we classify rain in a window convective if more than 35% raining PR pixels therein are convective. Otherwise, we call rain in the window stratiform. It may be noted that the classification as defined above is not precise. Within a window rain rates may be comprised of both convective and stratiform types, and thus it is not possible to define a precise threshold. The threshold used here only intends to divide all the windows into two categories where their averaged properties are close to convective or stratiform rain types. A slight change in the threshold would not likely to change the averaged properties of two rain categories. This issue will be revisited later when we analyze the impact of lowering and raising the above threshold for convective-stratiform classification on derived results. It may further be noted that convective-stratiform classification also depends upon the PR rain rates classification scheme. There have been studies showing a reasonable agreement between TRMM PR and ground-based or airborne radar derived convective-stratiform classification (Heymsfield et al., 2000; Schumacher and Houze, 2000; Liao et al., 2001). Schumacher and Houze (2003) pointed that such factors as the scanning geometry, wavelength and sensitivity of PR, the change of radar reflectivity profiles with height, and the Z e -R 11

13 relationship, may also be responsible for introducing retrieval uncertainty in PR convective and stratiform rain statistics. They suggested that PR is possibly over-classifying rain as stratiform. This is another reason that we choose the threshold for window type at 35% (instead of 50%) of fraction of convection to allow more numbers of windows to be defined as convective. b. Fractional Rain Cover The FRC in each 0.25 x0.25 window is computed separately for convective and stratiform rain, and for ocean, land and coastal regions. For a clear representation of the relationship between FRC and window averaged rain rate (R av ), we have averaged R av in each 0.5% of FRC bins. For brevity, in Fig. 4 we only show the globally-averaged FRC versus R av plot for January 1998 data. Plots for other months during all the three years of study show the same general pattern, only with slightly different slopes. Figure 4 provides the general nature of the monthly averaged FRC-R av relation for convective and stratiform rain. The figure shows a clear distinction in FRC-R av relation between convective and stratiform rain, i.e., a steeper slope for stratiform than for convective rain. The plots in Fig 4 can be fitted to a function with the following form: FRC = 100 [1 a exp( brav)], (1) where FRC is in %, R av is in mm h -1, and a and b are regression coefficients. This function fits well to the plots shown in Fig. 4, as well as to similar plots for other months. The value of a is always close to 1, i.e., when FRC=0, R av =0. Rearranging (1), it follows ( 100 FRC ) = a0 a1rav log +, (2) 12

14 where a 0 =log(a 100) [log() is natural log], and a 1 =-b. Thus the slope a 1 in (2) defines the characteristics of the FRC versus R av relationship. Given the same value of surface rain rate, a smaller value of a 1 corresponds to a greater fractional rain coverage. In Figs. 5a and 5b, the global distribution of the slope a 1 averaged in every 10 x10 latitude-longitude boxes over 3 years is shown, separately for convective and stratiform rain. The calculation is carried out by first computing a 1 in each 0.25 x0.25 window using FRC and R av, then averaging the a 1 s over every 10 x10 boxes. The 10 x10 averaging area is chosen to ensure a sufficient number of windows within for developing a stable FRC-R av relationship. The distinction in characteristics between convective and stratiform rain as shown in Fig. 4 is further signified by Fig. 5, in which the value of a 1 is distinctly different between convective and stratiform rain. It generally varies from 0.5 to 0.2 for convective, but from 1.8 to 0.8 for stratiform rain over most part of the land and ocean. There is also a clear spatial variation of the slope for both convective and stratiform rain. For convective rain, the slope (a 1 ) in the FRC-R av relationship is larger over land than over ocean (Fig. 5a). The smallest value lies over oceanic areas away from the land regions. This implies that for a given R av, rainy area within the window is larger over ocean than over land. Thus rain rates in the rainy area need to be higher over land than those over ocean to keep the same window averaged rain rate. The land-ocean difference in precipitation behavior has been investigated in several earlier studies. For example, Liu and Fu (2001) studied the difference in rain profiles over land and ocean using TRMM PR observations. They found that while stratiform rain profiles are similar with each other regardless surface type, there exist significant differences in the convective profiles between over land and ocean. Specifically, for the same surface rain rate, the height of rain rate maximum in convective profiles is 3-4 km higher over land than over ocean, 13

