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Grant W. Petty

Ship reports of present weather obtained from the Comprehensive Ocean–Atmosphere Data Set are analyzed for the period 1958–91 in orderto elucidate regional and seasonal variations in the climatological frequency, phase, intensity, and character of oceanic precipitation. Specific findings of note include the following:

The results of this study suggest that many current satellite rainfall estimation techniques may substantially underestimate the fractional coverage or frequency of precipitation poleward of 50° latitude and in the subtropical dry zones. They also draw attention to the need to carefully account for regional differences in the physical and spatial properties of rainfall when developing calibration relationships for satellite algorithms.

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Grant W. Petty

Abstract

A new conceptual and computational basis is described for renormalizing the single-scatter and extinction properties (optical depth, single-scatter albedo, and scattering phase function or asymmetry parameter) of a three-dimensionally inhomogeneous cloud volume or layer so as to describe a radiatively equivalent homogeneous volume or layer. The renormalization may allow area-averaged fluxes and intensities to be efficiently computed for some inhomogeneous cloud fields using standard homogeneous (e.g., plane parallel) radiative transfer codes.

In the Independently Scattering Cloudlet (ISC) model, macroscopic “cloudlets” distributed randomly throughout a volume are treated as discrete scatterers, analogous to individual cloud droplets but with modified single-scatter properties due to internal multiple scattering. If a volume encompasses only cloudlets that are optically thin, the renormalized single-scatter properties for the volume revert to the intrinsic values and the homogeneous case is recovered.

Although the ISC approach is based on a highly idealized, and therefore unrealistic, geometric model of inhomogeneous cloud structure, comparisons with accurate Monte Carlo flux calculations for more realistic random structures reveal surprising accuracy in its reproduction of the relationship between area-averaged albedo, direct transmittance, diffuse transmittance, and in-cloud absorptance. In particular, it describes the approximate functional dependence of these characteristics on the intrinsic single-scatter albedo when all other parameters are held constant. Moreover, it reproduces the relationship between renormalized single-scatter albedo and renormalized optical thickness derived independently, via a perturbative analysis, by other authors. Finally, the ISC model offers a reasonably intuitive physical interpretation of how cloud inhomogeneities influence area-averaged solar radiative transfer, including the significant enhancement of in-cloud absorption under certain conditions.

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Grant W. Petty

Abstract

Using stringent criteria pertaining to rain-cloud optical thickness and horizontal extent, 3203 multichannel microwave observations of heavy, widespread tropical precipitation over ocean were selected from 9 months of global Special Sensor Microwave Imager (SSM/I) data. These observations subsequently were found to be associated almost exclusively with stratiform rain areas in tropical cyclones. Because of the restrictions on optical thickness and spatial extent, the mean multichannel microwave brightness temperatures and their interchannel covariances are presumed to be determined primarily by the vertical microphysical structure of the rain clouds. The distribution of the above observations in seven-dimensional channel space is characterized concisely using principal component analysis. It is found that only three independent variables are sufficient to explain 97% of the variance in the correlation matrix. This result suggests that the radiometrically important microphysical properties of these rain clouds are strongly interdependent. The most significant eigenvector of the observation correlation matrix corresponds to variable scattering at high frequencies by ice aloft. Its spectral dependence is accurately given by ν 1.76, where ν is the microwave frequency. This empirical result constrains the effective mean sizes of ice particles responsible for observed passive microwave scattering in rain clouds and provides a plausible empirical basis for accurately predicting the magnitude of scattering effects by ice at non-SSM/I microwave frequencies. There are also qualitative indications that this mode of brightness temperature variability is poorly correlated with surface rain rate in this study sample. The empirical results presented herein are expected to be of value for the validation and improvement of microphysical assumptions and optical parameterizations in forward microwave radiative transfer models. Companion papers describe the actual retrieval of effective rain-cloud microphysical properties from the observed multichannel radiances.

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Grant W. Petty

Abstract

Shannon entropy has long been accepted as a primary basis for assessing the information content of sensor channels used for the remote sensing of atmospheric variables. It is not widely appreciated, however, that Shannon information content (SIC) can be misleading in retrieval problems involving nonlinear mappings between direct observations and retrieved variables and/or non-Gaussian prior and posterior PDFs. The potentially severe shortcomings of SIC are illustrated with simple experiments that reveal, for example, that a measurement can be judged to provide negative information even in cases in which the postretrieval PDF is undeniably improved over an informed prior based on climatology. Following previous authors’ writing mainly in the data assimilation and climate analysis literature, the Kullback–Leibler (KL) divergence, also commonly known as relative entropy, is shown to suffer from fewer obvious defects in this particular context. Yet, even KL divergence is blind to the expected magnitude of errors as typically measured by the error variance or root-mean-square error. Thus, neither information metric can necessarily be counted on to respond in a predictable way to changes in the precision or quality of a retrieved quantity.

