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- Author or Editor: Ziad S. Haddad x

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## Abstract

In this paper, an analytical treatment of the atmospheric remote sensing problem of determining the raindrop size distribution (DSD) with a spaceborne multifrequency microwave nadir-looking radar system is presented. It is typically assumed that with two radar measurements at different frequencies one ought to be able to calculate two state variables of the DSD: a bulk quantity, such as the rain rate, and a distribution shape parameter. To determine if this nonlinear problem can indeed be solved, the DSD is modeled as a Γ distribution and quadratic approximations to the corresponding radar–rain relations are used to examine the invertibility of the resulting system of equations in the case of two as well as three radar frequencies. From the investigation, it is found that for regions of DSD state space multiple solutions exist for two or even three different frequency radar measurements. This should not be surprising given the nonlinear coupled nature of the problem.

## Abstract

In this paper, an analytical treatment of the atmospheric remote sensing problem of determining the raindrop size distribution (DSD) with a spaceborne multifrequency microwave nadir-looking radar system is presented. It is typically assumed that with two radar measurements at different frequencies one ought to be able to calculate two state variables of the DSD: a bulk quantity, such as the rain rate, and a distribution shape parameter. To determine if this nonlinear problem can indeed be solved, the DSD is modeled as a Γ distribution and quadratic approximations to the corresponding radar–rain relations are used to examine the invertibility of the resulting system of equations in the case of two as well as three radar frequencies. From the investigation, it is found that for regions of DSD state space multiple solutions exist for two or even three different frequency radar measurements. This should not be surprising given the nonlinear coupled nature of the problem.

## Abstract

Current passive-microwave rain-retrieval methods are largely based on databases built offline using cloud models. Since the vertical distribution of hydrometeors within the cloud has a large impact on upwelling brightness temperatures, a forward radiative transfer model can associate microwave radiances with different rain scenarios. Once such a database is available, to estimate the rain from measured brightness temperatures, one would look for the rain scenarios in the database whose associated radiances are closest to the measurements. To understand the uncertainties in this process, the authors have restricted their attention to tropical ocean cases and analyzed the marginal and joint distributions of the radiances observed by the Tropical Rainfall Measuring Mission (TRMM) satellite’s passive-microwave imager and of those in the databases used in the TRMM passive rain retrieval. The authors also calculated the covariances of the rain profiles and brightness temperatures in the TRMM passive-microwave database and derived a simple parametric model for the conditional variance, given measured radiances. These results are used to characterize the uncertainty inherent in the passive-microwave retrieval.

## Abstract

Current passive-microwave rain-retrieval methods are largely based on databases built offline using cloud models. Since the vertical distribution of hydrometeors within the cloud has a large impact on upwelling brightness temperatures, a forward radiative transfer model can associate microwave radiances with different rain scenarios. Once such a database is available, to estimate the rain from measured brightness temperatures, one would look for the rain scenarios in the database whose associated radiances are closest to the measurements. To understand the uncertainties in this process, the authors have restricted their attention to tropical ocean cases and analyzed the marginal and joint distributions of the radiances observed by the Tropical Rainfall Measuring Mission (TRMM) satellite’s passive-microwave imager and of those in the databases used in the TRMM passive rain retrieval. The authors also calculated the covariances of the rain profiles and brightness temperatures in the TRMM passive-microwave database and derived a simple parametric model for the conditional variance, given measured radiances. These results are used to characterize the uncertainty inherent in the passive-microwave retrieval.

## Abstract

This paper addresses the problem of finding a parametric form for the raindrop size distribution (DSD) that 1) is an appropriate model for tropical rainfall, and 2) involves statistically independent parameters. Such a parameterization is derived in this paper. One of the resulting three “canonical” parameters turns out to vary relatively little, thus making the parameterization particularly useful for remote sensing applications. In fact, a new set of Γ drop-size-distribution-based *Z-R* and *k-R* relations is obtained. Only slightly more complex than power laws, they are very good approximations to the exact radar relations one would obtain using Mie scattering. The coefficients of the new relations are directly related to the shape parameters of the particular DSD that one starts with. Perhaps most important, since the coefficients are independent of the rain rate itself, the relations are ideally suited for rain retrieval algorithms.