15 and convective rain over land are significantly deeper than those over ocean. Their results are in agreement with Zipser and Lutz (1994), who found that convective cells over land are deeper and with stronger vertical velocity compared to those in oceanic regions. The studies involving lightening observations also corroborate this fact (Jorgensen and LeMone, 1989; Lucas et al., 1994). Schumacher and Houze (2003) found that stratiform rainfall rates tends to be evenly distributed over land and oceans ( mm h -1 ), whereas convective rain rates are highest over continents (> 10 mm h -1 ) and moderate over oceans (5-7 mm h -1 ). They further noted that convection over ocean has higher sustainability by warm, moist boundary layer with weak diurnal variability as compared to that over land. LeMone and Zipser (1980), Zipser and Lutz (1994), and Nesbitt et al. (2000) also reported weaker convective rain rates over tropical oceans than over land. The land-ocean difference in horizontal variability of convective rain rates as revealed by this study is another aspect of all the features that distinguish convective rain events between over land and over ocean. For stratiform rain, the FRC-R av relationship seems to have a latitudinal biasing with smaller value of slopes away from the equator (Fig. 5b). A possible explanation to the different slopes for stratiform rain in tropics and extratropics lies in the difference of their formation processes. Extratropical stratiform precipitation occurs mainly associated with cyclones and fronts, in which clouds are primarily formed by large-scale lifting. In tropical regions, however, stratiform precipitation occurs in anvil clouds and the region of older convection where vertical motion is weaker and precipitating particles drift downward (Houze, 1997). There are some exceptions to latitudinal biasing in stratiform rain. For example, in Fig. 5b in the west coasts of South Africa and South America the latitudinal biasing is less prominent. Although we do not ascribe any specific reason to this behavior with full confidence, we note that these areas have 14

16 much less stratiform rain fraction as shown in Fig.2 and described by Schumacher et al. (2004) compared to other areas in the same latitudinal belt. So the exceptions as they appear might be due to the less number of window samples. Similar to Fig.5, Figs. 6 and 7 show the seasonal variation in FRC-R av relationship for convective and stratiform rain, respectively, which are generated using 3 years of observations for four seasons with three months each, viz., DJF: December, January, February, MAM: March, April, May, JJA: June, July, August and SON: September, October, November. It appears to have some seasonal variability, related to the seasonal shifting of the global rain patterns. However, the seasonal variation does not disturb the general behavior, i.e., the major difference for convective rain variability arises from surface types (land or ocean) while latitudinal difference dominates the stratiform rain s horizontal variability. c. Conditional Probability Distribution Function The conditional PDFs of rain rates (in logarithmic scale) in a window of 0.25 x 0.25 are first computed for convective rain up to R av = 70 mm h -1 and for stratiform rain up to R av = 30 mm h -1, and then averaged in 10 x10 boxes, separately for convective and stratiform rain over three different surface types. Figure 8 shows the 3-year globally averaged conditional PDFs of rain rates for a moderate R av of 5 to 10 mm h -1. The PDFs in this figure represent the general shape of sub-pixel rain rate distribution; PDFs for other R av s follow a similar pattern. It is observed that the conditional PDFs of sub-pixel rain rates, when plotted in log-scale, follow a shape that resembles to the normal distribution. In other words, the sub-pixel rain rates approximately obey a lognormal distribution. The similarity of the PDFs of rain rates to the lognormal distribution is further discussed by their skewness and kurtosis below. 15

17 The shape of the conditional PDFs of rain rates for convective rain is different from that for stratiform rain. The distribution for stratiform rain is more peaked than that for convective rain with steeper slopes on both sides of the median. Additionally, for convective rain, the PDF shape for rain rates over land is different from that over ocean with oceanic rain rate PDF being more peaked. On the other hand, for stratiform rain, the shape of the PDFs is not significantly different between land and oceanic regions. This behavior of PDFs is in agreement with the behavior of FRCs as discussed earlier, where it was noted that for stratiform rain the FRC-R av relationship does not have a visible dependence upon underlying surfaces; it rather depends upon latitudes. The PDFs presented in Fig. 8 represents the globally averaged feature of the horizontal variability of rain rates. To describe this variability in a global perspective, the global map of PDFs of rain rates is shown in Fig. 9 for R av = 5-10 mm h -1. The scales of the abscissa (rain rate in logarithmic scale) and ordinate (PDF in percent) are from 3 to 5 and from 0 to 7%, respectively, in all the PDF plots. Again, it is universally true that the convective PDFs are broader than its stratiform counterpart. Within the same rain type, one can also see the differences among PDFs in various regions in terms of their standard deviation, skewness and peakedness, which are described below in more detail. Next, we investigate three attributes characterizing the PDFs of rain rates: standard deviation, skewness and kurtosis; all are calculated using logarithmic rain rates, log(r). A higher standard deviation indicates a broader distribution of rain rates within the window. Skewness for a normal distribution is zero. A negative value of skewness indicates that the logarithmic rain rate distribution is skewed to left (i.e., lower than modal rain rates are more populated) and a positive value of skewness indicates that the distribution is skewed to right. The kurtosis 16

