A Space-Filling Algorithm to Extrapolate Narrow ... - AMS Journals

VOLUME 25

JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY

NOVEMBER 2008

A Space-Filling Algorithm to Extrapolate Narrow-Swath Instantaneous TRMM Microwave Rain-Rate Estimates Using Thermal IR Imagery ANA P. BARROS

AND

KUN TAO

Pratt School of Engineering, Duke University, Durham, North Carolina (Manuscript received 17 April 2007, in final form 13 December 2007) ABSTRACT A space-filling algorithm (SFA) based on 2D spectral estimation techniques was developed to extrapolate the spatial domain of the narrow-swath near-instantaneous rain-rate estimates from Tropical Rainfall Measuring Mission (TRMM) precipitation radar (PR) and TRMM Microwave Imager (TMI) using thermal infrared imagery (Meteosat-5) without making use of calibration or statistical fitting. A comparison against rain gauge observations and the original PR 2A25 and TMI 2A12 estimates in the central Himalayas during the monsoon season (June–September) over a 3-yr period of 1999–2001 was conducted to assess the algorithm’s performance. Evaluation over the continental United States was conducted against the NCEP stage IV combined radar and gauge analysis for selected events. Overall, the extrapolated PR and TMI rainfall fields derived using SFA exhibit skill comparable to the original TRMM estimates. The results indicate that probability of detection and threat scores of the reconstructed products are significantly better than the original PR data at high-elevation stations (⬎2000 m) on mountain ridges, and specifically for rainfall rates exceeding 2–5 mm h⫺1 and for afternoon convection. For low-elevation stations located in steep narrow valleys, the performance varies from year to year and deteriorates strongly for light rainfall (false alarm rates significantly increase). A preliminary comparison with other satellite products (e.g., 3B42, a TRMM-adjusted merged infrared-based rainfall product) suggests that integrating this algorithm in currently existing operational multisensor algorithms has the potential to improve significantly spatial resolution, texture, and detection of rainfall, especially in mountainous regions, which present some of the greatest challenges in precipitation retrieval from satellites over land, and for hydrological operations during extreme events.

1. Introduction Limitations due to inadequate spatial resolution, infrequent and/or irregular sampling, and basic technical difficulties with remote sensing of the structure of the atmosphere in continental regions (i.e., effects of terrain complexity, surface heterogeneity, etc.) produce large uncertainties in satellite-based precipitation estimates, especially at short time scales (subdiurnal) and over small areas (⬍1000 km2; e.g., Kummerow et al. 2004; Bell et al. 2001; Barros et al. 2000, 2006; among others). For hydrological applications and water resources management, the ability to estimate the spatial location of rainfall correctly with regard to the river basin boundaries—that is, placing rainfall in the catchment or

Corresponding author address: Dr. Ana P. Barros, CIEMAS 2457, Box 90287, Duke University, Durham, NC 27708. E-mail: [email protected]

watershed where it occurs—is essential to close the water budget. In operational hydrology, especially in the case of flood forecasting, the requirement is especially important for heavy rainfall. Whereas rainfall intensity, timing, and duration are also critical for hydrometeorological prediction, it is practically impossible to achieve useful monitoring or predictive skill if rainfall is not detected in the right catchment basin (Kim and Barros 2001). This challenge is particularly acute in regions where the topography is complex and the representative ground-based rainfall observations are scarce. The precipitation radar (PR) and the Tropical Rainfall Measuring Mission (TRMM) Microwave Imager (TMI) on board the TRMM satellite provide rainfall estimates at high spatial resolution. However, low revisit frequency and relatively narrow swaths (⬃350 km for the PR), along with terrain clutter effects on radarbased retrieval of rainfall, undercut its effectiveness for applications. Precipitation products such as nearly instantaneous rainfall rates (TRMM PR 2A25 and TMI

DOI: 10.1175/2008JTECHA1019.1 © 2008 American Meteorological Society

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2A12) tend to severely underestimate precipitation at high elevations even during heavy rainstorms (e.g., Lang and Barros 2002; Barros et al. 2006). Nevertheless, satellite-based sensors provide the only viable platform for monitoring precipitation over large regions, and especially in remote mountainous regions over the range of elevations, and at spatial resolutions consistent with the complexity of orographic precipitation phenomena. Thermal infrared imagery at very high spatial and temporal resolutions is available nearly globally (⬃60°N–60°S) from geosynchronous satellites (http:// www.ssec.wisc.edu). Satellite platforms such as TRMM that carry microwave sensors fly at low earth orbit (LEO) to achieve good spatial resolution, but the trade-off is comparatively poor temporal sampling (satellite revisit times are on the order of several hours at least) with limited time-varying spatial coverage (each orbit spans a different geographical subregion) as discussed above. To reconcile the combined requirements of spatial coverage and high spatial and temporal sampling, much research has been directed toward exploring the correlation between rainfall and cloud cover in the formulation of satellite-based rainfall estimation algorithms (e.g., Richards and Arkin 1981; Adler and Negri 1988; Arkin and Ardanuy 1989; Ferriday and Avery 1994; Kummerow and Giglio 1995; Levizzani et al. 1996; Ferraro 1997; Miller et al. 2001; Kummerow et al. 2001; Marzano et al. 2004; Joyce et al. 2004; Hong et al. 2004, 2005; Huffman et al. 2007; among others). The goal of this work is to address the challenge posed by near-instantaneous narrow-swath observations. The specific objective is to generate spatial fields of rain-rate estimates at high spatial and temporal resolution, hereafter referred to as “extrapolated,” over large regions and without making use of calibration or statistical fitting. We aim to develop a simple and yet robust methodology that can capture the observed space–time variability of precipitation fields including diurnal cycle statistics, and produce consistent rainfall fields from one overpass to the next, from one region to another, while maintaining generality in the formulation. For this purpose, a space-filling algorithm (SFA) based on spectral estimation techniques was developed that expands the spatial domain of near-instantaneous TRMM rain-rate estimates by relying on thermal infrared imagery as support space for extrapolation in the frequency domain. This is an intermediate step in addressing the challenge that temporal sampling poses currently in the case of TRMM sensors, and it will continue for future applications such as in the advent of Global Precipitation Measurement Mission (GPM) products (http://www.gpm.gsfc.nasa.gov). The working

