Uncertainty of estimates of monthly areal rainfall for

WATER RESOURCES RESEARCH, VOL. 32, NO. 2, PAGES 373-388, FEBRUARY 1996

Uncertainty of estimates of monthly areal rainfall for temporally sparse remote observations Matthias Steiner1 Department of Atmospheric Sciences, University of Washington, Seattle Abstract. The uncertainty of monthly areal mean rainfall estimates, caused by a finite sampling time resolution, is investigated making extensive use of data collected by tippingbucket rain gauge networks at Darwin, Australia, and Melbourne, Florida. Using a subsampling methodology, the uncertainty is studied for complete area-covering observations occurring at regular time intervals. The analysis indicates that the sampling uncertainty is constrained by the rainfall depth, the sampling frequency, and the domain size. Taking into account that a rain gauge network does not truly reflect the sampling uncertainty one would encounter using complete area-covering observations, such as provided by radar or satellite, a rule of thumb is established for estimating the average sampling uncertainty as a function of the above factors. The estimated root mean square error E, expressed as a percentage of the monthly rainfall amount Rs, is found to be inversely proportional to the rain amount and the domain size A but proportional to the sampling time interval AT and can be approximated by E — 8.5 X 103 X R~°'6A~°'5kT. Within this framework the various results reported in the literature from studies based on using other data and different spatial scales are found to be consistent with our results and among each other. The scaling of our results to timescales other than a month is indicated. of the rather limited temporal resolution of many spaceborne observations and the intermittency of rainfall in space and Quantitative knowledge of the global water budget is of key time, such estimates of areal rainfall accumulations are uncerimportance to the understanding of the Earth's climate system. tain. For example, the satellite designed for the Tropical RainCondensation of water vapor releases latent heat into the at- fall Measuring Mission (TRMM), a joint Japan-United States mosphere, and this energy is a principal driver for the global- space program to study the spatial and temporal distribution of scale circulations in the troposphere. The mean hydrological the latent heat release to the tropical atmosphere [Simpson, cycle is important, but its variability in space and time is crucial 1988; Simpson et al., 1988], has a revisiting time of slightly less to explain changes in the circulation system on timescales of than 12 hours, i.e., roughly two visits per day. For the lowdays to decades or longer. Rainfall in the tropics and subtrop- inclination TRMM satellite the error due to sparse sampling is ics is particularly important, since about two thirds of the a critical issue. According to an analysis of the payload instruglobal precipitation falls within these climatic belts [Sellers, ments and their associated errors by Wilheit [1988], the sam1969; Baumgartner and Reichel, 1975; Jaeger, 1976; Legates and pling is the largest single component of the error budget for Willmott, 1990; Spencer, 1993]. The observation of precipita- TRMM. tion on a global scale, however, is difficult because significant For the purpose of this study we are interested in the accuportions of the Earth are occupied by extensive oceans and racy of monthly areal mean rainfall estimates as a function of the frequency of observations. Results of similar studies on tropical rain forests. Satellite-based observations present the best possible mode small space scales and timescales are reported by Hudlow and by which global precipitation measurements can be made. Sat- Arkell [1978], Huebner et al. [1986], and Fabry et al. [1994] ellite observations generally provide excellent spatial coverage among others. We assume an ideal situation with complete each time the satellite passes over a given location; however, area-covering observation for each visit, regularly spaced in spaceborne observations encounter different types of prob- time, and perfect instruments providing accurate rain rates. lems, such as instrumental accuracies (i.e., technical accuracy Problems arising due to irregularities in the space-time samand quality of the measurement's physical relation to rainfall), pling pattern, for example, are discussed by Zeng and Levy the spatial representation (i.e., complete area-covering obser- [1995]. Traditional approaches reported in the literature estivations or only partial visits), and the temporal representation mate the sampling errors as a function of the space and time (i.e., snapshots at regular or irregular time intervals). Because correlation characteristics of the rain fields to be encountered. North [1988] reviews some of these studies to gauge the imx Now at Water Resources Program, Department of Civil Engineer- portance of time gaps in the construction of space-time avering and Operations Research, Princeton University, Princeton, New aged rainfall estimates, conducted using both data taken from Jersey. GATE, the Global Atmospheric Research Program (GARP) Copyright 1996 by the American Geophysical Union. Atlantic Tropical Experiment [Kuettner et al., 1974; Hudlow and Patterson, 1979], and stochastic models of rain rates based Paper number 95WR03396. 0043-1397/96/95WR-03396$05.00 upon these data. 1.

Introduction

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STEINER: UNCERTAINTY OF MONTHLY AREAL RAINFALL ESTIMATES

In this study we approach the problem differently. Employing highly time-resolved data sets collected by rain gauge networks, we use an extensive subsampling exercise to determine how much uncertainty, on the average, is introduced to monthly areal rainfall estimates as a function of decreasing temporal representation. Ten months of ground-level rain gauge data obtained from Darwin, Australia, and another 2 months of data collected in central Florida are analyzed by the subsampling methodology described in section 2.2. These data cover a wide range of monthly rain accumulations and can therefore be used to study the scaling of the sampling problem with rain amount (section 3.2). Considering how the results obtained by using a rain gauge network of point measurements compare to the results one would obtain using complete areacovering (e.g., radar) observations, the area represented by the rain gauge network can be estimated. This anchor is needed to combine our results of the scaling with rainfall depth with the spatial scaling behavior pointed out by North and Nakamoto [1989], Bell et al [1990], and Graves et al [1993] to establish a framework that describes the uncertainty for monthly areal rainfall estimates as a function of the rainfall depth, domain size, and sampling frequency. Using this framework, we can compare results obtained by different studies using data from different climatological regions (section 4). We can show that results previously considered controversial can be explained and that they are consistent with our results and also among each other (sections 4.3 and 4.4).