## Abstract

This paper addresses the problem of finding a parametric form for the raindrop size distribution (DSD) that 1) is an appropriate model for tropical rainfall, and 2) involves statistically independent parameters. Such a parameterization is derived in this paper. One of the resulting three “canonical” parameters turns out to vary relatively little, thus making the parameterization particularly useful for remote sensing applications. In fact, a new set of Γ drop-size-distribution-based *Z-R* and *k-R* relations is obtained. Only slightly more complex than power laws, they are very good approximations to the exact radar relations one would obtain using Mie scattering. The coefficients of the new relations are directly related to the shape parameters of the particular DSD that one starts with. Perhaps most important, since the coefficients are independent of the rain rate itself, the relations are ideally suited for rain retrieval algorithms.

## Abstract

The ensemble square root Kalman filter (ESRF) is a variant of the ensemble Kalman filter used with deterministic observations that includes a matrix square root to account for the uncertainty of the unperturbed ensemble observations. Because of the difficulties in solving this equation, a serial approach is often used where observations are assimilated sequentially one after another. As previously demonstrated, in implementations to date the serial approach for the ESRF is suboptimal when used in conjunction with covariance localization, as the Schur product used in the localization does not commute with assimilation. In this work, a new algorithm is presented for the direct solution of the ESRF equations based on finding the eigenvalues and eigenvectors of a sparse, square, and symmetric positive semidefinite matrix with dimensions of the number of observations to be assimilated. This is amenable to direct computation using dedicated, massively parallel, and mature libraries. These libraries make it relatively simple to assemble and compute the observation principal components and to solve the ESRF without using the serial approach. They also provide the eigenspectrum of the forward observation covariance matrix. The parallel direct approach described in this paper neglects the near-zero eigenvalues, which regularizes the ESRF problem. Numerical results show this approach is a highly scalable parallel method.

## Abstract

The ensemble square root Kalman filter (ESRF) is a variant of the ensemble Kalman filter used with deterministic observations that includes a matrix square root to account for the uncertainty of the unperturbed ensemble observations. Because of the difficulties in solving this equation, a serial approach is often used where observations are assimilated sequentially one after another. As previously demonstrated, in implementations to date the serial approach for the ESRF is suboptimal when used in conjunction with covariance localization, as the Schur product used in the localization does not commute with assimilation. In this work, a new algorithm is presented for the direct solution of the ESRF equations based on finding the eigenvalues and eigenvectors of a sparse, square, and symmetric positive semidefinite matrix with dimensions of the number of observations to be assimilated. This is amenable to direct computation using dedicated, massively parallel, and mature libraries. These libraries make it relatively simple to assemble and compute the observation principal components and to solve the ESRF without using the serial approach. They also provide the eigenspectrum of the forward observation covariance matrix. The parallel direct approach described in this paper neglects the near-zero eigenvalues, which regularizes the ESRF problem. Numerical results show this approach is a highly scalable parallel method.

## Abstract

Quantifying time-averaged rain rate, or rain accumulation, on subhourly time scales is essential for various application studies requiring rain estimates. This study proposes a novel idea to estimate subhourly time-averaged surface rain rate based on the instantaneous vertical rain profile observed from low-Earth-orbiting satellites. Instantaneous rain estimates from the Tropical Rainfall Measuring Mission (TRMM) Precipitation Radar (PR) are compared with 1-min surface rain gauges in North America and Kwajalein atoll for the warm seasons of 2005–14. Time-lagged correlation analysis between PR rain rates at various height levels and surface rain gauge data shows that the peak of the correlations tends to be delayed for PR rain at higher levels up to around 6-km altitude. PR estimates for low to middle height levels have better correlations with time-delayed surface gauge data than the PR’s estimated surface rain rate product. This implies that rain estimates for lower to middle heights may have skill to estimate the eventual surface rain rate that occurs 1–30 min later. Therefore, in this study, the vertical profiles of TRMM PR instantaneous rain estimates are averaged between the surface and various heights above the surface to represent time-averaged surface rain rate. It was shown that vertically averaged PR estimates up to middle heights (~4.5 km) exhibit better skill, compared to the PR estimated instantaneous surface rain product, to represent subhourly (~30 min) time-averaged surface rain rate. These findings highlight the merit of additional consideration of vertical rain profiles, not only instantaneous surface rain rate, to improve subhourly surface estimates of satellite-based rain products.