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hypothesis is that IR imagery provides information content of rainfall fields at the macroscale, which is eliminated by sampling along narrow swaths, and therefore it will improve the fidelity of spatial structure in satellite-based rainfall algorithms. Whereas the algorithm is demonstrated for the case of TRMM rainfall, the fundamentals of the methodology are general and therefore the approach should be useful for any other application facing similar space–time sampling challenges.

2. Concept and algorithm formulation The formulation of the SFA combining microwave and infrared observations to extrapolate the spatial extent of near-instantaneous regional swaths over limited areas is described next. In the current implementation, half-hourly Meteosat-5 thermal IR imagery and the TRMM data are merged in the frequency domain using a spectral estimation approach adapted from ideas in digital image restoration practice (e.g., Yegnanarayana et al. 1990; Diethorn and Munson 1991; Plevritis and Macovski 1995; Ferreira 1996; Strohmer 1997; Tsao 2001; Gonzalez and Woods 1992). Levizzani et al. (1996) proposed to extrapolate microwave rainfall estimates by using the former to calibrate infrared instantaneous rain-rate estimates. This idea was applied by Marzano et al. (2004), who further addressed the temporal interpolation due to differences in timing between microwave and infrared observations. Joyce et al. (2004) developed a technique that relies on motion vectors derived from successive IR images (at half-hourly intervals) to advect spatial features of passive microwave rainfall estimates. In their technique, referred to as the Climate Prediction Center (CPC) morphing method (CMORPH), the morphology of the rainfall features evolves between microwave observations using a linear-weighted interpolation scheme. An integrated end-to-end approach to the integration of passive microwave and infrared satellite data and rain gauge–based calibration was developed by Huffman et al. (2007). Whereas this work shares with previous research the basic concept of extrapolating microwave rainfall estimates using thermal infrared data, the strategy is very different as we pursue a general mathematical approach without requirements for rain-rate calibration or site-specific statistical adjustments. It is understood that postprocessing of the extrapolated fields including the applications of filters and bias correction should further reduce local errors and improve accuracy, but development of a full operational algorithm is out of the scope of this work. The SFA proposed here should be useful for existing algorithms as an early step to improve the spatial support of

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microwave estimates before they are combined (Huffman et al. 2007) or advected and morphed (Joyce et al. 2004). The SFA is demonstrated through an application in the northern Indian Subcontinent combining spatial and temporal features of Meteosat-5 thermal infrared imagery and TRMM PR 2A25, and also TMI 2A12 rainfall (see section 2 for data description). Algorithm performance is assessed via comparison among rain rates before (original TRMM products) and after extrapolation (SFA extrapolation) first, and against rain gauge observations in the central Himalayas during June–September (JJAS) over the 3-yr period 1999– 2001 using data from the hydrometeorological network first introduced by Barros et al. (2000). Clearly, given the lack of high spatial resolution rain gauge networks in the region of study, including the lack of runoff data that could be used to assess the water balance at the catchment scale, it is impossible to know what is the true rainfall in the Himalayas as a whole, and thus we must limit ourselves to the region where data are available. To address this limitation, we also present results over the southern United States that are evaluated against National Centers for Environmental Prediction (NCEP) stage IV rainfall analysis at 4-km resolution (http://www.emc.ncep.noaa.gov/mmb), and TRMM 3B42 rainfall products at 0.25° ⫻ 0.25° resolution

ˆ R IR_swath_e共k, l 兲 ⫽

1 共2N ⫺ 1兲共2N ⫺ 1兲

(http://trmm.gsfc.nasa.gov/3b42.html). Like Marzano et al. (2004), our approach is to demonstrate that the expanded fields are physically and statistically consistent with the original TRMM products, but in this case without calibration or multilayered postprocessing. The underlying premise is that, if the performance against existing observations is comparable among the extrapolated and original swath rainfall fields, the expansion is successful.

a. Methodology The formulation of the SFA is presented in terms of the PR 2A25 rain-rate estimates. The notation convention used hereafter is as follows: PR refers to the original 2A25 rainfall product; IR refers to the original infrared data; the subscript swath refers to an extracted overpass from either dataset; the subscript ext refers to the extrapolated PR rain field. Let IR(x, y) and PRext(x, y) denote the ergodic N ⫻ N random fields of IRand PRext at point (x, y), respectively. The overpass imagery of the original PR 2A25 rainfall product, namely PRswath(x, y), is a fraction of a single realization of PRext(x, y). Correspondingly, all pixels along the PR swath in the IR imagery are extracted and denoted as IRswath(x, y). The autocorrelation function estimator of IRswath(x, y) is

2N⫺1 2N⫺1

兺 兺 IR*

swath_e共x,

x⫽0

where IRswath_e(x, y) is the extended IRswath(x, y), and the symbol (*) denotes the conjugate (for a complex number z ⫽ a ⫹ jb, z* ⫽ a ⫺ jb is the conjugate of z). The extended images (denoted with subscript e) corre-

兺 兺 Rˆ

k⫽0

共1兲

spond to 2N ⫻ 2N images obtained by padding zeros around the original images. The periodogram estimator of the power spectral density is obtained by