2. Methodology and Data 2.1. Rain Gauge Networks The data analysis and discussion for this study are largely based on data collected by a rain gauge network in the vicinity of Darwin, Australia. For comparison, however, 2 months of data collected by extensive rain gauge networks around Melbourne, Florida, are processed similarly. The rain gauge networks of Darwin and Melbourne comprise two types of rain gauges: accumulation and tippingbucket devices. Because of the required high temporal resolution, however, only data recorded by the tipping-bucket rain gauges are used for this study. The gauge data (recorded tips) were processed to a 1-min time resolution and archived at the NASA Goddard Space Flight Center as part of the TRMM ground validation program [Thiele, 1988; Thiele et al., 1992]. For this study we use only data from gauges that have complete records (i.e., no known data gaps for reasons of instrument failure, etc.) for the months of interest. Details of the rain gauge networks are given below. 2.1.1. Darwin, Australia. Darwin lies just south of the tropical "maritime continent," defined as the islands and seas of the Indonesian-Malaysian archipelago [Ramage, 1968]. The region has a monsoonal climate [Holland, 1986; Keenan et al., 1988] with pronounced wet and dry seasons. The rainy season normally begins in late December and extends through March. The mean large-scale flow consists of a cyclonic gyre in the lower troposphere and an anticyclonic gyre in the upper troposphere. The general pattern undergoes synoptic-scale spatial and temporal variations accounting for the "break" and "active" periods of the monsoon. During active periods, the center of the cyclonic gyre is near or south of Darwin, and the low-tropospheric flow on the north side of the cyclone has a long westerly oceanic fetch. During such times, precipitation in the vicinity of Darwin is heaviest

and most widespread. During breaks in the monsoon the cyclonic gyre is centered north of Darwin, and a dry continental flow dominates the region. The precipitation seen on the Darwin radar, located at Berrimah (12°27'26"S and 130°55'31"E; about 20 km east-southeast of Darwin), is of three main types: 1. During active monsoon periods the precipitation typically occurs in oceanic mesoscale convective systems (MCSs), which consist of ensembles of convective lines and stratiform regions [Mapes and Houze, 1992, 1993]. These systems are similar to the tropical oceanic MCSs observed in GATE [Houze and Betts, 1981] and the Monsoon Experiments (MONEX) [Johnson and Houze, 1987]. 2. During break periods the precipitating cloud systems are less frequent and are typical of a continental origin. They may be intense and are often in the form of squall lines with trailing stratiform precipitation [e.g., Drosdowsky, 1984]. 3. Pronounced diurnally forced thunderstorms locally known as "Hectors" [Keenan et al., 1989, 1990; Simpson et al., 1993] occur over Bathurst and Melville Islands to the north of Darwin (Figure 1). Keenan and Carbone [1992] discuss a variety of precipitation systems observed in the vicinity of Darwin. The data used from the Darwin rain gauge network were collected during three rainy seasons: 1987/1988,1989/1990, and 1993/1994. A total of 10 months were analyzed using the methodology described in section 2.2. Table 1 presents a compilation of the gauge sites used in the data analysis for each particular month. The rain gauge locations relative to the radar site at Berrimah are indicated in Table 1 and shown in Figure 1. The gauge locations of the Darwin network are all within a range of 130 km (Table 1) and about evenly distributed over the land and islands. Since only half the area covered by the radar umbrella out to a radius of 130 km is over land (there are no rain gauges over the ocean), as indicated by the solid range ring in Figure 1, the area represented by the rain gauge network may be estimated to about 26,500 km2. This estimate of the area represented by the gauge network is needed to anchor the scaling of uncertainty with domain size, as discussed in section 3.2, which then enables a comparison of the results obtained on different spatial scales (section 4). The sensitivity of the results to the above estimate of the area represented by the gauge network will be discussed later on (section 3.2). 2.1.2. Melbourne, Florida. The climate of Florida ranges from temperate to subtropical conditions in the northern interior portion of the state to the tropical conditions found on the Florida Keys. The latitude, proximity to the Atlantic Ocean and the Gulf of Mexico, and numerous lakes are chief factors in controlling Florida's climate [National Oceanic and Atmospheric Administration (NOAA), 1980]. Rainfall in Florida varies both in the annual amount and in the seasonal distribution. Except for the northwestern portion of the state, the average year can be divided into two seasons: a short "rainy season" and a long, relatively dry season. In central and south Florida, generally more than 50% of the precipitation for an average year falls during the 4-month "rainy season," June through September. This is reflected in Florida's thunderstorm statistics, which show that June, July, and August are the peak months [Neumann, 1971]. Furthermore, central Florida has the highest annual number of days with thunderstorms of the United States [Court and Griffiths, 1986]. In northwest Florida, there is a secondary rainfall maximum in late winter and in early spring. Summer rainfall is produced mostly by showers or thunderstorms. These storms are triggered by horizontal, low-level


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Distance East of Radar [km] Figure 1. Geographical map of the Darwin region indicating the radar and rain gauge network. The land and islands within the 400 km by 400 km domain centered at the Darwin radar are heavily shaded. The dotted range rings indicate equal distances from the radar site (Berrimah, indicated by the open triangle) and are 50 km apart. The rain gauge locations, as given in Table 1, are shown by the solid triangles. The solid range ring with a radius of 130 km indicates the maximum distance rain gauges are located from the radar site.

convergence resulting from sea breezes entering the peninsula from both sides and only rarely are associated with fronts or large-scale disturbances [Byers andRodebush, 1948; Burpee and Lahiff, 1984; Blanchard and Lopez, 1985]. The rainfall of these sea-breeze-triggered multicellular storms is often heavy but usually lasts only for a couple of hours. Rainfall exhibits a strong diurnal variation with >90% occurring during the afternoon to early evening. The data used in this study were collected during the months of July and August 1993 by rain gauges located within the Melbourne radar range. The tipping-bucket gauge data are from the Kennedy Space Center and the St. John's Water Management District networks. Table 2 presents a compilation of the gauge sites used in the data analysis for both months. The rain gauge locations relative to the Melbourne radar site (28°06'47"N and 80°39'16"W) are shown in Figure 2. The area represented by the Melbourne gauge networks used in this study may be estimated to —18,700 km2, which is about half of the domain indicated by the box in Figure 2 (150 km by 250 km). The sensitivity of the results to this area estimate will be discussed in section 3.2. 2.2. Sampling Scenario

The rainfall rates recorded in the continuous rain gauge traces are sampled at various time intervals to investigate the uncertainty of the estimated mean rainfall introduced by decreasing sampling time resolution. The monthly areal mean

rainfall accumulation, the quantity of interest here, is estimated from the samples taken. It is assumed that these samples would be representative for a period corresponding to the sampling time interval. In addition, the continuous recordings of the rain gauge network are assumed to represent the true areal mean rainfall. The values obtained from lower time resolution sampling are only estimates thereof. The continuous recordings of each rain gauge in the network are partitioned into 15-min rainfall accumulation amounts. The contemporaneous values of all the gauges together form a block of information. With 96 such blocks for each day we obtain between 2688 and 2976 blocks per month, depending on the month under investigation. From these blocks of information, any sampling scenario can be examined by extracting a subset of N blocks and using only those blocks to estimate the mean rain rate for the month. The standard procedure is to express the selected 15-min samples as an hourly rain rate (in millimeters per hour). Each 15-min rain rate, R(15 min)site /? is assumed to be representative of the entire time period between samples (AT). The rainfall accumulation for the month at each gauge site Rs^ is obtained by multiplying the monthly mean rain rate (obtained by a linear average of all the samples selected) by the number of hours of a particular month (e.g., 696 hours for February 1988)

