1. Introduction

The definition of statistical–empirical algorithms is not necessarily easy to determine. To envisage these algorithms as being a mere statistical relationship between two parameters (the passive microwave data and the ground truth data) would be too simplistic. Similarly, at the other extreme, rainfall retrievals using a physical model cannot solely rely upon radiative transfer theory. There is a full spectrum of algorithms that have evolved over several decades of research, some having been developed from relatively straightforward theories, through to more complex, multichannel algorithms, and finally to the models that use the radiative transfer models in combination with cloud models to replicate the cloud processes that produce the satellite-observed brightness temperatures. The boundary between each group of algorithms is not easily defined as there is a certain amount of overlap between the groups. Indeed, some algorithms actually sit astride boundaries, with the same equation being used with each group, but the parameters used within the algorithm determined either by statistical or by modeling techniques [e.g., the polarization-corrected temperature algorithm; see Kidd (1998) and Spencer et al. (1989)]. It can also be argued that many statistical–empirical algorithms have been based upon the results of modeling but have been fine-tuned by calibration against ground data, while most physical models have been calibrated (or used some form of ground truth data) at some stage of their development (Wilheit et al. 1994).

To understand the rationale behind the statistical–empirical algorithms, it is first necessary to outline the basic relationships between the passive microwave observed brightness temperatures and rain intensity as defined by the calibration data.

2. Physical rationale

The basis of many of the statistical–empirical algorithms is determined by the suitability of certain channels to respond to the precipitation-sized particles. Early investigations of radiative transfer at passive microwave frequencies showed that precipitation-sized particles modify the upwelling radiation from the surface (see Hess 1959). This modification may take the form of augmentation due to additional radiation being emitted or by a reduction due to the scattering of the upwelling radiation stream by the precipitation sized particles.

The factors determining the type of attenuation encountered are the size of precipitation particle, the phase of the particle (ice or liquid), and the wavelength of the radiation. Fowler et al. (1979) noted that when the atmospheric particle size was very small in relation to the wavelength of the radiation, attenuation of the radiation stream would be by absorption and reemission. However, when the size of the particle approaches the wavelength of the radiation, scattering increases and plays a dominant role in the extinction of the upwelling radiation.

The amount of upwelling radiation from the surface is also critical to the choice of algorithm. Wilheit et al. (1976) noted that at 19.35 GHz the emissivity of the earth varies considerably more than at infrared wavelengths. One of the features of passive microwave radiometry is the difference between land and ocean surfaces. This contrast is determined by the emissivity of the surface. An ocean surface has a low but relatively constant emissivity resulting in low brightness temperatures. Land surfaces, however, have a higher and more variable emissivity resulting in high brightness temperatures. For many techniques the difference between the land and ocean emissivities is such that two separate algorithms need to be used, one for each surface.

Initial algorithm development relied on low-/medium-frequency measurements (19.35 GHz on ESMR-5 and 37 GHz on ESMR-6). At these low/medium frequencies, the size of the atmospheric particles is not usually great enough to induce scattering (at least at the large spatial resolutions involved). Techniques, therefore, relied upon the amount of extra radiation being emitted by the particles against a radiometrically coldocean surface. These techniques are not suitable for over-land applications due to the high surface emissions effectively masking the precipitation attenuation.

Techniques over land surfaces have to rely upon the scattering processes and therefore tend to be confined to the medium/high frequencies. Savage and Weinman (1975) noted that while the 19.35-GHz channel on the ESMR-5 instrument did not detect an observable decrease in brightness temperatures over land at high rainfall intensities, the 37-GHz channels did, making it feasible to detect precipitation over land surfaces. Rodgers (1981) noted, however, that while the precipitation could be qualitatively mapped the prospect for quantitative mapping was less promising. The subsequent inclusion of the 85-GHz channel on the Special Sensor Microwave/Imager (SSM/I) instrument significantly increased the potential for mapping rainfall over land surfaces (see Barrett et al. 1988).

3. Algorithm selection

Table 1 shows the typical range of fundamental algorithms. The simplest algorithms involve a simple brightness temperature or polarization relationship to establish rainfall rates. At the lower frequencies, this is confined to oceans and relies on the increase of brightness temperatures with increasing rainfall rates. Over land, high frequencies may be used to detect decreases in brightness temperatures due to scattering of radiation. The main drawback is that over water the brightness temperature values reach an asymptotic level before recurving, making high rain rates have similar brightness temperatures to lower rainfall rates due to the emission of radiation from the rain droplets themselves. However, good use has been made of a simple brightness-temperature to rain-rate technique for climatological analysis of rainfall (e.g., Barrett et al. 1991). Over land, thescattering of the radiation may not be directly linked to the rainfall itself, but may be more indicative of the ice at the top of the precipitating cloud system, which may be significant in large convective situations (Spencer et al. 1989).