冋

2N⫺1 2N⫺1

ˆ P IR_swath_e共u, ␷ 兲 ⫽

y兲IRKswath_e共x ⫹ k, y ⫹ l 兲 ,

y⫽0

IR_swath_e共k,

l 兲 exp ⫺j

l⫽0

册

2␲ 共uk ⫹ ␷ l 兲 . 2N

共2兲

Similarly, the cross-correlation function estimator of IRswath_e(x, y) and PRswath_e(x, y) is ˆ R IR_PR_swath_e共k, l 兲 ⫽

1 共2N ⫺ 1兲共2N ⫺ 1兲

2N⫺1 2N⫺1

兺 兺 IR*

swath_e共x,

x⫽0

y兲PRswath_e共x ⫹ k, y ⫹ l 兲 ,

共3兲

冋

共4兲

y⫽0

and the periodogram cross power spectrum estimator is 2N⫺1 2N⫺1

PˆIR_PR_swath_e共u, ␷兲 ⫽

兺 兺 Rˆ

k⫽0

l⫽0

IR_PR_swath_e共k,

l 兲 exp ⫺j

册

2␲ 共uk ⫹ ␷ l 兲 . 2N

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Finally, the spectral transfer function between IRswath_e(x, y) and PRswath_e(x, y) is estimated as follows: H 共u, ␷兲 ⫽

ˆ P IR_PR_swath_e共u, ␷ 兲 . ˆ P 共u, ␷兲

共5兲

IR_swath_e

Based on the assumptions of wide-sense stationarity and ergodicity in the IR support region, the cross power spectrum of the IRe(x, y) and the extrapolated PRext_e(x, y) can be estimated from the following relationship: ˆ Pˆ IR_PRext_e共u, ␷兲 ⫽ P IR_e共u, ␷ 兲 ⫻ H 共u, ␷ 兲,

共6兲

where Pˆ IR_e(u,␷) is the power spectral density estimator of IRe(x, y). For operational applications of the SFA, a scaling analysis should be conducted to determine the appropriate dimensions of the IR support for each location, which is the spatial scale of validity of the stationarity assumption, which should vary with the weather regime. From the 2D correlation theorem, the Fourier transform (indicated by the superscript F) of the extrapolated extended PR is given by PRFext_e共u, ␷兲 ⫽

Pˆ IR_PRext_e共u, ␷兲 . 关IRFe 共u, ␷兲兴*

共7兲

The extrapolated PRext(x, y) image is finally derived by applying the inverse Fourier transform and by eliminating and extracting the relevant (N ⫻ N ) portion from the extended matrix: 2N⫺1 2N⫺1

PRext_e共x, y兲 ⫽

兺 兺 PR

u⫽0

冋

exp j

␷⫽0

F ext_e共u,

册

2␲ 共ux ⫹ ␷y兲 . 2N

␷兲 共8兲

This methodology is deliberate in avoiding the use of parameters or constraints that do not stem rigorously from the extrapolation in the digital Fourier domain. Note that one implicit limitation in the application of the transfer function concept is the implication of linear-shift invariant transformations, which is acceptable in this case because we focus on near-instantaneous extrapolation of the rainfall fields and not on long-term integrations. It will be shown that, on the whole, it provides the required extended spatial support without deteriorating the overall statistical skill of the original estimates. The underlying premise is that if the performance against existing observations is comparable among the extrapolated and original swath rainfall fields, then the algorithm is successful.

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b. Datasets The datasets used in this work consist of half-hourly Meteosat-5 thermal IR brightness imagery at 0.1° spatial resolution obtained originally from the Space Science and Engineering Center at the University of Wisconsin—Madison (http://www.ssec.wisc.edu) and TRMM PR 2A25 and TMI 2A12 version 6 data (http:// daac.gsfc.nasa.gov/precipitation/TRMM_README) with a ground sampling distance on the order of 4 and 5 km, respectively. These fields were remapped to 0.1° ⫻ 0.1° to match the pixel resolution and geography of available IR data. Rain gauge observations from the TRMM hydrometeorological network in central Nepal were utilized to assess algorithm performance (see Fig. 1 for geographical location; Barros et al. 2000). Highquality data from nine high-elevation stations (⬎2000 m elevation) and eight low-elevation stations were available consistently during the 3-yr period of study (Fig. 1). In addition, data from the Global Energy and Water Cycle Experiment (GEWEX) Asian Monsoon Experiment (GAME)–Tibet Syangboche (Everest base camp) were also used for independent evaluation. To demonstrate the algorithm, the region for extrapolation of the PR swath was selected from the Meteosat-5 data to cover a large region with consistent continental orographic precipitation regime identified by Barros et al. (2004, 2006), and large enough to include a sufficient number of overpasses to permit adequate statistical evaluation of the results. This corresponds to a 149 ⫻ 149 IR image that covers the area of 20.1°–34.9°N, 72.5°–87.3°E. As noted previously, subsequent development of the algorithm will require more careful evaluation of the extent of the support region for spatial extrapolation including scaling behavior as a function of carefully delineated storm regimes (see, e.g., Barros et al. 2004). The TRMM 3B42 (http://trmm.gsfc.nasa.gov/3b42. html) precipitation product consists of 3-hourly (0.25° ⫻ 0.25°) rainfall estimates obtained by merging multisensor calibrated passive microwave data and infrared precipitation estimates, which are combined and subsequently rescaled to match monthly rain gauge amounts (Huffman et al. 2007). Strict comparison of rainfall estimates from the space-filling and the 3B42 algorithms is not possible even at nominal times, as there is no calibration or rescaling applied to the SFA results. The extrapolated PR fields obtained with the SFA algorithm should be viewed as a possible intermediate step in the 3B42 algorithm. We show selected examples to illustrate the potential benefit of increased spatial resolution obtained from the extrapolated PR data (active microwave). The NCEP stage IV rainfall analysis (http://www.emc.