18 represents the peakedness. Kurtosis for a normal distribution is 3. A value of kurtosis greater (less) than 3 indicates a more peaked (flatter) distribution than the normal distribution. In high R av bins, the number of observations in a 10 x10 box is small (Fig. 1), especially in boxes where there is climatologically less precipitation, which makes the statistics unstable. To maintain statistical stability, we analyzed only those bins whose total number of observations in a 10 x10 box is more than 500. The skewness and kurtosis for all the 10 x10 boxes in all R av bins are shown in Fig. 10. A normal distribution of the logarithmic rain rates would result in a kurtosis of 3 and a skewness of 0. The result shows that the majority of PDFs have the value of skewness between 1 and 1, and kurtosis between 2 and 4, indicating that the logarithmic rain rates distributed in a 0.25 x0.25 window follow a pattern similar to the normal distribution. Some other features of the PDFs can also been seen from the figure. For example, most of the PDFs have negative value of skewness (i.e., skewed toward the left); most of PDFs for the stratiform rain are narrower than lognormal distribution (with kurtosis greater than 3), while more than half of convective PDFs are wider than lognormal distribution (with kurtosis less than 3). Skewness and kurtosis are also correlated. An increase in the absolute value of skewness corresponds to an increase in the kurtosis (peakedness) of the distributions. In other words, the more peaked distributions are also more skewed. The global distributions of the standard deviation and skewness of rain rates within 0.25 x0.25 windows are shown in Figs. 11 and 12 for convective and stratiform rain, respectively. For convective rain (Fig.11), there is a clear land-ocean contrast with larger values of the standard deviation of rain rates over land, especially for low R av s (<15 mm h -1 ). The skewness shows that PDFs are generally more skewed over land and at high rain rates. That is, 17

19 similar to FRC discussed in previous subsection, the conditional PDFs of convective rain rates also show different characteristics between over land and ocean. At high R av s (>15 mm h -1 ), while maintaining the land-ocean contrast, standard deviation shows some latitudinal variation as well with slightly lower values at higher latitudes over oceans. There is a general trend of increase in standard deviation with R av when R av is low, while the rise becomes leveled at very high rain rates. Similar to standard deviations, skewness also shows some latitudinal dependence at rain rates higher than 15 mm h -1. For stratiform rain (Fig. 12), the standard deviation of sub-window rain rates increases with the increase of R av. Lower values of standard deviation appear at climatologically low rain rate regions. The magnitude of standard deviation has a latitudinal variation with lower values at higher latitudes. There are also indications of land-ocean difference at high rain rates, which will be investigated more thoroughly later. Both skewness and kurtosis show some latitudinal variation although only the distribution of skewness is shown in Fig. 12. When R av is lower than 5 mm h -1, the PDFs are skewed toward higher rain rates. But they become to be skewed toward lower rain rates as R av increases. This behavior is the same as that for convective rain. The values of kurtosis (not shown) indicate that the conditional PDFs for stratiform rain are more peaked than those for convective rain. Seasonal changes of the standard deviation, skewness and kurtosis of conditional PDFs are also examined (not shown). We did not find a clear regularity of their dependence on seasons, although their global distribution appears to have some seasonal variation, likely being related to the intra-annual movement of major rain zones. The results presented above indicate that the spatial distribution of rain rates within a 0.25 x0.25 window possesses a strong land-ocean contrast for convective rain and a latitudinal 18

