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NASA NTRS2000

Improving Global Analysis and Short-Range Forecast Using Rainfall and Moisture Observations Derived from TRMM and SSM/I Passive Microwave Instruments

The Global Precipitation Mission, a satellite project under consideration as a follow-on to the Tropical Rainfall Measuring Mission (TRMM) by the National Aeronautics and Space Agency (NASA) in the United States, the National Space Development Agency (NASDA) in Japan, and other international partners, comprises an improved TRMM-like satellite and a constellation of 8 satellites carrying passive microwave radiometers to provide global rainfall measurements at 3-hour intervals. The success of this concept relies on the merits of rainfall estimates derived from passive microwave radiometers. This article offers a proof-of-concept demonstration of the benefits of using, rainfall and total precipitable water (TPW) information derived from such instruments in global data assimilation with observations from the TRMM Microwave Imager (TMI) and 2 Special Sensor Microwave/Imager (SSM/I) instruments. Global analyses that optimally combine observations from diverse sources with physical models of atmospheric and land processes can provide a comprehensive description of the climate systems. Currently, such data analyses contain significant errors in primary hydrological fields such as precipitation and evaporation, especially in the tropics. We show that assimilating the 6-h averaged TMI and SSM/I surface rainrate and TPW retrievals improves not only the hydrological cycle but also key climate parameters such as clouds, radiation, and the upper tropospheric moisture in the analysis produced by the Goddard Earth Observing System (GEOS) Data Assimilation System, as verified against radiation measurements by the Clouds and the Earth's Radiant Energy System (CERES) instrument and brightness temperature observations by the TIROS Operational Vertical Sounder (TOVS) instruments. Typically, rainfall assimilation improves clouds and radiation in areas of active convection, as well as the latent heating and large-scale motions in the tropics, while TPW assimilation leads to reduced moisture biases and improved radiative fluxes in clear-sky regions. Ensemble forecasts initialized with analyses that incorporate TMI and SSM/I rainfall and TPW data also yield better short-range predictions of geopotential heights, winds, and precipitation in the tropics. This study offers a compelling illustration of the potential of using rainfall and TPW information derived from passive microwave instruments to significantly improve the quality of 4-dimensional global datasets for climate analysis and weather forecasting applications.

Hou, Arthur Y.↗
NASA NTRS2001

On Rainfall Modification by Major Urban Areas: Observations from Space-borne Rain Radar on TRMM - Part 1

This study represents one of the first published attempts to identify rainfall modification by urban areas using satellite-based rainfall measurements. Data from the first space-based rain-radar, the Tropical Rainfall Measuring Mission's (TRMM) Precipitation Radar, are employed. Analysis of the data enables identification of rainfall patterns around Atlanta, Montgomery, Nashville, San Antonio, Waco, and Dallas during the warm season. Results reveal an average increase of -28% in monthly rainfall rates within 30-60 kilometers downwind of the metropolis with a modest increase of 5.6% over the metropolis. Portions of the downwind area exhibit increases as high as 51%. The percentage chances are relative to an upwind CONTROL area. It was also found that maximum rainfall rates in the downwind impact area can exceed the mean value in the upwind CONTROL area by 48%-116%. The maximum value was generally found at an average distance of 39 km from the edge of the urban center or 64 km from the center of the city. These results are consistent with METROMEX studies of St. Louis almost two decades ago and more recent studies near Atlanta. Future work will investi(yate hypothesized factors causing rainfall modification by urban areas. Additional work is also needed to provide more robust validation of space-based rain estimates near major urban areas. Such research has implications for urban planning, water resource management, and understanding human impact on the environment.

Shepherd, J. Marshall↗
NASA NTRS2001

A TRMM-Calibrated Infrared Technique for Convective and Stratiform Rainfall: Analysis and Validation