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Table 1. Rain Gauge Network Around Darwin, Australia 1989/1990

1978/1988 Rain Gauge Site Berrimah (radar location) Darwin Airport Lee Point AWS Mandorah Jetty McMinns Lagoon Strangway Noonamah Gunn Point Prison Gunn Point Prison Farm Koolpinyah Koolpinyah Station Delisaville Pioneer Creek Southport AWS Charles Point Charles Point Transmitter Belleville Park Gunn Point Crab Farm D-scale DS03 D-scale DS10 D-scale DS22 Darwin River Dam Humpty Doo Navy D-scale DS08 Woolner Finnes River Dum In Mirrie Batchelor Airstrip Pickertaramoor Bathurst Island Airstrip Nguiu La Belle Airstrip Mt. Bundy Point Stuart Abbs. Mt. Ringwood Annaburroo Old Point Stuart Swim Creek Plains Goodall Mine Maxwells Creek Litchfield Channel Point Snake Bay Old Snake Bay Airstrip Cape Fourcroy Pirlimgimpi Garden Point Airstrip Gauges used per month (complete records only)

Direction, deg

Distance, km

0 304 344 275 119 130 138 28 32 75 74 248 188 176 278 283 188 17 108 111 116 173 113 108 81 222 251 171 356 336 336 213 165 99 148 122 83 83 150 340 204 228 346 345 308 335 335

0 7 13 18 19 22 22 27 27 28 28 28 30 32 33 33 34 35 39 39 42 42 43 44 59 62 63 67 77 83 83 87 89 91 91 96 97 97 97 107 119 118 118 119 124 129 130

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Only stations with complete monthly records are used in the analysis and compiled in this table (indicated by x's). The gauge sites are listed according to their relative distance from the radar installation at Berrimah, with the closest sites listed first. The geographical locations of the sites are shown in Figure 1. Abbreviations are D, December; J, January; F, February; M, March.

The monthly areal mean rainfall is then estimated as the linear average among all the sites of the network giving each rain gauge equal weight. The areal mean rainfall might be obtained by more sophisticated averaging procedures, for example, by using a isohyetal method or by geometric means (triangles or Thiessen polygons). However, the method applied to derive the areal mean rainfall has little effect on the estimates of the sampling uncertainties. For simplicity we use the linear average of the values obtained at the gauge sites.

3. Sampling Problem for Monthly Areal Mean Rainfall 3.1. Uncertainty as a Function of Sampling Time Interval 3.1.1. Darwin, February 1988. The data collected during February 1988 by the Darwin rain gauge network illustrate the procedure used. The most accurate monthly rainfall estimate is obtained by using every 15-min sample. The areal mean accumulation is 225 mm for February 1988 (Table 3). If we use a subset of all the samples, we obtain a different value. To illus-


trate the variability of the possible estimates that could be obtained by using a subset of the samples, we may take only one sample per day, always taken at the same time (cf. midnight). We subsequently repeat the calculation taking the sample at other times of the day (midnight + 15 min, + 30 min, + 45 min, etc.). As a result we obtain an indication of the areal mean diurnal rainfall cycle (over land) in the vicinity of Darwin for this particular month (Figure 3). Each dot in Figure 3 represents a different monthly areal rainfall estimate based on using only one 15-min sample per day, always taken at the indicated time. The heavy horizontal line at 225 mm indicates the true monthly areal mean rainfall accumulation. The areal mean rainfall is minimum during the morning (0800-1300 LST; local standard time for Darwin is UTC plus 9.5 h) and maximum in the afternoon. To obtain an estimate of the uncertainty associated with the sampling time interval, various sampling frequencies are examined. All the different estimates based on taking only one sample per day, as discussed above (Figure 3), result in the range of values plotted at the sampling time interval of 24 hours in Figure 4. Increasing the sampling frequency from one to two samples per day, we obtain the range of values plotted at the sampling time interval of 12 hours, as shown in Figure 4. These estimates of monthly rainfall range from -120 to 430 mm, depending on what time of day the samples are taken. Using a sampling interval of 8 hours (Figure 4), the spread of estimates is reduced somewhat to —125-390 mm. Further reduction of the spread is seen as the sampling interval is successively reduced. For a sampling interval of 1 hour the spread is from —205 to 240 mm, only slightly deviating from the true mean of 225 mm for February 1988 (Table 3); that is, all the estimates calculated with an hourly sampling frequency produce a monthly areal mean rainfall estimate within about 10% of the true accumulation. The scatter of the points in Figure 4 measures the uncertainty of any estimate obtained with a given regular sampling frequency. The dark (light) shaded area at each sampling time interval represents the range of one (two) standard deviation^) from the mean based on all the data, as indicated by the heavy horizontal line. The sloping dashed lines enclose approximately the estimates that are within one standard deviation of the mean accumulation. For February 1988 the slopes of these lines indicate that the uncertainty of the monthly areal mean rainfall estimate, expressed as the standard deviation (root mean square error) of estimates based on all possible regularly spaced sampling intervals, increases roughly 3.5% per hour increase in sampling time interval. In this particular case, at least eight samples per day would be necessary, on the average, to estimate the monthly areal mean rainfall accumulation within 10% accuracy. The sampling-based uncertainty, however, is also a function of the rainfall depth, as will be discussed in section 3.2.2. 3.1.2. Other months from Darwin and Melbourne. A total of 10 months of data collected by the Darwin rain gauge network and another 2 months from Melbourne, Florida, were processed as discussed above. Figures 5, 6, and 7 show the results from Darwin's rainy season in 1987/1988, 1989/1990, and 1993/1994, respectively. The analyses of the two summer months in 1993 for Melbourne are shown in Figure 8. The panels on the left present the monthly areal mean diurnal rainfall signature (similar to Figure 3), while the panels on the right illustrate the sampling uncertainty (similar to Figure 4). The pattern of the diurnal rainfall cycle at Darwin exhibits