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FIG. 1. Map of region of study including location of rain gauge network used for assessing the statistical performance of the SFA in central Nepal. Numbers indicate location of ridge stations where standard hydrometeorological towers were installed.

ncep.noaa.gov/mmb/ylin/pcpanl) consists of mosaicked gridded (4-km resolution) fields from the regional hourly/ 6-hourly multisensor (radar and gauges) precipitation analyses (MPEs) produced by the 12 River Forecast Centers (RFCs) operated by the National Weather Service over the continental United States. Quality control takes places in two phases: first locally at each RFC where the observations are collected, and second at NCEP within an hour of receiving any new hourly/6hourly data from one or more RFC.

c. Algorithm implementation and demonstration The SFA was implemented in two distinct phases as described below using as reference the PR data. The process is exactly the same for the TMI rain-rate fields. For each time step when a TRMM overpass was available within the region of study, the closest half-hourly Meteosat-5 image was obtained to delimit the working IR field. (Note that IR data would be available at higher temporal resolution in an operational context.)

Step 1: Two distinct masks of the delimited IR image are derived corresponding to cold and warm clouds, respectively: 1) the cold mask is obtained by retaining only those pixels with cloud-top temperature less than or equal to 235 K, which will be used to extrapolate convective rainfall; and 2) the warm mask is obtained by retaining only the pixels with cloud-top temperatures higher than 235 K, which will be used for extrapolating warm rain. Step 2: The original PR 2A25 swath is subsequently filtered to meet the commonsense constraint that absence of cloudiness implies absence of rain. In addition to the regular IR swath, two pseudo-IR swaths (one “cold” and one “warm” IR swath) are created by retaining only the pixels that fall within the region defined by the original PR 2A25 overpass, and cold and warm IR masks, respectively. Because of the challenges posed by light rainfall issues and low-level clouds, further thresholding of the IR image closest to the time of TRMM over-

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FIG. 2. Full-sequence demonstration of the application of the SFA algorithm at 2200 UTC 12 Jun (Julian day 163) 1999: (a) normalized Meteosat-5 IR image (pixel value divided by maximum value in the image); (b) cold cloud mask (step 1); (c) empirical probability distribution of probability of PR 2A12 rainfall occurrence as a function of cloud-top temperature (step 2); (d) empirical probability distribution of the probability of PR 2A12 rainfall as a function of cloud-top temperature (step 2); (e) IR swath extracted from cold cloud mask (step 2); and (f) normalized PR 2A25 swath (step 2).

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FIG. 2. (Continued) (g) Scatterplot of rain rates for cold pixels belonging to the PR swath before (PRoriginal) and after (PRext) application of the SFA algorithm (step 3); (h) PR swath using the cold cloud mask as extrapolation support (step 3); (i) IR swath extracted from the warm cloud mask (step 2); (j) scatterplot of rain rates for warm pixels belonging to the PR swath before (PRoriginal) and after (PRext) application of the SFA algorithm (step 3); and (k) PR swath using the warm cloud mask as extrapolation support (step 3).

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FIG. 3. (a) Final synthesis of extrapolated PR instantaneous rainfall estimates on 2200 UTC 12 Jun (Julian day 163) 1999 obtained by combining the cold and warm cloud rainfall fields extrapolated using the SFA (step 4); (b) TRMM 3B42 rainfall product nominally at 2100 UTC 12 Jun 1999; (c) TRMM 3B42 rainfall product nominally at 0000 UTC 13 Jun 1999; and (d) scaling behavior of the fields shown in (a), (b), and (c) expressed in terms of variance vis-à-vis area in log–log scale.

pass in the region may be necessary in operational applications. This is done here using a simple adaptive algorithm on a case-by-case basis by searching the PR 2A25 swath and examining the joint probability distribution of IR cloud-top temperatures and PR rainfall estimates. The temperature value corresponding to the peak of the joint probability distribution is selected as the warm cloud threshold. Typically, a maximum cloud-top temperature of 255 K was adopted, although this value was much higher in 2001 (up to 270 K), when largescale weather systems were conspicuously absent, and shallower cloud systems prevailed (e.g., Lang and Barros 2002; Barros and Lang 2003; Magagi and Barros 2004). The importance of the degree of correlation between shallow (stratiform) clouds

with or without embedded convection and rainfall rates will be addressed when we discuss the performance of the extrapolated fields. Step 3: Calculate the transfer function according to Eq. (5) for each cold and warm rain conditions and apply each separately to the entire original IR support image (149 ⫻ 149). Warm and cold rain field extrapolated fields are derived by performing an inverse Fourier transform. Step 4: Finally, the cold and warm extrapolated fields are combined and the synthesis large-area extrapolated rainfall field is obtained. Step-by-step examples of the application of the algorithm are presented in detail for the onset of the monsoon at 2200 UTC 12 June 1999 (Figs. 2 and 3).