A satellite infrared technique with passive microwave calibration has been developed for estimating convective and stratiform. rainfall. The Convective-Stratiform Technique, calibrated by coincident, physically retrieved rain rates from the TRMM Microwave Imager (TMI), has been applied to 30 min interval GOES infrared data and aggregated over seasonal and yearly periods over northern South America. The diurnal cycle of rainfall, as well as the division between convective and stratiform rainfall is presented. For the period Jan-April 1999, analysis revealed significant effects of local circulations (river breeze, land/sea breeze, mountain/valley) on both the total rainfall and it's diurnal cycle. Results compared well (a one-hour lag) with the diurnal cycle derived from TOGA radar-estimated rainfall in Rondonia. The satellite estimates revealed that the convective rain constituted 24% of the rain area while accounting for 67% of the rain volume. Estimates of the diurnal cycle (both total rainfall and convective/stratiform) for an area encompassing the Amazon Basin (3 x 10(exp 6) square km) were in phase with those from the TRMM Precipitation Radar, despite the latter's limited sampling. Results will be presented comparing the yearly (2000) diurnal cycle for large regions (including the Amazon Basin), and an intercomparison of January-March estimates for three years, 1999-2001. We hope to demonstrate the utility of using the TRMM PR observations as verification for infrared estimates of the diurnal cycle, and as verification of the apportionment of rainfall into convective and stratiform components.

Negri, Andrew↗
NASA NTRS2001

Validation of TRMM Satellite Rainfall Products Over Oklahoma for a Three Year Period (1998-2000)

This study evaluates the level 2 and level 3 rainfall products from the Tropical Rainfall Measuring Mission gridded distribution of 118 high resolution tipping buckets from the Oklahoma Mesonet. The Tropical Rainfall Measuring Mission, a joint satellite mission between NASA and the National Space Development Agency (NASDA) of Japan, was designed to estimate global precipitation between 40 S and 40 N latitude. The TRMM satellite, consisting of two main precipitation sensors, a passive microwave (TMI) and precipitation radar (PR) sensors, was launched in 1997. Although the great advantage of TRMM has been its ability to sample precipitation globally over the tropical oceans in places where ground sensors do not exist, regional points over land still offer the best opportunity for validating these rain estimates. For this study, the gauge data was gridded at I x I degree resolution between 34 and 36 N and -95 and -100 W. The location of Oklahoma is somewhat unique, in that, it is located near the turning point for the satellite. This study investigates effects of temporal sampling on the satellite measurements and the resulting rainfall bias observed between space and ground based sensors. The first part of the study analyzes the monthly rainfall statistics of the 3AI2 (TMI), the 3A25 (PR) and the 31142 (combined sensors) with the rain gauges. It uses regression techniques and a computation of the probability density function (PDF) and cumulative probability density function (CDF) for each sensor to evaluate the effects of temporal sampling on the satellite products and the sufficiency of using two overpasses a day to estimate rainfall on a monthly scale. In further probing this issue, the study also looked at the direct coincidence of ground and satellite measurements by comparing instantaneous comparisons obtained from the level 2 orbital track data, e.g. 2A12, 2A25. Lastly, the effects of temporal sampling were further studied by sub-sampling the gauge data only at overpass times. This methodology provides a quantitative way of separately inferring the fraction of bias due to temporal sampling and the rainfall algorithm. In addition to showing the salient effects of temporal sampling, the study also revisits the historical problem of correlating point gauge estimates with area estimates, though, the problem is ameliorated by averaging and gridding the data to the area resolution scale of the satellite.

Fisher, Brad↗
NASA NTRS2001

A TRMM-Calibrated Infrared Technique for Convective and Stratiform Rainfall: Analysis and Validation

A satellite infrared technique with passive microwave calibration has been developed for estimating convective and stratiform rainfall. The Convective-Stratiform Technique, calibrated by coincident, physically retrieved rain rates from the TRMM Microwave Imager (TMI), has been applied to 30 min interval GOES infrared data and aggregated over seasonal and yearly periods over northern South America. The diurnal cycle of rainfall, as well as the division between convective and stratiform rainfall is presented. For the period Jan-April 1999, analysis revealed significant effects of local circulations (river breeze, land/sea breeze, mountain/valley) on both the total rainfall and it's diurnal cycle. Results compared well (a one-hour lag) with the diurnal cycle derived from TOGA radar-estimated rainfall in Rondonia. The satellite estimates revealed that the convective rain constituted 24% of the rain area while accounting for 67% of the rain volume. Estimates of the diurnal cycle (both total rainfall and convective/stratiform) for an area encompassing the Amazon Basin (3 x 10(exp 6) sq km) were in phase with those from the TRMM Precipitation Radar, despite the latter's limited sampling. Results will be presented comparing the yearly (2000) diurnal cycle for large regions (including the Amazon Basin), and an intercomparison of January-March estimates for three years, (1999-2001). We hope to demonstrate the utility of using the TRMM PR observations as verification for infrared estimates of the diurnal cycle, and as verification of the apportionment of rainfall into convective and stratiform components.