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Table 2. Rain Gauge Network Around Melbourne, Florida, Consisting of Tipping-Bucket Gauges of the Kennedy Space Center (KSC) and the St. John's Water Management District (STJ) Networks 1993 Rain Gauge Site STJ-0410 STJ-0126 STJ-0285 STJ-0241 STJ-0122 STJ-0265 STJ-0263 STJ-0231 STJ-0215 STJ-0206 KSC-0033 STJ-0260 KSC-0025 STJ-0225 STJ-0120 KSC-0023 KSC-0028 STJ-0185 KSC-0022 STJ-0106 STJ-0271 KSC-0020 STJ-0511 STJ-0507 STJ-0526 KSC-0017 KSC-0014 STJ-0101 KSC-0011 STJ-0313 KSC-0007 KSC-0005 STJ-0294 KSC-0002 STJ-0277 STJ-0275 'STJ-0474 STJ-4197 STJ-4997 STJ-2172 STJ-2176 STJ-2157 STJ-2181 STJ-2186 STJ-2231 STJ-2202 STJ-2226 STJ-2266 STJ-3002 STJ-1140 STJ-1236 STJ-1138 STJ-3037 STJ-1258 STJ-1243 STJ-1130 Gauges used per month (complete records only)

•

Direction, deg

Distance, km

150 151 262 208 155 292 236 203 160 189 8 214 358 313 171 360 18 158 355 195 327 1 183 192 174 358 5 182 358 338 351 356 332 354 316 331 331 300 301 301 301 302 300 300 301 301 301 301 304 332 327 335 311 317 329 335

11 19 25 26 28 28 29 31 33 33 34 35 35 37 38 39 41 43 44 45 45 48 49 49 50 50 52 53 55 57 60 61 64 64 68 76 86 104 111 118 118 118 119 119 119 120 120 120 121 145 151 151 154 162 164 193

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Only stations with complete monthly records are used in the analysis and compiled in this table. The gauge sites are listed according to their relative distance from the NEXRAD WSR-88D installation at Melbourne, with the closest sites listed first. The geographical location of the sites is shown in Figure 2. Abbreviations are J, July; A, August.


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Distance East of Radar [km] Figure 2. Geographical map of the central Florida region indicating the radar and rain gauge network. The land and islands within the 400 km by 400 km domain centered at the Melbourne radar site are heavily shaded. The dotted range rings indicate equal distances from the radar site and are 50 km apart. The rain gauge locations, as given in Table 2, are shown by the triangles. The box (150 km by 250 km) is used to estimate the area represented by the gauge network (see text for details).

significant variability. It ranges from months with a very pronounced diurnal signature (e.g., February 1988 or January 1990) to months with almost none (e.g., March 1990). This variability results from the different precipitation systems occurring in the vicinity of Darwin during break and active monsoon periods [Keenan and Carbone, 1992] and when the active phases of the monsoon take place. Most of the months under investigation exhibited a pronounced rainfall maximum in the

afternoon, which is typical for convection over land in this region. Some months showed a secondary maximum in the early morning hours (Figures 5a, 5c, and 7a), which is typical for rainfall resulting from large MCSs [Gray and Jacobson, 1977; Hendon and Woodberry, 1993]. The distinct peaks in the midafternoon (—1500 LST), for example, clearly visible in the monthly mean diurnal rainfall signature of February 1988 (Figures 3 and 5e) and January 1994 (Figure 7a), are the result of

Table 3. Uncertainties as a Function of Sampling Frequency Based on Tipping-Bucket Rain Gauge Data Collected in the Vicinity of Darwin, Australia and Melbourne, Florida

Site Darwin

Melbourne

Month

Year

Number of Gauges

December January February March December January February March January February July August

1987 1988 1988 1988 1989 1990 1990 1990 1994 1994 1993 1993

22 22 22 22 22 24 24 21 22 25 41 54

Monthly Areal Mean Rainfall, mm

Uncertainty Increase, % hour"1

378.1 194.0 225.3 254.7 157.7 335.6 170.8 212.6 210.4 474.0 89.7 109.1

2.4 3.3 3.4 2.8 4.4 2.6 3.9 2.8 4.1 1.9 6.3 5.7

.

The last column expresses the average increase of uncertainty of the estimated monthly areal rainfall per hour decrease in sampling time resolution. See text for further details.

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intense connective thunderstorms (Hectors) over the islands north of Darwin. This type of convection develops frequently and always at about the same time in the midafternoon [Keenan et al., 1990]. The 10 months investigated from Darwin exhibit noticeably different monthly areal mean diurnal rainfall signatures, as shown by Figures 5, 6, and 7. As a result of the variable large-scale conditions these months went through a variety of break and active monsoon periods which determined the rainfall patterns. The monthly areal mean rainfall accumulations (over land), estimated by the Darwin rain gauge network, cover a range of values from as low as —150 mm to close to 500 mm (Table 3). In contrast, the summer precipitation in central Florida is influenced to a large extent by sea-breeze convection: thunderstorms developing in the afternoon, triggered by a low-level convergence resulting from sea breezes entering the peninsula from both sides. The monthly areal mean diurnal rainfall signatures revealed by the gauge networks around Melbourne for July and August 1993 are similar. Both months exhibit a pronounced rainfall maximum in the afternoon (1200-2000 EST; local time for Melbourne is eastern standard time, i.e., UTC minus 5 hours); however, the maximum in July appears as a single peak (at -1500 EST), while several maxima show up in August. The monthly rainfall totals are roughly 100 mm for both months (Table 3). The magnitude of the sampling uncertainty, as illustrated by the right-side panels of Figures 5-8, can be expressed as a function of the rainfall depth. The horizontal dotted lines marking the 50% uncertainty are intersected by the sloping dashed lines at a higher sampling frequency for smaller monthly areal rain accumulations.

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Figure 3. Rain gauge-based, monthly areal mean rainfall accumulation estimated from one 15-min sample per day, as a function of the time of day the sample is taken (solid line with dots). This figure reflects the monthly area-average diurnal rainfall signature over land for February 1988 at Darwin. The heavy horizontal line indicates the true monthly areal mean rainfall accumulation for this particular month and site based on the continuous rain gauge record.