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Results from the expanded PR images for two other randomly selected onset overpasses in 2000 (Fig. 4) and 2001 (Fig. 5) are also presented to illustrate the SFA for a diversity of scenarios. The original IR brightness image at 2200 UTC 12 June 1999 and its corresponding cold mask are shown in Figs. 2a and 2b, respectively. Figures 2c and 2d display, respectively, the statistical distribution of rainfall occurrence versus cloud-top temperatures (Fig. 2c) and the rainfall probability distribution versus cloud-top temperature along the PR swath (Fig. 2d). As it can be seen from Fig. 2c, the top temperature of rainy clouds ranges from 187 to 300 K, and the peak of the distribution of rainfall occurrence is around 255 K. Although it seems that there are more rainfall occurrences in the warm cloud region ( ⱖ 235 K), the cold cloud region has much higher rainfall probabilities (Fig. 2d). Moreover, for cloud-top temperatures above 255 K, both rainfall and rainfall probability distributions decrease jointly, which supports the selection of the 255-K threshold for the warm cloud mask (235 K ⱕ IR ⱕ 255 K). The IR–PR swath pairs for the reconstruction based on the cold cloud support region (cold cloud mask) are shown in Figs. 2e,f. The region of the reconstructed PR field that corresponds to Fig. 2f geographically is shown in Fig. 2h, and the scatterplot in Fig. 2g allows an evaluation of the extrapolated fields against the original (remapped) PR 2A25 estimates for collocated pixels. Similarly, the reconstruction of the warm rain fields is illustrated by Figs. 2i–k. Note the difference in rainfall intensities for the cold and warm cloud cases in Figs. 2g,j. The final synthesized large-area image that results from merging the cold and warm rain fields is shown in Fig. 3a. Two examples of the 3B42 3-hourly multisensor rainfall estimates at 2100 UTC 12 June and 0000 UTC 13 June are shown also in Figs. 3b and 3c, respectively, to facilitate visual comparison with the SFA synthesis field at an intermediate time. The most important and noticeable difference is the spatial extent and granularity of the rainfall fields including the squall lines over the Tibetan Plateau, larger features with substantially higher rainfall rates and higher variance over a wide range of scales. This is captured by the scaling analysis of variance versus area in the Fig. 3d, which shows that for spatial scales up to 300 km (9 ⫻ 104 km2), the extrapolated PR rainfall estimates exhibit linear scaling behavior in contrast with the 3B42 estimates that show much smaller variance and a scaling break at 100 km (⬃104 km2). (Note that for areas larger than 105 km2 the size of the domain begins to affect the calculations.) Selected steps in the application of the SFA to a second overpass in 2000 are shown in Figs. 4a–g. Visual inspection and intercomparison of Figs. 2g,j and 4d,e

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show that the reconstructed rain-rate images are similar to their original PR images over the common geographical region spanned by the TRMM swath as expected, with a better match in the 2000 case when cloud cover is limited. This is in contrast to the 1999 case that coincides with a monsoon onset system. The number of swath pixels with light rainfall is larger in the extrapolated field than in the original PR 2A25 (wet bias). This should be expected in part because the demonstration relies strictly on standard digital image restoration techniques, and because there is no optimization of the threshold used to define the warm cloud mask. Despite the wet bias for light rainfall, the agreement is very close for higher rain rates (5 mm h⫺1 and above), and in fact it improves with increasing rainfall rate, which suggests that the method may be particularly useful for capturing the full spatial manifestation of extreme events. Whereas developing a full operational algorithm is not part of the goal of this work, elimination of the wet bias for light rainfall via use of a postprocessing filtering algorithm should be straightforward using simple thresholding, adaptive rescaling by calibration against observations, and morphing approaches, among others.

3. Algorithm evaluation a. Comparison with rain gauge observations in the Himalayas The challenges posed by rain-rate estimation from satellite-based sensors over land generally, and in mountainous regions such as the Himalayas in particular are well known. One-on-one, point-to-pixel (rain gauge-to-satellite) comparisons are afflicted by measurement uncertainty, spatial resolution discrepancies, and validity of the approximations adopted in the retrieval algorithms. To avoid the high variability associated with short temporal scales, evaluation of remote sensing rainfall products is generally conducted at monthly to annual temporal scales. However, because of the nature and objectives of the proposed spacefilling algorithm, it is important that we focus on nearinstantaneous fields in this manuscript (effectively, the duration of the overpass). For the purpose of assessing the algorithm, we will rely on a suite of statistical measures traditionally used in evaluation of weather forecasts (see also Barros et al. 2000). Given a certain rainfall threshold, and using a two-way contingency table of satellite-based rainfall estimates versus rain gauge records, the data are organized into four classes: an instance when both satellite product and rain gauge observations match or exceed a specified rain-rate threshold is a hit (H ); when the opposite happens, the

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FIG. 4. Partial sequence of the application of the SFA algorithm at 1900 UTC 15 Jun (Julian day 165) 2000: (a) normalized Meteosat-5 IR image (pixel value divided by maximum value in the image); (b) empirical probability distribution of probability of PR 2A12 rainfall occurrence as a function of cloud-top temperature (step 2); and (c) empirical probability distribution of the probability of PR 2A12 rainfall as a function of cloud-top temperature (step 2).

estimate is classified as a miss (M ) if the rain gauge observation matches the criterion, and a false alarm (FA) otherwise. The statistical measures are obtained by combination of these classes as follows: false alarm rate [FR ⫽ FA/(H ⫹ FA), 0 ⱕ FR ⱕ1], probability of detection [PD ⫽ H/(H ⫹ M), 0 ⱕ PD ⱕ 1], threat score [TS ⫽ H/(H ⫹ FA ⫹ M), 0 ⱕ TS ⱕ 1], and the Heidke skill score [HSS ⫽ |2 ⫻ (Z ⫻ H ⫺ FA ⫻ M)/{[(H ⫹ FA) ⫻ (Z ⫹ FA)] ⫹ [(M ⫹ H) ⫻ (M ⫹ Z)]}|, ⫺1 ⱕ HSS ⱕ 1], where Z is the overall number of zeros (when neither product nor observations match the specified threshold criterion). In addition, rain-rate bias is also calculated. A Heidke skill score of zero implies no better skill than random chance, whereas a TS of 0.33 indicates that the

assessment criterion is met at least 50% of the time. These statistical measures should be viewed as indicators of relative performance and not absolute quantitative descriptions. For further details and insights on the use of these statistical measures, please see the see the postings of the National Precipitation Verification Unit (NPVU; http://www.hpc.ncep.noaa.gov/npvu). For the actual determination of the statistical scores, the strategy consists of first finding the nearest neighbor to each rain gauge location in the extrapolated rainrate field and in the remapped original PR or TMI data. When both the reconstructed and original fields have an estimate for the same gauge, than that gauge is included in the evaluation. Therefore, depending on the