Negri, Andrew↗
NASA NTRS2002

A TRMM-Calibrated Infrared Technique for Global Rainfall Estimation

The development of a satellite infrared (IR) technique for estimating convective and stratiform rainfall and its application in studying the diurnal variability of rainfall on a global scale is presented. The Convective-Stratiform Technique (CST), calibrated by coincident, physically retrieved rain rates from the Tropical Rainfall Measuring Mission (TRMM) Precipitation Radar (PR), is applied over the global tropics during 2001. The technique is calibrated separately over land and ocean, making ingenious use of the IR data from the TRMM Visible/Infrared Scanner (VIRS) before application to global geosynchronous satellite data. The low sampling rate of TRMM PR imposes limitations on calibrating IR-based techniques; however, our research shows that PR observations can be applied to improve IR-based techniques significantly by selecting adequate calibration areas and calibration length. The diurnal cycle of rainfall, as well as the division between convective and stratiform rainfall will be presented. The technique is validated using available data sets and compared to other global rainfall products such as Global Precipitation Climatology Project (GPCP) IR product, calibrated with TRMM Microwave Imager (TMI) data. The calibrated CST technique has the advantages of high spatial resolution (4 km), filtering of non-raining cirrus clouds, and the stratification of the rainfall into its convective and stratiform components, the latter being important for the calculation of vertical profiles of latent heating.

Negri, Andrew J.↗
NASA NTRS2002

The Effects of Amazon Deforestation on Rainfall

This study begins with the hypothesis that heavily deforested regions will experience increased surface heating, leading to local circulations that will ultimately enhance the rainfall, or at least, change the pattern of diurnal evolution of rainfall. This would be an important finding because several modeling studies have concluded that widespread deforestation would lead to decreased rainfall. Towards that end rain estimates from a combined GOES infrared/TRMM microwave technique were analyzed with respect to percent forest cover from Landsat data (courtesy of TRFIC at Michigan State University) and GOES visible channel data over a deforested area in Rondonia (southwest Brazil). Five 1" x 1" areas of varying forest cover were examined during the onset of the wet season in Amazonia (Aug-Sept), when the effects of the surface would not be dominated by large-scale synoptic weather patterns. Preliminary results revealed that: maximum rainfall fell in most deforested area; heavily forested areas received the least rainfall; cumulus cloud development initiated at borders; the amplitude of the diurnal cycle of precipitation was a function of th surface cover. Further work will be presented detailing effects of land surface cover on the GOES infrared-measured surface heating, GOES visible observed cumulus development, thunderstorm initiation based on the location of temperature minima in the infrared data, and estimated rainfall and its diurnal cycle from a combined GOES/TRMM technique. Rainfall estimates derived from non-geosynchronous microwave observations (i.e. Goddard Profiling Algorithm, GPROF) will also be examined.

Starr, David OC.↗
NASA NTRS2003

A TRMM-Calibrated Infrared Technique for Global Rainfall Estimation

This paper presents the development of a satellite infrared (IR) technique for estimating convective and stratiform rainfall and its application in studying the diurnal variability of rainfall on a global scale. The Convective-Stratiform Technique (CST), calibrated by coincident, physically retrieved rain rates from the Tropical Rainfall Measuring Mission (TRMM) Precipitation Radar (PR), is applied over the global tropics during summer 2001. The technique is calibrated separately over land and ocean, making ingenious use of the IR data from the TRMM Visible/Infrared Scanner (VIRS) before application to global geosynchronous satellite data. The low sampling rate of TRMM PR imposes limitations on calibrating IR- based techniques; however, our research shows that PR observations can be applied to improve IR-based techniques significantly by selecting adequate calibration areas and calibration length. The diurnal cycle of rainfall, as well as the division between convective and ~ t i f m rainfall will be presented. The technique is validated using available data sets and compared to other global rainfall products such as Global Precipitation Climatology Project (GPCP) IR product, calibrated with TRMM Microwave Imager (TMI) data. The calibrated CST technique has the advantages of high spatial resolution (4 km), filtering of non-raining cirrus clouds, and the stratification of the rainfall into its convective and stratiform components, the latter being important for the calculation of vertical profiles of latent heating.