## Abstract

Quantifying time-averaged rain rate, or rain accumulation, on subhourly time scales is essential for various application studies requiring rain estimates. This study proposes a novel idea to estimate subhourly time-averaged surface rain rate based on the instantaneous vertical rain profile observed from low-Earth-orbiting satellites. Instantaneous rain estimates from the Tropical Rainfall Measuring Mission (TRMM) Precipitation Radar (PR) are compared with 1-min surface rain gauges in North America and Kwajalein atoll for the warm seasons of 2005–14. Time-lagged correlation analysis between PR rain rates at various height levels and surface rain gauge data shows that the peak of the correlations tends to be delayed for PR rain at higher levels up to around 6-km altitude. PR estimates for low to middle height levels have better correlations with time-delayed surface gauge data than the PR’s estimated surface rain rate product. This implies that rain estimates for lower to middle heights may have skill to estimate the eventual surface rain rate that occurs 1–30 min later. Therefore, in this study, the vertical profiles of TRMM PR instantaneous rain estimates are averaged between the surface and various heights above the surface to represent time-averaged surface rain rate. It was shown that vertically averaged PR estimates up to middle heights (~4.5 km) exhibit better skill, compared to the PR estimated instantaneous surface rain product, to represent subhourly (~30 min) time-averaged surface rain rate. These findings highlight the merit of additional consideration of vertical rain profiles, not only instantaneous surface rain rate, to improve subhourly surface estimates of satellite-based rain products.

## Abstract

The rainfall retrieved using the Tropical Rainfall Measuring Mission (TRMM) precipitation radar (PR) depends on estimating the radar signal path-integrated attenuation using the surface reference technique (SRT). This technique assumes uniform ocean surface backscattering background at precipitation scale and attributes the difference in measured surface backscattering cross sections inside and outside of storms to path-integrated attenuation. Since surface wind is the major environmental variable that controls the strength of the ocean backscattering, it is very desirable to examine the impact of surface wind variation on the retrieved attenuation. To this end, we examined the feasibility of retrieving ocean surface winds from TRMM PR data for the benefit of the surface reference technique. A geophysical model function, a forward model, is developed based on ocean surface wind speed retrieved from TRMM Microwave Imager (TMI) data. A fieldwise wind-retrieval procedure, an inverse model, is formulated using maximum likelihood estimation. Comparison of the conventional SRT with the path-integrated radar attenuation derived using the wind field approach shows an rms difference of 1–2 dB, which is consistent with previous study based on data collected from the Jet Propulsion Laboratory Airborne Rain-Mapping Radar (ARMAR). In addition, there is excellent agreement between wind fields retrieved from TRMM PR and TMI data.

## Abstract

The rainfall retrieved using the Tropical Rainfall Measuring Mission (TRMM) precipitation radar (PR) depends on estimating the radar signal path-integrated attenuation using the surface reference technique (SRT). This technique assumes uniform ocean surface backscattering background at precipitation scale and attributes the difference in measured surface backscattering cross sections inside and outside of storms to path-integrated attenuation. Since surface wind is the major environmental variable that controls the strength of the ocean backscattering, it is very desirable to examine the impact of surface wind variation on the retrieved attenuation. To this end, we examined the feasibility of retrieving ocean surface winds from TRMM PR data for the benefit of the surface reference technique. A geophysical model function, a forward model, is developed based on ocean surface wind speed retrieved from TRMM Microwave Imager (TMI) data. A fieldwise wind-retrieval procedure, an inverse model, is formulated using maximum likelihood estimation. Comparison of the conventional SRT with the path-integrated radar attenuation derived using the wind field approach shows an rms difference of 1–2 dB, which is consistent with previous study based on data collected from the Jet Propulsion Laboratory Airborne Rain-Mapping Radar (ARMAR). In addition, there is excellent agreement between wind fields retrieved from TRMM PR and TMI data.