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FIG. 4. (Continued) (d) Scatterplot of rain rates for cold pixels belonging to the PR swath before (PRoriginal) and after (PRext) application of the SFA algorithm (step 3); (e) scatterplot of rain rates for warm pixels belonging to the PR swath before (PRoriginal) and after (PRext) application of the SFA algorithm (step 3); and (f) final synthesis of extrapolated PR instantaneous rainfall estimates at 1900 UTC 15 Jun 15 (Julian day 165) 2000 obtained by combining the cold and warm cloud rainfall fields extrapolated by the SFA algorithm (step 4).

geometry of the overpass, the number of gauges as well as which gauges are used for the calculation will vary from time to time. Two assessment criteria were used: 1) rain/no rain, and 2) rainfall intensity ⬎0.5 mm h⫺1. The total number of samples that match either criterion 1 or 2 at the high-elevation gauge locations is 345; at low elevations the number is 304. The goal is to deter-

mine whether the space-filling algorithm generally replicates the statistical performance of the original PR and TMI products, and therefore can be used without fear of deteriorating existing operational algorithms. Although the number of samples is adequate (standard error ⬀ 1/公N, where N is the total number of samples) and this number is typical of the challenges faced with

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FIG. 5. Partial sequence of the application of the SFA algorithm at 2200 UTC 12 Jun (Julian day 163) 2001: (a) normalized Meteosat-5 IR image (pixel value divided by maximum value in the image); (b) scatterplot of rain rates for cold and warm pixels (synthesis) belonging to the PR swath before (PRoriginal) and after (PRext) application of the SFA algorithm (step 4); and (c) final synthesis of extrapolated PR instantaneous rainfall estimates at 2200 UTC 12 Jun (Julian day 165) 2001 obtained by combining the cold and warm cloud rainfall fields extrapolated using the SFA algorithm (step 4).

evaluation of satellite data in remote reways preferable for statistical analysis. Independently of the approach, one common challenge in satellite rainfall estimation is that robust statistical performance as measured by high probability of detection (PD), low false alarm ratios (FR), and high skill scores (TS and HSS) is only achieved at coarse spatial and temporal resolutions. This is well illustrated by comparing for example monthly FR (⬃0.1) at 1° ⫻ 1° resolution from Turk et al. (2003) with FR (⬃0.5–

0.7) at 0.1° ⫻ 0.1° in Kidd et al. (2003). Here, the evaluation of algorithm performance was conducted from three different perspectives. First, all available reconstructed images over the 3-yr period of 1999 to 2001 are utilized for evaluation against rain gauge observations and original PR datasets. To eliminate the need for long tables, the results are presented in the form of scatter X versus Y relative skill diagrams (RSD) as shown conceptually in Fig. 6. The statistical indices are represented by numbers to reduce diagram clutter as

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FIG. 6. RSD to summarize visually the analysis of the relative statistical performance or two X and Y fields: if X exhibits an improvement on FR and Bias, the numbers 1 (FR) and 5 [Bias, (mm h⫺1)] will be above the diagonal line that indicates no change; if X exhibits an improvement on PD, TS, and/or HSS, the numbers 2, 3, and 4 will be below the red diagonal line. The objective of this plot is to facilitate visual intercomparison of the statistical indices as opposed to using long alphanumeric tables.

follows: 1 ⫽ FR, 2 ⫽ PD, 3 ⫽ TS, 4 ⫽ HSS, and 5 ⫽ Bias. If the statistical measures of the extrapolated (X ) and original fields (Y ) are equivalent, then the numbers are very closely aligned with the 1:1 red diagonal. If there is an improvement in FR or Bias, the numbers 1 and 5 fall above the “no change” diagonal line. It follows that an improvement in PD, TS, and/or HSS will be reflected by having the numbers 2, 3, and 4 below the “no change” diagonal line. Global results are summarized in Fig. 7a for the three years, at high elevations and for the two different rainfall-rate thresholds. Figure 7b summarizes the SFA per-

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formance for each year separately for the high elevations, and for rain rates exceeding 0.5 mm h⫺1. A careful survey shows that generally there is an improvement of PD, TS, and HSS for the extrapolated fields, but the opposite is true for FR and Bias, and the latter increases significantly for light rainfall as discussed earlier (⬍2–3 mm h⫺1; Fig. 7a). The more detailed year-byyear analysis shows that except for Bias (Fig. 7b), the extrapolated fields show consistently improved results for all other scores in 1999 and 2000, whereas there is a significant decrease of skill for 2001. This is relevant because of the very different characteristics of monsoon rainfall regimes in each of the three years (e.g., Barros and Lang 2003; Lang and Barros 2002). The 2001 monsoon season in the central Himalaya was weaker than that in the two previous years, with fewer large-scale monsoon systems reaching the region and overall weaker convective activity (e.g., Barros et al. 2004), which may explain at least in part the performance of the algorithm, and the difficulty of estimating warm rainfall from shallow clouds and embedded convection (Lang and Barros 2002). Barros et al. (2004) and Barros and Lang (2003) point out unique features of the diurnal cycle of rainfall in central Nepal, including a nocturnal peak between 0000 and 0300 local standard time (LST) that accounts for about one-third of all monsoon rainfall, and a late afternoon–early evening peak associated with afternoon convection. To assess whether the performance was robust across the diurnal cycle and especially during the periods most relevant for hydrological applications (that is when rainfall intensity is higher), the data were classified into five distinct classes as a function of time of day (class 1 ⫽ 0000–0500 LST; class 2 ⫽ 0500–

FIG. 7. RSD: (a) 3-yr summary (1999–2001) for all rain gauges and two distinct rain-rate thresholds (TH), and (b) year-to-year summary (1999–2001) for all high-elevation rain gauges for rain rates above 0.5 mm h⫺1.