20 dependence for stratiform rain. To further examine the land-ocean and latitudinal variability, we average the standard deviations in every 5 mm h -1 R av bin for each 10 latitudinal belt, separately for convective and stratiform rain, and for rain over land and ocean. The results are shown in Fig. 13. It is noted that for convective rain above R av =5 mm h -1 over ocean there exists a latitudinal variation with smaller values of standard deviation at higher latitudes. Over land, however, the variability trend is reversed with slightly larger value of standard deviation at higher latitudes. Figure 13c shows that the land-ocean contrast is always high at high latitudes. At low latitudes (15 S-15 N), this contrast decreases with the increase of R av and even reverses its sign at very high rain rates (>25 mm h -1 ). For stratiform rain, there is no clear trend indicating land-ocean contrast (Fig. 13f), but there does exist a strong latitudinal variation with lower values of standard deviation at higher latitudes. Over ocean, the latitudinal contrast increases with increasing rain rates. Over land, the latitudinal contrast is significant only for R av above 15 mm h -1. The land-ocean contrast in the conditional PDFs of convective rain rates and the latitudinal variation in the conditional PDFs of stratiform rain rates can possibly be explained with the same reasoning as the similar disparities of FRC in Fig. 5 are explained. The convection over oceanic regions is less intensive than that over land regions, and hence has higher sustainability. This results in lower standard deviation of rain rates over ocean. The latitudinal dependence of standard deviation of stratiform rain rates is likely due to the different mechanisms by which stratiform clouds are formed near the equator and at higher latitudes. That is, stratiform clouds near the equator are mostly attached to convective cells formed by thermodynamically buoyant updraft, while those at higher latitudes are mostly formed by largescale lifting. 19

21 Further analysis of the PDFs of rain rates is conducted by dividing data into the following two latitudinal regions: tropics between 20 N and 20 S, and extratropics S or N, separately for rain over land and ocean. Figs. 14 and 15 show the conditional PDFs of rain rates for selected 1 mm h -1 R av bins for convective and stratiform rain, respectively. For convective rain with R av < 5 mm h -1 or R av > 45 mm h -1, the PDFs for all 4 regions have a close resemblance. For 5 mm h -1 <R av <25 mm h -1, while PDFs for tropical and extratropical land rain rates are similar, there exists a visible difference between PDFs for land and ocean, and between ocean rain themselves. For 25 mm h -1 <R av < 45 mm h -1, all PDFs agree well except for that of tropical ocean. PDFs of stratiform rain rates are quite similar when R av < 10 mm h -1, while their difference between tropical and extratropical rain becomes evident for higher R av s. Below 15 mm h -1, which counts most of the stratiform rain, there is little difference between PDFs over land and ocean given the same latitudinal region. The PDF difference due to surface type becomes significant when R av is larger than 19 mm h -1, particularly for extratropical region. In section 3a, we have explained why the ratio 0.35 of convective to total number of raining pixels is utilized to define rain type of a window. To examine whether varying this threshold significantly alters the results described above, we further generate a test dataset, in which the convective windows includes only those with 90% pixels inside being convective, and the stratiform windows with 90% pixels inside being stratiform. The rest of the windows, which are about 22.06% for stratiform rain and 90.33% for convective rain are excluded for analysis. The comparison between standard deviation resulted from this and that from the 0.35-ratio definition is shown in Fig. 16. In this figure, the standard deviations are averaged over all possible R av s. It is seen that the patterns of standard deviation resulted from the two definitions does not differ qualitatively. In other words, the general results described in earlier sections will 20

22 not change even if we use a more stringent definition for convective and stratiform windows. The drawback for using a stringent definition is that we have to exclude a significant part of the available data, and could not be able to obtain stable statistics of FRCs and PDFs for some regions and some R av ranges. d. Comparison with surface radar data In this section, we compare results derived from TRMM PR with those from data of surface radars at TRMM validation sites. The surface radars are located at Darwin, Guam, Houston, Kwajalein, and Melbourne (Florida). Of the 5 radars, three are located in tropics and two in sub-tropical regions. The locations of these radars and the time period during which data are used in the analysis are provided in Table 1. The surface radars provide hourly rain maps as well as rain type over 300 x 300 km 2 area with 2 km pixel resolution. In processing the surface radar data, we define a window of the size 30 x 30 km 2. The FRCs and conditional PDFs of rain rates within a window are calculated and averaged using data for the entire period of observations. For comparison, the TRMM PR derived FRCs and PDFs of rain rates in a window of 0.25 x0.25 are averaged for 3 years in a 10 x10 area encompassing each surface radar site. Figures 17 and 18 show the comparison of FRCs (Fig.17) and PDFs (Fig.17) between those derived from surface and space radars. It is shown that while PR derived FRC-R av relationship and PDFs are quite close to that from surface radars located at Darwin, Guam and Houston, there are marked deviations for radars located at Melbourne and Kwajalein. For radars located at Melbourne and Kwajalein, the conditional PDFs for convective rain are found in better agreement with PR results than those for stratiform rain. Despite of some of the disagreement between surface and satellite observations, the differences of FRCs and PDFs between 21