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Grant W. Petty

Abstract

Land and ship surface synoptic reports of nondrizzle intensity precipitation in progress were matched with 3596 nearly coincident full disk 4-km resolution infrared images from the GMS-5 geostationary satellite, covering 18 calendar months, in order to derive regional and seasonal estimates of the contribution of relatively warm-topped clouds to the total time in precipitation.

Minimum infrared temperatures of 273 K or warmer were found to be associated with 20%–40% of the surface reports of nondrizzle precipitation over much of the ocean east of Australia during all four seasons. Similar or even larger fractions were found during December–March over parts of Indochina, southern China, and the adjacent South China Sea. Although reports of precipitation of moderate or heavy intensity were found to be associated more often with colder cloud tops, there were still regions for which a substantial fraction of these reports were associated with relatively warm clouds. These results suggest at least a potential for significant regional and seasonal biases in satellite infrared or passive microwave scattering based estimates of global precipitation.

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Grant W. Petty

Abstract

A highly simplified, yet meteorologically realistic and flexible, parametric model is described for generating hydrometeor profiles and other environmental properties relevant to microwave radiative transfer calculations in quasi-stratiform rain clouds. With this model, it is possible, via 19 adjustable parameters, to vary cloud and environmental properties, including hydrometeor size distributions and densities, in a continuous, yet self-consistent, fashion and to assess the impact of these changes on computed multichannel microwave brightness radiances. It is also possible to utilize gradient descent methods to find plausible combinations of cloud properties that explain observed multichannel microwave radiances in rain clouds. Potential applications of the above model include 1) gaining insight into effective microphysical properties, for microwave radiative transfer purposes, of actual precipitating cloud systems and 2) accurate extrapolation of observed microwave multichannel radiances in rain clouds to the frequencies and viewing angles of new microwave sensors.

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Grant W. Petty and Ke Li

Abstract

A new approach to passive microwave retrievals of precipitation is described that relies on an objective dimensional reduction procedure to filter, normalize, and decorrelate geophysical background noise while retaining the majority of radiometric information concerning precipitation. The dimensional reduction also sharply increases the effective density of any a priori database used in a Bayesian retrieval scheme.

The method is applied to passive microwave data from the Tropical Rainfall Measuring Mission (TRMM), reducing the original nine channels to three “pseudochannels” that are relatively insensitive to most background variations occurring within each of seven surface classes (one ocean plus six land and coast) for which they are defined. These pseudochannels may be used in any retrieval algorithm, including the current standard Goddard profiling algorithm (GPROF), in place of the original channels. The same methods are also under development for the Global Precipitation Measurement (GPM) Core Observatory Microwave Imager (GMI).

Starting with the pseudochannel definitions, a new Bayesian algorithm for retrieving the surface rain rate is described. The algorithm uses an a priori database populated with matchups between the TRMM precipitation radar (PR) and the TRMM Microwave Imager (TMI). The explicit goal of the algorithm is to retrieve the PR-derived best estimate of the surface rain rate in portions of the TMI swath not covered by the PR. A unique feature of the new algorithm is that it provides robust posterior Bayesian probabilities of pixel-averaged rain rate exceeding various thresholds.

Validation and intercomparison of the new algorithm is the subject of a companion paper.

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Grant W. Petty and Wei Huang

Abstract

The four-parameter modified gamma distribution (MGD) is the most general mathematically convenient model for size distributions of particle types ranging from aerosols and cloud droplets or ice particles to liquid and frozen precipitation. The common three-parameter gamma distribution, the exponential distribution (e.g., Marshall–Palmer), and power-law distribution (e.g., Junge) are all special cases. Depending on the context, the particle “size” used in a given formulation may be the actual geometric diameter, the volume- or area-equivalent spherical diameter, the actual or equivalent radius, the projected or surface area, or the mass.

For microphysical and radiative transfer calculations, it is often necessary to convert from one size representation to another, especially when comparing or utilizing distribution parameters obtained from a variety of sources. Furthermore, when the mass scales with Db, with b < 3, as is typical for snow and ice and other particles having a quasi-fractal structure, an exponential or gamma distribution expressed in terms of one size parameter becomes an MGD when expressed in terms of another. The MGD model is therefore more fundamentally relevant to size distributions of nonspherical particles than is often appreciated.

The central purpose of this paper is to serve as a concise single-source reference for the mathematical properties of, and conversions between, atmospheric particle size distributions that can expressed as MGDs, including exponential and gamma distributions as special cases.