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FIG. 8. RSD for rain rates above 0.5 mm h⫺1: (a) diurnal cycle for high-elevation stations, and (b) diurnal cycle for low-elevation stations.

0900 LST; class 3 ⫽ 0900–1200 LST; class 4 ⫽ 1200– 1700 LST; class 5 ⫽ 1700–2400 LST), and the performance indices were calculated separately for each class. Finally, the data were examined with regard to the diurnal cycle for the three periods of the day when rainfall more frequently occurs (classes 1, 4, and 5), thus there were never fewer than 70 samples at any time (Figs. 8a,b). Note the superior scores of the SFA extrapolation for afternoon rainfall, in particular on the mountain ridges (high-elevation locations). The PR 2A25 and the extrapolated PR rainfall estimates at (0.1° ⫻ 0.1°) compare favorably with the TMI 2A12 except for Bias and FR as shown in Fig. 9. An application of the SFA to the TMI2A12 data proper leads to a somewhat ambiguous outcome in that the extrapolated PR fields consistently outperform the extrapolated TMI fields except for the HSS, and yet the original TMI over the region of interest has better probability of detection and threat scores that any of the other rainfall estimates (Fig. 9). Local comparisons during the 1999 intense observation period (IOP) at the Everest base camp exhibit significant gains in PD (0.4 versus 0.1) and TS (0.27 versus 0.11); less so in HSS (0.19 versus 0.15). However, the false alarm rate FR is very high (0.6 versus 0.1). These results suggest that some elevation-based corrections will be necessary to eliminate the wet bias for light (stratiform) rainfall. In an operational implementation of this algorithm, this effect must be balanced against systematic underestimation of orographic rainfall by TRMM rainfall products even during significant monsoon storms in the region as shown by Barros et al. (2006) using PR reflectivity fields. The objective of the space-filling algorithm is to ex-

pand individual PR overpasses to encompass a large region. Therefore, in this context, the basic performance requirements are that the extrapolated fields exhibit similar monitoring skill statistics as the original PR overpass fields. A survey of the results shows that, compared with the original PR products, the reconstructed PR rainfall fields have two features: 1) the performance of the extrapolated PR rain rates in 1999 is the best among these three years and slightly better than the original PR, but the performance in 2001 is not satisfactory; and 2) on the whole, extrapolated PR fields show a better performance at high-altitude stations than at low-altitude stations. The former illustrates the

FIG. 9. RSD for different fields as compared to the original TMI 2A12 swath for high-elevation stations in 1999 with rain-rate threshold of 0.5 mm h⫺1.

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influence of storm regime, the latter the influence of landform controls on orographic precipitation. This can be explained in part by the space–time resolution at which the extrapolation was performed (modulation by narrow valleys and small terrain features cannot be captured) and could be addressed by using IR imagery at higher spatial resolution and timing closer to the overpass time. Regarding the contrasts between the 1999 and 2001 performance, we note that for most 1999 IR– PR swath pairs the cold cloud region has a much higher rainfall probability than the warm cloud region. In addition, most rainfall occurrences take place within or near the cold cloud region and decrease very quickly above cloud-top temperatures around 255 K. Thus, we can expect a good balance among the performance indices of false alarm, hits and misses by setting the threshold of temperature at 255 K for the warm cloud– based extrapolation cannot be expected. On the contrary, most rainfall distribution curves and rainfall probability curves of the available IR–PR overpass pairs in 2001 show a large difference in statistical characteristics. In 2001, quite a few IR–PR pairs have higher rainfall probability in the warm cloud region rather than in the cold region, and the peak probability is often less than 0.5, which implies that a very high false alarm rate usually accompanies a high hit rate. For operational applications, this problem should best be handled by postprocessing, but this is out of the scope of this manuscript.

b. Comparison against NCEP stage IV radar and gauge analysis To evaluate the SFA against spatial fields of rainfall observations, the algorithm was applied over the southern United States where the NCEP radar and rain gauge analysis has good coverage. An example is provided in Figs. 10a–e at 1200 UTC 31 August 2003 over Texas and the Gulf of Mexico. A comparison of the TRMM PR overpass (Fig. 10b) against the NCEP observations (Fig. 10c) shows that the two overlap, and one immediate striking observation is that the difference between the NCEP and the PR 2A25 estimates over the ocean, with the former failing to capture two significant rainfall features, likely because of poor radar coverage, especially for shallow precipitation systems (see Fig. 5 in Maddox et al. 2002). These features will remain an important point of reference throughout the remainder of this case study. Over land, a survey of Fig. 10c (NCEP), Fig. 10d (TRMM 3B42), and Fig. 10e (synthesis PRext) show that there is substantial agreement for the major precipitation features, especially over Morgan City, Louisiana, with higher dispersion of light rainfall in the case