## Abstract

It is well known that there are significant deterministic ambiguities inherent in trying to determine the particular rain-rate profile that produced some given sequence of air- or spaceborne radar echo powers at a single attenuating frequency. For different combinations of radar data, formulas for the mutually ambiguous solutions are derived and the resulting ambiguities are quantified mathematically. When the given data consist of a single radiometer measurement together with a single-frequency set of range-gated echo powers, it is shown that several substantially different rain profiles can still realistically be considered solutions. On the other hand, if the data consist of a two-frequency set of echo powers, it is proven that the inversion problem generically has a unique solution.

## Abstract

It is well known that there are significant deterministic ambiguities inherent in trying to determine the particular rain-rate profile that produced some given sequence of air- or spaceborne radar echo powers at a single attenuating frequency. For different combinations of radar data, formulas for the mutually ambiguous solutions are derived and the resulting ambiguities are quantified mathematically. When the given data consist of a single radiometer measurement together with a single-frequency set of range-gated echo powers, it is shown that several substantially different rain profiles can still realistically be considered solutions. On the other hand, if the data consist of a two-frequency set of echo powers, it is proven that the inversion problem generically has a unique solution.

## Abstract

The raindrop size distribution (RDSD) is defined as the relative frequency of raindrops per given diameter in a volume. This paper describes a mathematically consistent modeling of the RDSD drawing on probability theory. It is shown that this approach is simpler than the use of empirical fits and that it provides a more consistent procedure to estimate the rainfall rate (*R*) from reflectivity (*Z*) measurements without resorting to statistical regressions between both parameters. If the gamma distribution form is selected, the modeling expresses the integral parameters *Z* and *R* in terms of only the total number of drops per volume (*N*
_{T}), the sample mean [*m* = *E*(*D*)], and the sample variance [*σ*
^{2} = *E*(*m* − *D*)^{2}] of the drop diameters (*D*) or, alternatively, in terms of *N*
_{T}, *E*(*D*), and *E*[log(*D*)]. Statistical analyses indicate that (*N*
_{T}, *m*) are independent, as are (*N*
_{T}, *σ*
^{2}). The *Z*–*R* relationship that arises from this model is a linear *R* = *T* × *Z* expression (or *Z* = *T*
^{−1}
*R*), with *T* a factor depending on *m* and *σ*
^{2} only and thus independent of *N*
_{T}. The *Z*–*R* so described is instantaneous, in contrast with the operational calculation of the RDSD in radar meteorology, where the *Z*–*R* arises from a regression line over a usually large number of measurements. The probabilistic approach eliminates the need of intercept parameters *N*
_{0} or *Z*–*R* relationship.

## Abstract

The raindrop size distribution (RDSD) is defined as the relative frequency of raindrops per given diameter in a volume. This paper describes a mathematically consistent modeling of the RDSD drawing on probability theory. It is shown that this approach is simpler than the use of empirical fits and that it provides a more consistent procedure to estimate the rainfall rate (*R*) from reflectivity (*Z*) measurements without resorting to statistical regressions between both parameters. If the gamma distribution form is selected, the modeling expresses the integral parameters *Z* and *R* in terms of only the total number of drops per volume (*N*
_{T}), the sample mean [*m* = *E*(*D*)], and the sample variance [*σ*
^{2} = *E*(*m* − *D*)^{2}] of the drop diameters (*D*) or, alternatively, in terms of *N*
_{T}, *E*(*D*), and *E*[log(*D*)]. Statistical analyses indicate that (*N*
_{T}, *m*) are independent, as are (*N*
_{T}, *σ*
^{2}). The *Z*–*R* relationship that arises from this model is a linear *R* = *T* × *Z* expression (or *Z* = *T*
^{−1}
*R*), with *T* a factor depending on *m* and *σ*
^{2} only and thus independent of *N*
_{T}. The *Z*–*R* so described is instantaneous, in contrast with the operational calculation of the RDSD in radar meteorology, where the *Z*–*R* arises from a regression line over a usually large number of measurements. The probabilistic approach eliminates the need of intercept parameters *N*
_{0} or *Z*–*R* relationship.