For illustrative purposes, snow particle size distributions published by Sekhon and Srivastava, Braham, and Field et al. are converted to a common representation and directly compared for identical snow water content, allowing large differences in their properties to be discerned and quantified in a way that is not as easily achieved without such conversion.

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Ralf Bennartz and Grant W. Petty

Abstract

This study investigates the effect of variable size distribution and density of precipitation ice particles on microwave brightness temperatures. For this purpose, a set of self-consistent relationships among rain rate, size parameters of an exponential drop size distribution, and the radar reflectivity–rain rate relations for frozen and liquid precipitation was derived. Further, a scaling factor was introduced that is the ratio between the average melted diameter of the frozen and liquid precipitation and allows the specification of different sizes of the frozen particles. For given radar observations, this method allows size distributions of frozen and liquid precipitation to be derived, which are then used as input for a radiative transfer model.

These relationships were used to perform Mie calculations for different precipitation rates and different types of hydrometeors (snow, graupel, and hail) and to investigate the dependence of their respective optical properties on rain rate as well as on the radar reflectivity. It is shown that, for a given rain rate, variations of particle density as well as of particle size may result in variations of the extinction coefficient by an order of magnitude. However, a comparison of volume extinction coefficients with radar reflectivities found that, for a broad range of particle sizes, the particle density has little effect in comparison with the liquid-equivalent size of the ice particles.

The proposed method was applied to cases of coincident Special Sensor Microwave Imager (SSM/I) and radar volume scans, the latter being provided by the Swedish Gotland radar. Microwave optical fields for all four SSM/I frequencies were derived from the radar data, and the observed brightness temperature was simulated using a three-dimensional Monte Carlo radiative transfer model. A comparison of scattering indices at 85 GHz derived from the SSM/I overpass with those derived from the model data found that the size of the precipitation-sized ice particles governs the variability of the scattering index. For the particular cases investigated here, the ice particle size varied considerably depending on the type of the precipitation event. For the case of an intensive thunderstorm, ice particles are roughly four times larger than liquid precipitation at the same rain rate, but the ice particles of a frontal system are inferred to be only 20% larger than liquid ice particles. A further evaluation of the relation between scattering index and radar-derived precipitation intensity above the freezing level found high correlations (0.8–0.9) for all precipitation events. However, the sensitivity of scattering index to precipitation intensity varies within a broad range for different types of precipitation events. A convective event had a sensitivity of 9 K (mm h−1)−1, but frontal and small-scale convective events showed sensitivities in the range between 20 and 50 K (mm h−1)−1.

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Grant W. Petty and Wei Huang

Abstract

Coupled-dipole approximation (CDA) calculations of microwave extinction and radar backscatter are presented for nonhomogeneous (soft) ice spheres and for quasi-realistic aggregates of elementary ice crystal forms, including both simple needles and real dendrites. Frequencies considered include selections from the Dual-Frequency Precipitation Radar (DPR; 13.4 and 35.6 GHz) and the Global Precipitation Measurement (GPM) Microwave Imager (GMI; 18.7, 36.5, and 89.0 GHz), both slated for orbit on the GPM mission.

The computational method is first validated against Mie theory using dipole structures representing solid ice spheres as well as stochastically generated “soft” ice spheres of variable ice–air ratio. Neither the traditional Bruggeman nor Maxwell Garnett dielectric mixing formula is found to correctly predict the full range of CDA results for soft spheres. However, an excellent fit is found using the generalized mixing rule of Sihvola with ν = 0.85.

The suitability of the soft-sphere approximation for realistic aggregates is investigated, taking into account the spectral dependence of backscatter and/or extinction per unit mass at key DPR and GMI frequencies. Even when spheres of nonequal mass are considered, there is no single combination of fraction and mass that simultaneously captures all the relevant radiative properties. All four aggregate models do, however, exhibit a predictable power-law dependence of the mass extinction coefficient on the total particle mass. Dual-frequency mass extinction ratios are only very weakly dependent on particle masses; moreover, the ratio is found to be approximately proportional to frequency raised to the power 2.5.

The dual-frequency backscatter ratio is found to be a predictable function of the aggregate mass for particles smaller than 3 mg. Above this size, the ratio is strongly sensitive to aggregate shape, a finding that raises concerns about the utility of dual-frequency backscatter ratio measurements whenever larger particles might be present in a volume of air.

The validity of the Rayleigh–Gans approximation applied to radar backscatter from snow aggregates was also examined. Although the dual-frequency backscatter ratio was reasonably well reproduced, the absolute magnitude was not.

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