8 Feb88

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50% uncertainty

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Sampling Time Interval [hours]

Figure 4. Uncertainty of monthly areal rainfall estimates as a function of the sampling time resolution. The results are based on the Darwin rain gauge network data of February 1988 and a sampling scenario as explained in the text. The heavy shaded area in each sampling time interval bin corresponds to the range of the mean plus/minus one standard deviation; the light-shaded area corresponds to the range of the mean plus/ minus two standard deviations. The heavy horizontal line indicates the true monthly areal mean rainfall accumulation for this particular month and site based on the continuous rain gauge record (225 mm). The 50% uncertainty range is shown by the dotted horizontal lines. The sloping dashed lines indicate that decreasing the sampling time resolution increases the uncertainty of the monthly areal rainfall estimate by roughly 3.5% per hour decrease in time resolution.

3.1.3. Determination of the average sampling uncertainty. The sloping dashed lines in Figures 5-8 indicate the average uncertainty of the monthly areal mean rainfall estimates as a function of the sampling frequency. It might be interesting to further investigate the variability of the rainfall estimates for different sampling frequencies with respect to the temporal autocorrelation of the observed precipitation systems. For the purpose of this study, however, we wish to focus on the average uncertainty increase with decreasing sampling frequency (assuming a linear dependence upon the sampling frequency). The slope of the dashed lines in Figures 5-8 is determined to enable a comparison of the results obtained for different months. For each sampling time interval an estimate of that slope is obtained by dividing the standard deviation (i.e., the root mean square error) of the different rainfall estimates in that sampling bin by the mean rainfall estimate and by the sampling time interval AT. Multiplied by 100%, the unit of such a slope estimate is given in percent per hour, i.e., an increase in uncertainty, expressed as a percentage of the monthly areal mean rainfall, per hour decrease in sampling frequency. The individual slope estimates for each sampling time, S(AT), are averaged using a weighted mean to derive the average slope S, indicated by the dashed lines in Figures 4-8

J_ W

(2)

where W is the sum of all the N investigated sampling time intervals. The weighting with sampling frequency gives the

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fewer slope estimates obtained for longer sampling intervals more weight compared to the more numerous estimates based on short sampling intervals. This procedure takes into account that the slope estimates based on the longer sampling time intervals are the more robust estimators of the average slope. It is noteworthy to point out that the Laughlin-type formula [Laughlin, 1981] like the one in equations (23) and (24) of Bell et al. [1990] predicts that the quantity S(AT) should increase linearly with AT for small ATs (i.e., AT small compared to the correlation time of the area-averaged rain) but that it should increase as (AT) 0 5 for large AT. The latter behavior is given in equation (30) of Bell et al. [1990]. The linear approach used in (2) above is likely to produce reasonable estimates since the "fit" is biased toward the data with larger AT. In addition, the subsequently derived formulas, for example, (5) and (6), may be used mostly with AT in the region where the fit has been performed (several hours to a day). However, (5) or (6) may tend to predict too rapid a growth in error if extended to still larger ATs (>24 hours). For each month of the Darwin and Melbourne data the average slope S, indicating the sampling uncertainty introduced by reducing the sampling frequency, is determined using the above procedure and compiled in Table 3. These uncertainties introduced per hour decrease in sampling time resolution cover a range of about 2% hour"1 to more than 6% hour"1. The largest values are found for Melbourne, Florida.

3.2. Uncertainty as a Function of Monthly Rainfall Amount and Domain Size

3.2.1. Scaling with domain size. To compare the results obtained using the Darwin rain gauge network with those of the Melbourne data and to compare these results with those of previous studies of the sampling problem, we have to address how the uncertainties scale with the domain size. Studies by North and Nakamoto [1989], Bell et al. [1990], and Graves et al. [1993] indicate that the uncertainty introduced by a finite sampling frequency decreases with increasing spatial scale. They show that the root mean square (RMS) error E scales with area A according to

E*A-* (3) where a is 0.5. This is in close agreement with the results of Huff [1970], who found a power factor a of 0.56. To compare the Melbourne gauge data-based results with those of the Darwin rain gauge network, the sampling uncertainties listed in Table 3 for July and August 1993 have to be scaled by a factor of 0.84, the square root of the ratio of the area represented by the Melbourne (18,700 km2) and the Darwin (26,500 km2) rain gauge networks. The resulting sampling uncertainties for Melbourne, scaled to the domain size represented by the Darwin rain gauge network, are 5.3% hour"1 for July and 4.8% hour"1, respectively, for August. Assuming a 10% error in the area represented by the Melbourne gauge

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networks, these values may be accurate to within ±0.3% hour"1. 3.2.2. Scaling with rain amount. Figure 9 shows the Darwin sampling uncertainties, as listed in Table 3, together with the adjusted Melbourne values (open diamonds) as a function of the corresponding monthly areal mean rainfall amounts. The magnitude of the sampling-related, average uncertainty is a function of the rain accumulation. Months exhibiting a large areal mean rainfall have a much smaller sampling uncertainty for a given frequency of observation than months with low rain accumulations. This scaling with rainfall depth, shown by the dashed curve in Figure 9, can be approximated by

the true order of magnitude of the sampling problem one would encounter by using complete area-covering observations (i.e., radar or satellite). Point observations generally exhibit more variability than area observations, which introduces additional uncertainty depending on the rain gauge network density. On the other hand, the satellite (or radar) will measure instantaneous instead of 15-min rain rates. Since the variance of the 15-min rain rate process is smaller than that of the instantaneous rain rate process, the use of 15-min averages tends to underestimate the sampling error. Hereafter, we attempt to estimate the magnitude by which the Darwin rain gauge network overestimates the sampling uncertainties compared to complete area-covering radar observations. S = 83(1.0 mm/Rs)° (4) Using radar data, an estimate of the overestimation of the where Rs is the monthly rain accumulation (in millimeters). sampling problem by rain gauge networks can be obtained. Equation (4) describes the average scaling behavior of the Comparing the average sampling uncertainties obtained by sampling uncertainty with rain amount; however, there is some using all the radar data over land (there are no rain gauge sites variability of the data about the dashed curve in Figure 9. This over the ocean) with those obtained by using the radar inforvariability, quantified by an RMS error of less than 0.5% mation at the rain gauge sites only (closest pixels), the rain hour"1, may result from differences in the space and time gauge network at Darwin appears to overestimate the sampling intermittency and diurnal cycle of rainfall from month to problem by about a factor of 1.6, as will be shown below. This month. For example, the data collected in March 1990 exhibit result is based on roughly 2 months of radar data collected below average sampling uncertainty, which has to be attributed during the Australian rainy season in 1993/1994 by the Darwin to the absence of a diurnal rainfall signature for that particular radar and a rain gauge network configuration of 26 sites. The month (Figure 6g). first time period comprises 25 days of radar information (vol3.2.3. Area versus point observations. The study of the ume scans every 10 min) of December 1993 and January 1994, sampling uncertainties using rain gauge data may not reflect while the second has 20 days of data collected in February and