The problem with a single channel technique is one of temperature variations. Although surface variations do not influence the radiances emanating from the top of high-intensity precipitation systems, the setting of the rain–no rain boundary is greatly affected by the surface temperature. Figure 1 shows the season variations of the brightness temperature for the 85-GHz vertically polarized channel for 1 yr of data. It can be observed that the average brightness temperature for the nonraining (radar identified) surface varies from just under 279 to 263 K, while that of the 0.01–0.50 mm h⁻¹ rain rate varies from 270.5 to 259.5 K. Clearly the setting of a fixed rain–no rain threshold for temporal and global studies would not be appropriate. To counteract the effect of temperature variations, multichannel algorithms have been developed. The frequency difference and polarization algorithms both fall into a dual-channel category. These algorithms reduce the effects of surface temperature variations and therefore can be considered to be useful for global and temporal studies.

The frequency-difference algorithm utilizes the difference between a low-frequency channel minus a high-frequency channel (of the same polarization). The rationale behind this technique relies on the higher-frequency channel having a greater response to scattering from precipitation compared to the low-frequency channel. While the vertical (polarization) channels are normally used, there are situations where the horizontal channels must be used, such as over desert areas (see Barrett et al. 1991). This technique, while being fundamentally simple, has proved accurate: in the third Algorithm Intercomparison Project (AIP-3) (Ebert 1996), a frequency-difference algorithm using the 19- and 85-GHz channels (vertically polarized) proved to be one of the best algorithms submitted. Polarization techniques are limited to over-water retrievals of precipitation and are based upon the depolarization of the upwelling radiation by the intervening cloud and rain. The use of dual polarizations at the same frequency should eliminate the effects of surface temperature variations, although insensitivity to high rainfall rates is a main drawback of this technique.

Polarization-corrected temperatures (PCTs) are generated by using dual polarizations at a single frequency. The basis of this algorithm has been explored initially by Weinman and Guetter (1977) and subsequently exploited by Grody (1984), Kidd (1988), and Spencer et al. (1989). The algorithm essentially removes the effectsof surface emissivity, making the atmospheric attenuations more prominent; thus the algorithm may be used over both land and ocean, and, perhaps more importantly, over coastal areas where mixed fields of view would normally preclude the extraction of rainfall information. While Weinman and Guetter, and Spencer relied upon modeling techniques to establish algorithm parameters, Grody and Kidd used objective approaches. However, Spencer mentions that although he calculated values for these parameters from radiative transfer models, he adjusted the values to compensate for the deficiencies in the model. The PCT approach is one example of where the statistical–empirical group and modeling group share common algorithms but with differing approaches to establishing the parameters involved.

The ultimate goal of the statistical–empirical algorithms is to develop a technique that may be used under all circumstances. While single/dual-channel algorithms are useful, they often need careful handling due to variations in surface temperatures, atmospheric constituents, and regional applications. By using multichannel approaches, some of the effects of, for example, water vapor may be reduced by using other channels more sensitive to water vapor than the main precipitation retrieval channels. However, as the complexity of these algorithms increases, outside factors can cause additional errors to be encountered with few benefits.

4. Discussion

The benefits and deficiencies of the statistical–empirical algorithms are many and will no doubt continue to generate debate for many years to come. This section will try to evaluate some of the advantages and related disadvantages of the statistical–empirical techniques over other current passive microwave algorithms.

Wilheit et al. (1994) provide a useful discussion on the different algorithms submitted for the PIP-1 exercise, noting three main groups of algorithms:

1. mainly empirical, using the minimal level of physical input;

2. mainly physical, those algorithms using a radiative transfer model; and

3. highly physical, involving the generation of hydrometeor profiles and subsequent derivation of rainfall rates.

Relationships between rainfall rate and brightness temperatures are similar whether derived via empirical means or via radiative transfer models. Figure 2 exemplifies this issue. The response of the 19-, 37-, and 85-GHz channels, over both land and ocean, derived from empirical–statistical procedures is essentially the same as that modeled by Wu and Weinman (1984). One of the major drawbacks of the empirical approach is that there is a preponderance of the low rainfall intensities, and the subsequent lack of high rainfall occurrences makes statistical relationships at the high rain rates meaningless. The model output, however, is not constrained by the distribution of the rainfall intensities; brightness temperatures may be calculated for any realistic rainfall rate. Nevertheless, the empirical algorithms do score well on other issues. While the models can simulate high-intensity events, the empirically derived relationships incorporate the properties of the sensing instrument, such as the field-of-view pattern (Ferraro and Marks 1995). The modeling of the antenna pattern is perhaps as great a problem to the model algorithms as the retrieval of rainfall itself. Many of these physical algorithms have resorted to empirical calibration to establish beam-filling corrections (Wilheit et al. 1994). In addition, most model-generated brightness temperature–rainfall rate relationships are illustrated for a homogeneous surface: land surfaces are highly variable and the modeling of precipitation over such surfaces is difficult (Ferraro and Marks 1995).