23 convective and stratiform rain are clearly brought out by surface radars as well. Like results presented in previous sections with PR based analysis, the surface radars also show a clear difference in the slope of FRC-R av relationship between convective and stratiform rain with a greater slope for stratiform rain. The difference is also evident in the form of conditional PDFs between the two rain types with broader spread and flatter shape for convective rain. However, the land-ocean difference in FRCs and PDFs is difficult to be noticed from the radar data analysis, as these radars exist either close to coast or on small islands. 4. Parameterization of Conditional PDF The conditional probability distribution functions as revealed by Figs. 8, 14 and 15 have a similar pattern to normal distribution. Since abscissa in these figures is log(r), the rain rates within a window nearly obey a lognormal distribution. The basic philosophy to introduce this section is to find the PDF of sub-pixel rain rates as a function of the pixel averaged rain rate R av. In rain rate 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 (e.g., Varma et al., 2004): T = [ 1 FRC ( Rav )] TB (0) + FRC( Rav ) TB ( x) PDF ( x, Rav dx (3) B ) where x=log(r); T B (0) is the brightness temperature at no-rain area; T B (x) is the brightness temperature at rain rate R. The retrieval problem is to estimate R av from observed T B at several frequencies. Clearly, The characteristics of the probability distribution of sub-pixel rain rates, as well as the fractional rain coverage 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), in which the numerical R 22

24 model provides rain rate (or rain water content) averaged over a model grid while the satellite observes the pixel averaged brightness temperature. Both the model grid and satellite pixel size are on the order to the window size we used in this study (~30 km). To connect model and observation variables, it is necessary to know the sub-pixel (or subgrid) distribution of rain rates. Similarly, the probability distribution function of a given variable is needed to account for its subgrid scale heterogeneity in general circulation models for calculating grid-averaged fluxes by numerical integration over the subgrid (bin) intervals (e.g., see Avissar, 1992; Famiglietti and Wood, 1994). The lognormal form of the conditional PDF of sub-window rain rate is in agreement with previous studies by several researchers (e.g., López, 1976, Varma et al., 2004). In this section, we fit the PDFs to a common function over all R av bins, separately for convective and stratiform rain, over land and oceans, and in tropical and extratropical regions. This is carried out by fitting a large number of possible functions, and then studying the correlation coefficient and the rootmean-square (rms) error of the fittings. After many attempts, we find that a double Gaussian function fits to all PDFs with reasonable accuracy. The form of the function is as follows: x x = i PDF (%) = exp 0.5, (4) i 1 bi bi where x = log(r). The PDF is the conditional probability distribution of log(r) in percentage for a given interval x+dx, and for a given window averaged rain rate R av. We fitted this function to observed PDFs for R av from 1 to 70 mm h -1 for convective rain, and 1 to 30 mm h -1 for stratiform rain. The two unknown parameters, x i and b i are related to the window averaged rain rate, R av. 23

25 The mathematical form of their relationship and the values of coefficients are summarized in Table 2. Figure 19 shows the comparison of the estimated probability from (4) with coefficients in Table 2 and the observed probability for all R av ranges. The correlation coefficient and rms error are provided in Table 3. It is shown that the probability calculated by the parameterized function agrees well with the observed one for all realistic values of convective and stratiform rain rates, indicating that the equation expressed by (4) reasonably represent the conditional probability distribution function of the real rain observed in the nature. 5. Conclusions The present study describes the features of two attributes of horizontal rain variability, viz., the FRC-R av relation and the conditional PDF of rain rates. The analysis is conducted by distinguishing between convective versus stratiform rain within an area comparable to the size of a satellite pixel of microwave radiometers and the grid size of present numerical weather prediction models. The results show that the first order variability in FRC-R av relationship is its rain type (convective or stratiform) dependence, followed by the second order variability due to their geographical locations. That is, the relation has a strong land-ocean difference for convective rain, whereas for stratiform rain, it has a latitudinal dependence. The conditional PDF of rain rates for all rain ranges has a similar distribution to the lognormal distribution. Similar to FRC, the first order variability of the conditional PDFs arises from the different rain types. Given the same area averaged rain rate, convective rain type corresponds to a broader conditional PDF of rain rates. The conditional PDFs for convective rain are found to have a strong land-ocean 24