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4. Discussion of Results 4.1. Sampling Uncertainty and Climate Figure 9 shows that the sampling problem for Darwin and Melbourne, Florida Melbourne can be described by the same scaling law in rainfall Aug93 amount. This suggests that the sampling uncertainty of a monthly rainfall estimate is constrained by the rainfall depth and the sampling time interval. The large-scale environment is Dec 89 a key factor determining the rainfall amount. The climatic regime defines the range of possible rainfall accumulations and O therefore limits the sampling problem for a given geographical \ 9 Darwin, Australia location. Large areal rainfall amount and reduced uncertainty for a given sampling frequency are associated with extensive active monsoonal periods at Darwin (e.g., December 1987, 0 January 1990, February 1994). Moderate or low rainfall depths O and larger sampling-based uncertainties occur in months with ID significant break periods (e.g., February 1988, December 1989, January 1994). In contrast to Darwin the climatic regime in central Florida is quite different, producing multicellular con200 300 400 500 600 vective storms that are triggered by sea and lake breezes and 100 are scattered over the peninsula. Though locally these storms Monthly Areal Mean Rainfall [mm] may produce significant rain amounts, the monthly areal mean rainfall depths are much lower than observed around Darwin. Figure 9. Scaling of the sampling uncertainty with rainfall amount. The results of the 10 months of rain gauge data from Because of the somewhat random distribution of the rainfall in Darwin, Australia, and the 2 months from Melbourne, Florida, central Florida and the lower rain amounts, the uncertainties are shown by the circles, as listed in Table 3. The space- resulting for a given sampling frequency are significantly larger adjusted values of the 2 Melbourne months are indicated by than for Darwin. However, more rain gauge and/or radar data, the open diamonds (see text for details). The dashed, fitted collected at other sites as well, will have to be analyzed simicurve represents the average scaling behavior of the sampling larly to validate this result. The inclusion of some parameters problem with rainfall amount, as given by (4). describing the large-scale environment may provide valuable additional information. The scaling of the sampling problem with rain amount is comparable to a scaling with the length of the rain accumulation time period (i.e., daily, weekly, monthly rainfall, etc.). The March 1994. Only days with almost complete records (i.e., less longer the time period of interest is the more rain might acthan 10% of radar volumes missing per day) are used for this cumulate, which in turn determines the magnitude of the samanalysis to avoid the introduction of additional uncertainty due pling problem. We choose to present our analysis on the basis to data gaps. The subsampling procedure applied to the 2-km of monthly rainfall accumulations, because this is a typical gridded radar data is the same as outlined in section 2.2 and timescale for climatological studies like TRMM [Simpson, used for processing the rain gauge data. 1988; Simpson et at., 1988]. However, if the timescale is During the first period of the radar observations at Darwin, changed from 1 month to some other time interval T, the the rainfall over land totaled —270 mm. Using the area- percent sampling error is expected to scale as roughly (30/ covering radar information over land (within a radius of 130 jT)° 5 , if T is measured in days. This scaling in time is prekm; see Figure 1), the average sampling uncertainty is esti- dicted, for example, by equation (25) of Bell et al. [1990]. Since mated to 1.9% hour" 1 (±0.1% hour" 1 ). The estimate of the (4) is written in terms of accumulated rain (or monthly mean sensitivity (value in parenthesis) of the sampling uncertainty to rain rate times 24 hours times the number of days) and has a the represented area is obtained by varying the radius of 130 R~°'6 behavior in it, the inverse square root dependence on T km to 123 km and to 136 km, corresponding to a 10% decrease is represented pretty well in that formula. and increase of the area covered, respectively. In comparison, using the radar information at the rain gauge locations only 4.2. Scaling With Rain Amount and Domain Size (closest pixel), the average uncertainty introduced by a reducThe scaling of the sampling uncertainty with rain amount tion in sampling frequency is estimated to be approximately 2.8% hour"1. The results of the second period are different in and domain size, as described by (4) and (3), respectively, can that the rainfall totaled -380 mm and the sampling uncer- be combined into a single relationship to estimate the average tainty is estimated to 1.0% hour"1 (±0.01% hour"1) for the uncertainty of monthly areal rainfall based on a given sampling complete area (over land) versus —1.7% hour"1 for the point frequency, rain amount, and spatial scale. Taking into account observations (at gauge sites). Though these are preliminary that rain gauge point observations tend to overestimate the results, they suggest that the overestimation of the sampling sampling uncertainty compared to complete area-covering obproblem by using the radar information at the gauge sites only, servations, as discussed in section 3.2.3, and assuming that the on the average, may be approximated by a factor of 1.6 ± 0.2. scaling with rain amount behaves similarly on different spatial This correction factor, however, is dependent on the density of scales, the combination of (3) and (4) results in the rain gauge network and the area assumed to be represented; it may also be dependent on the climatic regime. E =C (5)

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where C has a value of 52 ± 7 (this incorporates the uncertainty of the correction factor of 1.6) andA0 is 26,500 km2. Note that we take over the terminology used in (3) to express the sampling-related RMS error by the symbol E instead of S, the average slope parameter which, however, implicitly is an (average) RMS error. Figure 10 displays this scaling of sampling uncertainty with rain amount and spatial scale. The uncertainty for a given rain amount and domain size read from Figure 10, or calculated using (5), have to be multiplied by the sampling time interval AT to obtain an estimate of the uncertainty of the monthly areal rainfall. Therefore a rule of thumb for estimating the average sampling-related uncertainty to expect for a given sampling frequency, rainfall depth, and domain size is

E = 8.5 x io 3 x/?;°- 6 ^-°- 5 Ar

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where E is the RMS error expressed as a percentage of the monthly rainfall amount. Scaling the variability of the data in Figure 9 about (4), resulting from differences in the space and time characteristics of the rain fields observed and previously determined as —0.5% hour"1, to a domain size of 500 km by 500 km and taking into account a factor of 1.6 overestimation of the sampling problem by the rain gauge network, we obtain an uncertainty of 0.1% hour"1 for (5). For a TRMM-like configuration with two complete area-covering visits daily, the estimated sampling-based uncertainties using (6) are therefore likely to be accurate to within a couple of percents. On the other hand, the uncertainty of the factor 1.6 (±0.2), used to account for the overestimation of the sampling problem by the gauge network, results in a dispersion of the multiplicative factor in (6) of ±13%. Thus overall the values computed using (6) to estimate the average sampling uncertainty for a given rainfall depth, sampling frequency, and domain size are likely to be accurate to within ±15%. For example, the estimated standard deviation for a computed sampling uncertainty of E = 10% using (6) is 1.5%.