## Abstract

The significant ambiguities inherent in the determination of a particular vertical rain intensity profile from a given time profile of radar echo powers measured by a downward-looking (spaceborne or airborne) radar at a single attenuating frequency are well documented. Indeed, one already knows that by appropriately varying the parameters of the frequency are well documented. Indeed, one already knows that by appropriately varying the parameters of the reflectively-rain rate (*Z*–*R*) and/or attenuation-rain rate (*k*–*R*) relationships one can produce several substantially different rain-rate profiles that would produce the same radar power profile. Imposing the additional constraint that the path-averaged rain rate be a given fixed number does reduce the ambiguities but falls far short of eliminating them. While formulas to generate all mutually ambiguous rain-rate profiles from a given profile of received radar reflectivities have already been derived, there remains to be produced a quantitative measure to assess how likely each of these profiles is, what the appropriate “average” profile should be, and what the “variance” of these multiple solutions is. To do this, one needs to spell out the stochastic constraints that can allow us to make sense of the words “averaged” and “variance” in a mathematically rigorous way. Such a quantitative approach would be particularly well suited for such systems as the planned precipitation radar of the Tropical Rainfall Measuring Mission (TRMM). Indeed, one would then be able to use the radar reflectivities measured by the TRMM radar to estimate the rain-rate profile that would most likely have produced the measurements, as well as the uncertainty in the estimated rain rates as a function of range. Such an optimal approach is described in this paper.

## Abstract

The significant ambiguities inherent in the determination of a particular vertical rain intensity profile from a given time profile of radar echo powers measured by a downward-looking (spaceborne or airborne) radar at a single attenuating frequency are well documented. Indeed, one already knows that by appropriately varying the parameters of the frequency are well documented. Indeed, one already knows that by appropriately varying the parameters of the reflectively-rain rate (*Z*–*R*) and/or attenuation-rain rate (*k*–*R*) relationships one can produce several substantially different rain-rate profiles that would produce the same radar power profile. Imposing the additional constraint that the path-averaged rain rate be a given fixed number does reduce the ambiguities but falls far short of eliminating them. While formulas to generate all mutually ambiguous rain-rate profiles from a given profile of received radar reflectivities have already been derived, there remains to be produced a quantitative measure to assess how likely each of these profiles is, what the appropriate “average” profile should be, and what the “variance” of these multiple solutions is. To do this, one needs to spell out the stochastic constraints that can allow us to make sense of the words “averaged” and “variance” in a mathematically rigorous way. Such a quantitative approach would be particularly well suited for such systems as the planned precipitation radar of the Tropical Rainfall Measuring Mission (TRMM). Indeed, one would then be able to use the radar reflectivities measured by the TRMM radar to estimate the rain-rate profile that would most likely have produced the measurements, as well as the uncertainty in the estimated rain rates as a function of range. Such an optimal approach is described in this paper.

## Abstract

This paper describes a computationally efficient nearly optimal Bayesian algorithm to estimate rain (and drop size distribution) profiles, given a radar reflectivity profile at a single attenuating wavelength. In addition to estimating the averages of all the mutually ambiguous combinations of rain parameters that can produce the data observed, the approach also calculates the rms uncertainty in its estimates (this uncertainty thus quantifies the “amount of ambiguity” in the “solution”). The paper also describes a more general approach that can make estimates based on a radar reflectivity profile together with an approximate measurement of the path-integrated attenuation, or a radar reflectivity profile and a set of passive microwave brightness temperatures. This more general “combined” algorithm is currently being adapted for the Tropical Rainfall Measuring Mission.

## Abstract

This paper describes a computationally efficient nearly optimal Bayesian algorithm to estimate rain (and drop size distribution) profiles, given a radar reflectivity profile at a single attenuating wavelength. In addition to estimating the averages of all the mutually ambiguous combinations of rain parameters that can produce the data observed, the approach also calculates the rms uncertainty in its estimates (this uncertainty thus quantifies the “amount of ambiguity” in the “solution”). The paper also describes a more general approach that can make estimates based on a radar reflectivity profile together with an approximate measurement of the path-integrated attenuation, or a radar reflectivity profile and a set of passive microwave brightness temperatures. This more general “combined” algorithm is currently being adapted for the Tropical Rainfall Measuring Mission.