Mugnai et al. (1993) emphasizes that empirical algorithms have problems in retrieving consistently accurate values due to variations in the brightness temperature–rainfall rate relationship. However, from the results of the PIP-2 exercise (Smith et al. 1998), correlations between the physical and empirical algorithms are very good, while the correlations between the satellite algorithms and the ground truth are much worse. While the validation data and satellite data have numerous problems that could reduce the correlations, J. O. Bailey (1996, personal communication) suggests that a large proportion of the errors may be attributable to geolocation errors. Reduced resolution radar data (to match the SSM/I fields of view) were mislocated and then compared with the original data. Correlations between these two datasets were found to fall from a perfect correlation (1.00) for the collocated data to an average value of just 0.58 for a geolocation error of just one field of view (at 15-km resolution). The lowest correlation coefficient for a single field-of-view geolocation error was 0.01. The implication of this is that the statistical analysis of the datasets must be treated with caution if external errors are not to be incorporated into the results. The results of the AIP-3 exercise showed that correlations of 70%–80% were possible between the satellite measurements and ground truth data when good validation data exist. There is also an important lesson to be learned from these results in terms of calibration of algorithms. Existing techniques for the calibration of algorithms are always biased toward the lowrainfall intensities, and a method is needed where the histograms of the algorithm and the validation data form the basis of the calibration. This technique would eliminate the need for accurately located satellite and ground truth data.

The use of radiative transfer models offers a higher degree of sophistication over that of the empirical algorithms. The main drawback with the models is the degree to which their models represent reality. A radiative transfer equation cannot wholly model reality due to the limitations of the input data, limitations of our understanding of the radiative transfer processes, and the necessary approximations to those processes we do know about to make the model viable. Wilheit et al. (1977) notes that models often simplify the complex distribution of the size and shape of particles found within a precipitating cloud. Smith et al. (1992) highlight problems faced by the algorithms on four issues. First, algorithms based upon radiative transfer models or derived using statistical methods are purely hypothetical or derived from very limited observations of liquid water associated with precipitation. Second, algorithms “utilize arbitrary procedures to separate cloud liquid water into nonprecipitating and precipitating terms” (Smith et al. 1992). Third, the ice/water content distributions are unrealistically specified in model algorithms (and not considered in empirical algorithms); and last, the microphysical components are vertically distributed in only a few homogeneous layers in model algorithms. The development of all the current viable passive microwave techniques have all been based to some extent on the fundamental principles associated with radiative transfer theory, and, in many cases, the algorithms have undergone rigorous calibration, testing, and recalibration from several years of radar (and other) data. Indeed, the second issue raised by Smith concerning the delineation of the rain–no rain boundary has been thoroughly addressed by Kidd (1988) and Barrett and Kidd (1991). Many statistical–empirical algorithms have been developed to identify the rain–no rain boundary as part of their rainfall rate retrieval requiring few, if any, additional procedures to eliminate false rain signatures. Many of the physical algorithms [the Smith et al. (1992) algorithms included] use a rain–no rain identification procedure based upon empirical observations.

Wilheit et al. (1994) states that one disadvantage of the empirical techniques is the lack of stability, while those algorithms are soundly based upon physics and have physical mechanisms to adjust for changing circumstances. However, it must be pointed out, that for long-term changes in precipitation (such as climate change), the changes must already be known to be able to implement them in the physical algorithms. Many empirical algorithms have been developed to minimize the unwanted effects of changes in temperature, etc. Allalgorithms attempt to parameterize the relationship between the satellite measurements and the rain rates falling at the surface, be it through radiative transfer models, inversion techniques, or statistical–empirical methods. As such, they are all based upon physics and thus can be said to be vaguely flexible. Experience from the intercomparison exercises (e.g., PIP-1) shows that some algorithms designed using particular datasets [such as the Cal/Val algorithm; see Olson et al. (1989)] or specific models (e.g., Smith et al. 1992) perform better in some regions than in others. Therefore the suggestion is that specific algorithms should be used in specific regions: the modelers require a multitude of cloud models and hydrometeor profiles, while empirical algorithms require different calibration datasets. The problem is the same to some degree: all algorithms need different relationships depending upon the conditions prevalent at the time. The crucial part is determining which relationship of which model is most appropriate. The advantage of the statistical–empirical algorithms is that a number of relationships may be easily applied to evaluate the precipitation signature. In addition, some algorithms such as the D5K (see Kniveton 1994) use self-calibrating techniques to determine the rain–no rain boundary and the rain-rate relationship.

Perhaps the greatest advantage of the statistical–empirical algorithms over all other groups is the simplicity of their calculation. The use of few lines of code greatly helps the elimination of programing errors that are unavoidable in large, unwieldy inversion-type algorithms, computation speed is kept to a minimum, and sources of errors in the generated rainfall maps may be more easily traced. The speed of computation is vital when investigating real-time situations, quick-look images, or climatological studies, where time or size of dataset are important.

5. Conclusions

The advantages of statistical–empirical algorithms at present tend to outweigh the disadvantages, with the developers of these algorithms aware of the limitations imposed by the simplicity of the techniques involved. It is recognized that future rainfall algorithms will become more physically based so that additional parameters may be extracted, but improvements are unlikely without significant advances in the theoretical relationships (Petty and Katsaros 1990). Ultimately, the algorithms used are determined by the end-use requirements, and the ability of the algorithms to generate the desired results within a given time frame.