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of the PRext. Note the overall good agreement between the shape and spatial extent of the TRMM 3B42 and the PRext, whereas both underestimate rainfall rates by as much as 100% to the east of Morgan City on the east quadrant of the storm system. Over the Gulf of Mexico, the TRMM 3B42 also underestimates the maximum rain rate (recall there is a significant difference in original spatial resolution), but the PRext preserves the information originally contained in the PR 2A25 estimates. This behavior is well described by the scatterplots shown in Figs. 11a–c. The overall correlation between NCEP and TRMM 3B42 (Fig. 11a) is higher than between NCEP and PRext (Fig. 11b), though the difference is small given the failure to capture the oceanic rainfall in the NCEP analysis. The scatterplot between TRMM 3B42 and PRext further confirms the effect of the oceanic rainfall grid points in the correlation between the two fields. Again, these results suggest that is value in incorporating the SFA to improve the spatial resolution and contrast of microwave estimates. The scatterplots in Fig. 11 are discouraging in that they illustrate weaknesses in all datasets including the NCEP analysis. We must point out that the results are better at coarser resolution (e.g., the correlation improves by 30% for TRMM 3B42 when used at its nominal resolution of 0.25°) and that results are also better at 3-hourly, 6-hourly, or longer time scales in contrast with the near-instantaneous fields we are using here (see, e.g., Kidd et al. 2003; Joyce et al. 2004; Huffman et al. 2007; among others). Thus, the research question is how to improve the accuracy and fidelity of spatial structure at short time scales (ⱕ 1 h) and fine spatial resolutions, that is, for hydrometeorological applications and operational weather forecasting rather than climate research or applications. Figure 12 shows the spatial correlation structure of NCEP (Fig. 12a), TRMM 3B42 (Fig. 12b), and PRext (Fig. 12c). Similar to our previous analysis focusing on spatial scaling behavior (Fig. 3), the correlograms can be used to learn about the spatial structure of the rainfall fields. One critical difference relates to the spikiness of the NCEP and Prext correlograms, reflecting the patchy nature of real rainfall, which is captured in part by the SFA in the PRext. One characteristic both PRext and TRMM 3B42 exhibit is a significant rotation (⬃45° for TRMM 3B42 and ⬃30°) with respect to the major axis of the spatial correlation function with respect to that of NCEP (taken as 0°). This realignment of overall spatial variability in the microwave estimates generally (similar results can be found for TMI, not shown) as significant implications for the application of downscaling algorithms to remote sensing products, especially those that rely on preserving fundamental statistical

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FIG. 10. Partial sequence of the application of the SFA algorithm at 1200 UTC 31 Aug 2003: (a) Geostationary Operational Environmental Satellite-10 (GOES-10) IR brightness temperature field; (b) PR swath and PR 2A25 rainfall-rate estimates; (c) NCEP/Environmental Modeling Center (EMC) stage IV gridded hourly rainfall; (d) downscaled (linear interpolation) TRMM 3B42 rain rates; and (e) final synthesis of extrapolated PR instantaneous rainfall estimates.

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FIG. 11. Scatterplots of rain rates: (a) NCEP vs synthesis PRext, (b) NCEP vs downscaled TRMM 3B42, and (c) Prext vs TRMM 3B42.

and scaling behavior such fractal interpolation methods, for example (Bindlish and Barros 2000). We argue this is the result of narrow swaths (e.g., field of view) of satellite-based sensors that can excise significant portions of storm systems and therefore effectively set up a ceiling to the spatial fidelity of rainfall products. Coarser-resolution products alleviate this effect in part by averaging but lose discrimination skill. The SFA appears to perform better at preserving spatial structure, but it is still inadequate. Further research is necessary not only to expand the spatial support of satellite observations but also to emulate spatial structure.

4. Discussion It is not the goal of this work to replace or even suggest an alternative to such complex end-to-end algorithms as those developed by Huffman et al. (2007), Hong et al. 2005, Joyce et al. (2004), or Marzano et al. (2004). Rather, the objective is to propose an intermediate step that can lighten the burden of calibration and heuristics in the spatial combination of multisensor, multifrequency satellite data in existing frameworks, and thus contribute to improving their accuracy and statistical robustness at short time scales and high spatial resolution.

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FIG. 12. Spatial correlogram structure: (a) NCEP, (b) downscaled TRMM 3B42, and (c) Prext.

The extrapolated PR and TMI rainfall fields derived from this algorithm generally preserve the skill of the original data and have better detection probability in complex terrain. The results indicate that probability of detection and threat scores of the reconstructed products are significantly better than the original PR data at high-elevation stations (⬎2000 m) on mountain ridges, and especially under intense rainfall conditions. For low-elevation stations located in steep narrow valleys, the performance varies from year to year and deteriorates strongly in the early morning and morning hours for light rainfall (false alarm rates significantly increase). This can be explained in part by the coarser resolution of the expanded PR imagery that matches the resolution of the available IR imagery. Application

of the SFA to the TMI2A12 rainfall suggests that the integration or combination of extrapolated PR 2A25 (active microwave and infrared) and original TMI2A12 estimates (passive microwave) might produce a superior multifrequency instantaneous rainfall estimates from satellite observations and existing algorithms. A comparison with other satellite products (e.g., 3B42, a TRMM-adjusted merged infrared-based rainfall product) suggests that there should be significant potential benefits in terms of spatial resolution from integrating this algorithm in the currently existing operational multisensor algorithms combining IR and microwave satellite observations. Whereas more extensive evaluation of the algorithm and further refinement for implementation with operational algorithms is required, these

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first results indicate that this algorithm may be generally useful for deriving precipitation products, especially in mountainous regions, which present some of the greatest challenges in precipitation retrieval from satellites over land. Acknowledgments. This work was supported by NASA Grant NNG04GP02G with the first author through the Precipitation Measurement Missions program. The authors are grateful to Prabhakar Shrestha at Duke University and Eddie K. Harm at Harvard University for their help with data and algorithm testing. Two thoughtful reviewers provided very helpful comments and suggestions. The research algorithm is available by contacting the first author.

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