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Square Root of Domain Size [km] Figure 10. Scaling of the sampling-based uncertainty with rainfall amount and domain size, as described by (5). The contours represent points of equal sampling uncertainty, expressed as percentage uncertainty per hour decrease in sampling frequency. Shown are the contours 0.25% hour"1, 0.31.0% hour'1 (in 0.1% hour"1 steps), 1-3% hour"1 (in 0.5% hour"1 steps), and 3-15% hour"1 (in 1% hour"1 steps). The contours of 1, 5, 10, and 15% hour"1 are heavily outlined. The value read off the diagram for a given monthly rainfall amount and characteristic length of the domain has to be multiplied by the sampling time interval to obtain an estimate of the sampling uncertainty (in percent) of that monthly rainfall.

report that the multiyear average rainfall over Japan in June is about 300 mm, which is very close to the GATE rainfall. Using 20 days worth of radar data from central Florida, Seed 4.3. Comparison With Other Results and Austin [1990] report sampling uncertainties of about 22% Using either Figure 10 or (5) or (6) enables us to compare for a TRMM-like configuration. Scaling this result to a our results with previous findings reported in the literature. monthly basis (see section 4.1), the sampling uncertainty would For example, the mean rain rate for GATE (Phase I) is —0.45 be about 18%. This appears to be significantly larger than the mm hour"1 [Kedem et al., 1990; Bell et al., 1990], which results results which are based on the GATE data set. However, using in a monthly rain accumulation of about 330 mm. For a sam- the values for the mean rain rate (0.1 mm hour"1) and domain pling configuration like TRMM, with two visits daily and an size (124,000 km2) given by Seed and Austin, it can be shown area of 280 km by 280 km (GATE radar domain), we obtain an by means of the relationships established in section 4.2 that uncertainty increase of —0.9% per hour decrease in sampling their results are consistent with those studies mentioned above time interval. Using AT = 12 hours, the monthly areal rainfall using the GATE data set. Using (6) together with Seed and exhibits an uncertainty of—11% (±1.5%). This is in very good Austin's values, we predict an uncertainty of —22% (±3.5%) agreement with results reported by Laughlin [1981], McConnell for a monthly rainfall of 72 mm on a 350 km by 350 km domain. Graves et al. [1993] investigated the rainfall sampling proband North [1987], andNakamoto etal. [1990], who found errors of —8-10% for complete area-covering visits over the GATE lem over land (compared to ocean) using rain gauge data ship array. Using the GATE rainfall and (6) to estimate the collected during the Preliminary Regional Experiment for average sampling uncertainty for TRMM (AT = 12 hours) on STORM-Central (PRE-STORM), which took place in Maya domain of 500 km by 500 km, we obtain a sampling error of June 1985 in Kansas and Oklahoma. Making use of the stoabout 6% (±1%). This appears to be in excellent agreement chastic rainfall model of Waymire et al. [1984], they estimate a with the results reported by Weng et al. [1994] (e.g., see their sampling uncertainty for a TRMM-like satellite (500 km by 500 Figure 4), who used hourly radar-AMeDAS (Automatic Me- km domain, completely visited every 12 hours) of about 12% teorological Data Acquisition System, operational since 1988) for rainfall as observed during May-June 1985 over the PREcomposite data collected in June 1989 over Japan. Though STORM area. Comparing their results to those similarly obWeng et al. do not mention the areal mean rainfall depth of tained using the GATE data set (sampling uncertainty of June 1989, Old and Sumi [1994], also using the AMeDAS data, 10%), Graves et al. [1993] conclude that the sampling problem

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over land might be somewhat larger than over ocean, mainly caused by the larger variability in the PRE-STORM rain field compared to the GATE data. Since Graves et al. do not provide information about the mean rain rate observed during May-June 1985 over the Kansas-Oklahoma area, we make use of the rainfall amounts listed in the Climatic Division Data [NOAA, 1988] to estimate the monthly rain accumulations for the 2 months. Using those divisions that most closely represent the PRE-STORM domain covered by the mesonet stations (numbers 4-9 for Kansas and numbers 2, 4, 5, 7, and 8 for Oklahoma), we estimate an average rainfall accumulation of 71 mm for May and 142 mm for June 1985; the average over both months is 106 mm. According to (6), the sampling uncertainty for a TRMM-like satellite, completely visiting a domain of 500 km by 500 km every 12 hours, is expected to be of the order of -16% (±2.5%) for May, 10-11% (±1.5%) for June 1985, and 12-13% (±2%) for both months combined. This is in excellent agreement with the results of Graves et al. [1993]. Thus the differences in sampling uncertainties estimated for rainfall over land versus over ocean, given the same domain size and similar sampling frequency, can be attributed to a large extent to differences in rainfall depth. 4.4. Realistic Satellite Overflights Depending on the orbital configuration, satellites provide mostly partial visits for a given domain of interest. Partial visits, however, increase the sampling problem for monthly areal rainfall estimation. Assuming realistic overflights with partial visits, Shin and North [1988] and Bell et al. [1990], for example, investigate the sampling problem the TRMM satellite will encounter. Using the GATE data set, their results indicate that the sampling problem for partial visits is roughly a factor of 2 larger than assuming complete area-covering visits. Shin and North and Bell et al. estimate the sampling uncertainty of monthly rainfall over a 500 km by 500 km domain to be about 8-12%, more or less independent of the geographical latitude. In another study using hourly radar-AMeDAS composite data, Oki and Sumi [1994] analyze the sampling problem of the TRMM satellite over Japan. For monthly areal mean rainfall accumulations, ranging from 56 mm to 348 mm and assuming realistic satellite overflights, they estimate sampling uncertainties of -16-32% for a swath width of 200 km (TRMM precipitation radar, PR) and -11-29% for a swath width of 700 km (TRMM microwave imager, TMI). These values are somewhat larger than the estimates of Shin and North [1988] and Bell et al. [1990]. However, if we scale Oki and Sumi's sampling uncertainties to the rainfall depth observed during GATE (—330 mm per month), we obtain values of —10-18% for the PR and -8.5-13.5% for the TMI. These values are much closer to the results obtained by Shin and North [1988] and Bell et al. [1990]. Thus the data shown by Oki and Sumi [1994] also appear to follow the presented scaling rules. The seasonal dependence of the sampling problem shown by Oki and Sumi is a result of the seasonally varying rainfall amounts observed over Japan.