26 contrast for low to moderate window averaged rain rates (<20 mm h -1 ) at all latitudes, and for high window averaged rain rates (>20 mm h -1 ) outside the tropics. For stratiform rain, the conditional PDFs of rain rates exhibit a latitudinal variation at all rain ranges, although they also show a land-ocean contrast at higher window averaged rain rates (>15 mm h -1 ) outside tropics. Additionally, although it is less dominant, the spatial variability of PDFs also has some dependence on window (pixel) averaged rain rate. Finally, the conditional PDFs of (logarithmic) rain rates are modeled as a double Gaussian function. The coefficients of the Gaussian function were parameterized in terms of window averaged rain rate, R av. Comparison is made between satellite and surface radar data derived horizontal variability characteristics. Their similarities and discrepancies are presented. This study reveals the horizontal rain variability in a global perspective. While the horizontal variability of rain rates is important in its own right, the statistical results from this study can also have two useful applications. First, it may be helpful for developing and validating cumulus parameterizations in numerical weather prediction and climate models, the grid size of which is commonly much larger than a cumulus cell. Another application is in tackling the beam-filling problem associated with satellite microwave observations. Acknowledgement. TRMM data are provided by NASA Goddard Space Flight Center DAAC. Comments from three anonymous reviewers are very helpful. The research has been supported by NASA grant NNG04GB04G and DOE ARM grant DE-FG02-03ER

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34 Table Captions: Table 1: Geolocation and period of observations of surface radars used in the study. Table 2: Parametric equations of the coefficients in (4) for different regions and rain types. Table 3: Correlation coefficient and rms error in the best fit in the comparisons shown in Fig

35 Figure Captions: Figure 1: Number of window samples as a function of window averaged rain rate (R av ). Figure 2: Distribution of window samples (a) for convective rain and (b) stratiform rain. Figure 3: Percentage of convective pixels out of total number of raining pixels versus standard deviation of rain rates in 0.25 x0.25 windows from PR data from December 1997 to November 2000 over (a) ocean and (b) land, and (c) shows the difference of standard deviation between land and ocean. Figure 4: Fractional rain cover (FRC) versus window average rain rate (R av ) from PR data of January 1998 over ocean (circle), land (square) and coast (triangle) for convective and stratiform rain. Open symbols represent convective rain and solid symbols represent stratiform rain. Figure 5: Slope a 1 in equation (2), which represents the variation of the FRC-R av relationship. (a) for convective rain and (b) for stratiform rain. Figure 6: Same as Fig. 5a (convective rain) but divided to 4 seasons: (a) December, January, February, (b) March, April, May, (c) June, July, August, and (d) September, October, November. If number of window samples in a 10 x10 box is found less than 1000, the value in that box is not plotted. Figure 7: Same as Fig. 6 but for stratiform rain. Figure 8: Conditional probability distribution function (PDF) of rain rates for window averaged rain rate (R av ) of 5-10 mm h -1 over land (square), oceans (circle) and coast (triangle) from PR observations for (a) convective and (b) stratiform rain. 34

36 Figure 9: Global distribution of PDFs of rain rates for window averaged rain rate (R av ) of 5-10 mm h -1 for (a) convective and (b) stratiform rain. The scale of the abscissa and ordinate are from 3 to 5 and 0 to 7, respectively, in all PDF plots. Figure 10: Relation between kurtosis and skewness of conditional PDFs for window averaged rain rate (R av ) from 0 to 70 mm h -1 for convective rain (circle) and 0 to 30 mm h -1 for stratiform rain (triangle). Figure 11: Global distribution of standard deviation (left panel) and skewness (right panel) of the conditional probability distribution function of convective rain for window averaged rain rate (R av ) from 0 to 40 mm h -1. Figure 12: Same as Fig. 11 but for stratiform rain and for window averaged rain rate from 0 to 25 mm h -1. Figure 13: Zonally averaged standard deviation for convective rain over (a) ocean and (b) land, and for stratiform rain over (d) ocean and (e) land. The difference of standard deviation between over land and ocean is shown in (c) for convective rain and (f) for stratiform rain. Figure 14: Conditional probability distribution function of rain rates in convective rain for extratropical land (circle), extratropical ocean (square), tropical land (triangle) and tropical ocean (diamond). Figure 15: Same as Fig. 14 but for stratiform rain. Figure 16: Comparison of the distribution of standard deviation of rain rates when using 2 different definitions of window rain type. The convective to stratiform pixel ratio is > 90% in (a), > 35 % in (b), < 10% in (c) and < 35% in (d). 35