Negri, Andrew J.↗
NASA NTRS2004

Characteristics of Heavy Summer Rainfall in Southwestern Taiwan in Relation to Orographic Effects

On the windward side of southwestern Taiwan, about a quarter to a half of all rainfall during mid-July through August from 1994 to 2000 came from convective systems embedded in the southwesterly monsoon flow. k this study, the causes of two heavy rainfall events (daily rainfall exceeding 100 mm day over at least three rainfall stations) observed over the slopes and/or lowlands of southwestern Taiwan were examined. Data from European Center for Medium-Range Weather Forecasts /Tropical Ocean- Global Atmosphere (EC/TOGA) analyses, the rainfall stations of the Automatic Rainfall and Meteorological Telemetry System (ARMTS) and the conventional surface stations over Taiwan, and the simulation results from a regional-scale numerical model were used to accomplish the objectives. In one event (393 mm day on 9 August 1999), heavy rainfall was observed over the windward slopes of southern Taiwan in a potentially unstable environment with very humid air around 850 hPa. The extreme accumulation was simulated and attributed to orographic lifting effects. No preexisting convection drifted in from the Taiwan Strait into western Taiwan.

Chen, Ching-Sen↗
NASA NTRS2005

Interannual Rainfall Variability in the Tropical Atlantic Region

Rainfall variability on seasonal and interannual-to-interdecadal time scales in the tropical Atlantic is quantified using a 25-year (1979-2003) monthly rainfall dataset from the Global Precipitation Climatology Project (GPCP). The ITCZ measured by monthly rainfall between 15-37.5 deg W attains its peak as moving to the northernmost latitude (4-10 deg N) during July-September in which the most total rainfall is observed in the tropical Atlantic basin (17.5 deg S-22.5 deg N, 15 deg-37.5 deg W); the ITCZ becomes weakest during January-February with the least total rainfall as it moves to the south. In contrast, rainfall variability on interannual to interdecadal time scales shows a quite different seasonal preference. The most intense interannual variability occurs during March-May when the ITCZ tends to be near the equator and becomes weaker. Significant, negative correlations between the ITCZ strength and latitude anomalies are observed during boreal spring and early summer. The ITCZ strength and total rainfall amount in the tropical Atlantic basin are significantly modulated by the Pacific El Nino and the Atlantic equatorial mode (or Atlantic Nino) particularly during boreal spring and summer; whereas the impact of the Atlantic interhemispheric mode is considerably weaker. Regarding the anomalous latitudes of the ITCZ, the influence can come from both local, i.e., the Atlantic interhemispheric and equatorial modes, and remote forcings, i. e., El Nino; however, a direct impact of El Nino on the latitudes of the ITCZ can only be found during April-July, not in winter and early spring in which the warmest SST anomalies are usually observed in the equatorial Pacific.

Gu, Guojun↗
NASA NTRS2006

Weekend Effect" in Summertime U.S. Rainfall: Evidence for Midweek Intensification of Storms by Pollution

Persistent and strong dependence of rain rate on the day of the week has been found in Tropical Rainfall Measuring Mission (TRMM) satellite estimates of summer afternoon rainfall over the southeast U.S. and the nearby Atlantic from 1998 to 2005. Midweek (Tue-Thu) rain rates and rain area appear to increase over land, and this increase is accompanied by a corresponding diminution of rainfall over nearby waters. Reanalysis data from atmospheric models, suggest that there is a corresponding weekly variation in atmospheric winds consistent with the changes in rainfall. These variations are almost certainly caused by weekly variations in human activity. The most likely cause of the observed changes in rainfall is the well documented weekly variation in atmospheric pollution. Particulate pollution is highest in the middle of the week. Considerable observational and modeling evidence has accumulated concerning the effects of aerosols on precipitation. Most of this evidence relates to the suppression of precipitation by aerosols, but it has been argued that storms in highly unstable moist environments can be invigorated by aerosols, and some modeling studies seem to confirm this. The strong weekly cycle in rainfall observed over the southeast U.S. along with what appears to be dynamical suppression of rainfall over the nearby Atlantic, and the lack of an observable cycle over the southwest U.S., are consistent with this theory.