5.

Summary and Conclusions

Uncertainties in the estimation of monthly areal mean rainfall have been investigated by making use of data collected by tipping-bucket rain gauge networks at Darwin, Australia, and Melbourne, Florida. Using a subsampling methodology, the average uncertainty introduced to the monthly areal mean

rainfall estimates as a function of the sampling frequency is studied for full area-covering observations occurring at regularly spaced intervals in time. This is a central problem for estimating the accuracy of remote-sensing observations such as those made by satellites or radar. Data collected by rain gauge networks do not reveal the true sampling uncertainty representative for area-covering measurements. However, the investigation of the sampling problem using rain gauge data may provide an estimate of the upper limit of the sampling uncertainty introduced by a finite sampling frequency. Using radar data, an estimate of the overestimation of the sampling uncertainty by rain gauge point observations compared to complete area measurements can be obtained and taken into account in the analysis. In turn, this provides also an anchor for the area represented by the gauge network. However, we would like to recommend the use of area-covering radar data for studying the sampling-related uncertainty whenever possible, avoiding thus the introduction of additional uncertainty due to point observations. A key result of this analysis is that the sampling uncertainty scales with rainfall depth and spatial scale. For a given frequency of observations and domain size the sampling uncertainty is small for months with a large areal rainfall accumulation, while large uncertainties occur for months with moderate or small areal rainfall totals. Moreover, the sampling problem for rainfall observed at Darwin can be described similarly to the sampling problem encountered with Florida precipitation. The average uncertainty of a monthly areal rainfall estimate, introduced by a finite sampling frequency, therefore appears to be constrained by the rainfall depth, the sampling time interval, and the domain size of interest. A simple rule of thumb is presented to estimate the sampling uncertainty one has to expect, on the average, for a given sampling frequency, rain amount, and domain size. The estimated root mean square error E, expressed as a percentage of the monthly rainfall amount Rs, is inversely proportional to the rainfall amount and the domain size A but proportional to the sampling time interval AT

E = 8.5 x io3x#;°-6,4-0-5A:r The standard deviation of the estimated values for the sampling uncertainty E, expressed as a percentage of E, is of the order of 15%. For example, an estimated sampling uncertainty of E = 10% is accurate to within ±1.5%. Using this framework, it is shown that other results previously reported in the literature, including findings considered controversial, are consistent with our results and among each other. The presented framework seems to capture the essence of the sampling uncertainties of tropical or subtropical precipitation observed over the eastern Atlantic (GATE), central Florida, the maritime continent (Darwin, Australia), and Japan. Moreover, rainfall observed during PRE-STORM in the midlatitudes (Kansas and Oklahoma area) appears to follow the same rules as well. Differences encountered in studying the sampling problem of monthly areal rainfall at different geographical locations can mostly be explained by differences in the observed precipitation amounts. The large-scale environment is the key factor determining the rainfall patterns and their intermittency in space and time. The climatic regime defines the range of possible rainfall amounts and consequently limits the sampling uncertainty for a given geographical region. Knowledge of the potential range of rainfall accumulations is useful for designing


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Krishnamurti, pp. 298-353, Oxford Univ. Press, New York, AT sampling time interval; hours. 1987. T timescale of interest to sampling error Kedem, B., L. S. Chiu, and G. R. North, Estimation of mean rain rate: estimation; days. Application to satellite observations, /. Geophys. Res., 95, 1965W sum of all the sampling time intervals; 1972, 1990. hours. Keenan, T. D., and R. E. Carbone, A preliminary morphology of precipitation systems in tropical northern Australia, Q. J. R. Meteorol. Soc., 118, 283-326, 1992. Acknowledgments. The rain gauge and radar data have been pro- Keenan, T. D., G. J. Holland, M. J. Manton, and J. Simpson, TRMM vided by the TRMM Office under the leadership of Otto Thiele. The ground truth in a monsoon environment: Darwin, Australia, Aust. valuable discussions with Robert A. Houze and the thoughtful comMeteorol Mag., 36, 81-90, 1988. ments of Stephen J. Burges, Shuyi S. Chen, Chidong Zhang, Sandra E. Keenan, T. D., B. R. Morton, M. J. Manton, and G. J. Holland, The Yuter, and Christof Appenzeller are greatly appreciated. This work Island Thunderstorm Experiment (ITEX)—A study of tropical benefited considerably from several, very stimulating discussions with thunderstorms in the maritime continent, Bull. Am. Meteorol. Soc., Thomas L. Bell. The comments of two anonymous reviewers helped 70, 152-159, 1989. sharpen the discussion. Some computational assistance has been pro- Keenan, T. D., B. R. Morton, X. S. Zhang, and K. Nyguen, Some vided by Stacy R. Brodzik, Mark D. Albright, and Michael Steiner. characteristics of thunderstorms over Bathurst and Melville Islands Grace C. Gudmundson edited a previous version of the manuscript. near Darwin, Australia, Q. J. R. Meteorol. Soc., 116,1153-1172,1990. This work was supported by NASA grant NAG 5-1599 to the Univer- Kuettner, J. P., D. E. Parker, D. R. Rodenhuis, H. Hoeber, H. Kraus, sity of Washington and by NSF grant EAR-9528886 to Princeton and G. 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the orbital configuration of satellites devoted to rainfall measurements. However, detailed information about the intermittency in space and time, diurnal patterns, and other characteristics of the rainfall on a regional scale are needed for the interpretation of the observations' accuracy for specific situations. We choose to study the sampling problem for monthly rainfall observations; however, the presented rule of thumb might be useful for estimating sampling uncertainties on different timescales as well. On shorter timescales (e.g., weeks or days), less rainfall will accumulate and the predicted sampling uncertainty will therefore be larger. In contrast, on larger timescales (e.g., seasons or years) a smaller sampling uncertainty will be found. If the timescale is changed from 1 month to some other time interval T, the percent sampling error E is expected to scale as roughly (30/r)°- 5 , if T is measured in days.

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(Received April 11, 1995; revised September 27, 1995; accepted November 2, 1995.)