37 Figure 17: Comparison of FRC-R av relationship derived from PR with those from ground radar observations at Darwin, Guam, Houston, Melbourne (Florida) and Kwajalein. Circles and squares represent surface radar and PR derived values, respectively. Figure 18: Comparison of the conditional probability distribution functions derived from PR with those from ground radar observations at Darwin, Guam, Houston, Melbourne (Florida) and Kwajalein. Circles and squares represent surface radar and PR derived values, respectively. Figure 19: Comparison between observed probability and estimated probability from (4) and Table 2. 36

38 Table 1: Geolocation and period of observations of surface radars used in the study. Radar Location Period Darwin Guam Houston Kwajalein Melbourne (Florida) S, E N, E N W 8.72 N, E N, W Jan-Feb, 1998 Jul-Oct, 1998 Aug-Sep, 1999 Jan-Dec, 1998 Jul-Aug,

39 Table 2: Parametric equations of the coefficients in (2) for different regions and rain types. Regions b c R c c e i av i = i0 + i (1 R x av 0 d e d i i = d i + i ) Convective Rain Tropical Land c 10 = c 20 = d 10 =0 d 20= Tropical Ocean c 11 = c 21 = d 11 = d 21 = Extratropical Land c 12 = c 22 = d 12 = d 22 = Extratropical Oceans c 10 = d 10 =0 c 11 = do---- d 11 = do---- c 12 = d 12 = Stratiform Rain Tropical Land c 10 = 0 c 20 = d 10 =0 d 20= Tropical Ocean c 11 = c 21 = d 11 = d 21 = c 12 = c 22 = d 12 = d 22 = Extratropical Land c 10 = 0.59 c 20 = d 10 =0 d 20= c 11 = c 21 = d 11 = d 21 = c 12 = c 22 = d 12 = d 22 = Extratropical Land c 20 = d 20= do---- c 21 = do---- d 21 = c 22 = d 22 =

40 Table 3: correlation coefficient and rms error in the best fit in the comparisons shown in Fig. 18. Rain Type Latitudinal Surface Area Correlation Error (rms) Area coefficient (r) Convective Stratiform Extratropical Tropical Extratropical Tropical Land Ocean Land Ocean Land Ocean Land Ocean

41 Fig. 1: Number of window samples as a function of window averaged rain rate (R av ). 39

42 8x10 5 7x10 5 6x10 5 5x10 5 4x10 5 3x10 5 2x10 5 1x x x x x x x10 5 Fig. 2: Distribution of number of window samples for (a) convective rain (b) stratiform rain. 40

43 Fig. 3: Percentage of convective points out of total number of raining points versus standard deviation (SD) of rain rates in 30 km x 30 km windows from global PR data from December 1997 to November 2000, over (a) ocean (b) land (c) land and ocean difference. 41

44 Fig. 4: Fractional rain cover (FRC) versus average rain rate (R av ) from global PR-radar data of January 1998 over ocean (circle), land (square) and coast (triangle) for convective rain and stratiform rain. Open symbols represent convective rain and solid symbols represent stratiform rain. 42

45 Fig. 5: Slope a 1 in equation (2), which represents the variation of the FRC-R av relationship for (a) convective rain and (b) stratiform rain. 43

46 Fig. 6: Same as Fig. 5a (convective rain) but divided to 4 seasons, (a) December, January, February, (b) March, April, May, (c) June, July, August, and (d) September, October, November. 44

47 Fig. 7: Same as Fig. 6 but for stratiform rain. 45

48 Fig. 8: Probability distribution function of rain rates for window averaged rain rate (R av ) of 5-10 mm h -1 over land (square), oceans (circle) and coast (triangle) from PR global observations for (a) convective rain and (b) stratiform rain. 46

49 Fig. 9: Global distribution of PDF of rain rates for window-averaged rain rate (R av ) of 5-10 mm h -1 for (a) convective rain and (b) stratiform rain. The abscissa and ordinate are from 3 to 5 and 0 to 7, respectively, in all PDF plots. 47