Bell, Thomas L.↗
NASA NTRS2010

Ten-Year Climatology of Summertime Diurnal Rainfall Rate Over the Conterminous U.S.

Diurnal cycles of summertime rainfall rates are examined over the conterminous United States, using radar-gauge assimilated hourly rainfall data. As in earlier studies, rainfall diurnal composites show a well-defined region of rainfall propagation over the Great Plains and an afternoon maximum area over the south and eastern portion of the United States. Zonal phase speeds of rainfall in three different small domains are estimated, and rainfall propagation speeds are compared with background zonal wind speeds. Unique rainfall propagation speeds in three different regions can be explained by the evolution of latent-heat theory linked to the convective available potential energy, than by gust-front induced or gravity wave propagation mechanisms.

Matsui, Toshihisa↗
NASA NTRS2017

A Remote Sensing-Based Tool for Assessing Rainfall-Driven Hazards

RainyDay is a Python-based platform that couples rainfall remote sensing data with Stochastic Storm Transposition (SST) for modeling rainfall-driven hazards such as floods and landslides. SST effectively lengthens the extreme rainfall record through temporal resampling and spatial transposition of observed storms from the surrounding region to create many extreme rainfall scenarios. Intensity-Duration-Frequency (IDF) curves are often used for hazard modeling but require long records to describe the distribution of rainfall depth and duration and do not provide information regarding rainfall space-time structure, limiting their usefulness to small scales. In contrast, Rainy Day can be used for many hazard applications with 1-2 decades of data, and output rainfall scenarios incorporate detailed space-time structure from remote sensing. Thanks to global satellite coverage, Rainy Day can be used in inaccessible areas and developing countries lacking ground measurements, though results are impacted by remote sensing errors. Rainy Day can be useful for hazard modeling under nonstationary conditions.

Daniel B Wright↗
NASA NTRS2024

Vegetation Greening Mitigates the Impacts of Increasing Extreme Rainfall on Runoff Events

Future flood risk assessment has primarily focused on heavy rainfall as the main driver, with the assumption that projected increases in extreme rain events will lead to subsequent flooding. However, the presence of and changes in vegetation have long been known to influence the relationship between rainfall and runoff. Here, we extract historical (1850–1880) and projected (2070–2100) daily extreme rainfall events, the corresponding runoff, and antecedent conditions simulated in a prominent large Earth system model ensemble to examine the shifting extreme rainfall and runoff relationship. Even with widespread projected increases in the magnitude (78% of the land surface) and number (72%) of extreme rainfall events, we find projected declines in event-based runoff ratio (runoff/rainfall) for a majority (57%) of the Earth surface. Runoff ratio declines are linked with decreases in antecedent soil water driven by greater transpiration and canopy evaporation (both linked to vegetation greening) compared to areas with runoff ratio increases. Using a machine learning regression tree approach, we find that changes in canopy evaporation is the most important variable related to changes in antecedent soil water content in areas of decreased runoff ratios (with minimal changes in antecedent rainfall) while antecedent ground evaporation is the most important variable in areas of increased runoff ratios. Our results suggest that simulated interactions between vegetation greening, increasing evaporative demand, and antecedent soil drying are projected to diminish runoff associated with extreme rainfall events, with important implications for society.

soil moisture↗
DOE OSTI2024

Seasonal delay of Sahelian rainfall driven by an east–west contrast in radiative forcing in idealized CESM experiments

Recent studies suggest the observed seasonal delay of rainfall over the Sahel is mainly driven by anthropogenic aerosol forcing, which features a robust east–west contrasting changes in recent decades, with negative and positive top-of-the-atmosphere shortwave radiative forcing in South and East Asia (SA&EA) and in North America and Europe (NA&EU), respectively. Their individual effects on the Sahel rainfall annual cycle remain unclear. Here, by designing idealized sensitivity experiments based on Community Earth System Model (CESM1.2), we show that both negative radiative forcing over SA&EA and positive radiative forcing over NA&EU contribute to the delayed phase and enhanced amplitude of Sahel rainfall annual cycle. To understand the underlying physical processes, both the convective quasi-equilibrium (CQE) and atmospheric energetic frameworks are utilized. Both frameworks can well explain the seasonal delay in the rainfall annual cycle over the Sahel under the regional radiative forcings, with the CQE framework showing stronger explanatory power. Based on the insights from both frameworks, we conducted a moisture budget analysis and found that the negative radiative forcing over SA&EA causes an anomalous low-level anticyclonic circulation, which transports more moisture to the Sahel. As the anomalous anticyclone is stronger in summer and fall compared to spring, rainfall is delayed in the Sahel. Further, under the influence of the positive radiative forcing over NA&EU, more warming in the northern hemisphere causes northward transport of moisture to the northern tropics associated with the cross-equatorial transport of energy. The moisture transport increases relativehumidity over the Sahel and enhances the effective atmospheric heat capacity that delays the seasonal rainfall.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2024

Rainfall events stimulate episodic associative nitrogen fixation in switchgrass

Abstract Associative N 2 fixation (ANF) is widespread but poorly characterized, limiting our ability to estimate global inputs from N 2 fixation. In some places, ANF rates are at or below detection most of the time but occasionally and unpredictably spiking to very high rates. Here we test the hypothesis that plant phenology and rainfall events stimulate ANF episodes. We measured ANF in intact soil cores in switchgrass ( Panicum virgatum L.) in Michigan, USA. We used rain exclusion shelters to impose three rainfall treatments with each receiving 60 mm of water over a 20-day period but at different frequencies. We concurrently established a treatment that received ambient rainfall, and all four treatments were replicated four times. To assess the effects of plant phenology, we measured ANF at key phenological stages in the ambient treatment. To assess the effects of rainfall, we measured ANF immediately before and immediately after each wetting event in each treatment involving rainfall manipulation. We found that the previous day’s rainfall could explain 29% of the variation in ANF rates within the ambient treatment alone, and that bulk soil C:N ratio was also positively correlated with ANF, explaining 18% of the variation alone. Wetting events increased ANF and the magnitude of response to wetting increased with the amount of water added and decreased with the amount of inorganic N added in water. ANF episodes thus appear to be driven primarily by wetting events. Wetting events likely increase C availability, promote microbial growth, and make rhizosphere conditions conducive to ANF.

Vizza, Carmella (ORCID:0000000292690357)↗
DOE OSTI2025

Impact of Penetrating Kelvin Waves on MJO Rainfall Propagation Through Deep Convection Modulation

The Madden‐Julian Oscillation (MJO) is a dominant intraseasonal oscillation in the tropics. As MJOs propagate over the Indian Ocean, some of them can be penetrated by fast‐moving equatorial Kelvin waves originating to the west of the MJOs. Through a series of sensitivity experiments, we demonstrate that these penetrating Kelvin waves can modulate the eastward propagation speed of MJO rainfall. Deep convection in the western MJO rainfall envelope initiates more frequently and produces greater lifetime rainfall when a convectively active Kelvin phase is present, while the opposite occurs during a convectively suppressed Kelvin phase. This alteration in deep convection activity can lead to notable changes in the propagation of the large‐scale MJO rainfall envelope. The modulation of deep convection lifetime rainfall by penetrating Kelvin waves primarily occurs through their impact on deep convection rainfall area, which is closely related to the variations in lower‐tropospheric moisture induced by the Kelvin waves.

Chen, Xingchao [Pennsylvania State Univ., Universi↗
NASA NTRS1979

Relation between West Coastal rainfall and Nimbus 6 SCAMS liquid water data over the northeastern Pacific Ocean

A research study is described in which the application to rainfall prediction of cloud liquid water data obtained from the SCAMS experiment of Nimbus 6 is explored. The study area is the Pacific Northwest coast of the United States, where rainfall is produced by extratropical storms that approach from across the Pacific Ocean. SCAMS data related to cloud liquid water over the ocean, and coastal rainfall data, are analyzed for 20 different storm systems in the northeastern Pacific Ocean; these produced significant rainfall from Washington to central California during the period October 1975-March 1976. Results show that the distribution of storm-cloud water analyzed from the SCAMS data over the ocean foreshadows the distribution of coastal rainfall accumulated from the storm at a later time. It is concluded that passive microwave sensor measurements of cloud water over the ocean, when used in conjunction with numerical and other objective guidance, can be used to enhance the accuracy of predictions of coastal rainfall distribution. Limitations in the SCAMS measurements and in the data analysis and interpretation are noted.

Viezee, W.↗