Rain/no-rain discrimination

Past studies indicated that for most PM algorithms (such as those investigated in this study) better performance is expected when rain area delineation is performed prior to the retrieval (Smith et al. 1998). The main difficulty in passive microwave rain/no-rain discrimination over land originates in the high variability of ground emissivity. Ferraro et al. (1998) have described various complex procedures for rain/no-rain discrimination based on PM data from the Special Sensor Microwave Imager (SSM/I). We evaluated those procedures on a large TMI dataset and found that the differences between the SSM/I and TMI characteristics have noticeable effects on the discrimination performance. We elaborate on this issue in section 5, in which the performance of several of these procedures is evaluated in terms of rain estimation accuracy. A discrimination scheme is developed herein based on the available TMI and PR data. A set of descriptors, including some of those of Ferraro et al. (1998), are optimally combined within a back-propagation NN scheme to determine rain areas from TMI multifrequency data. A concise and rigorous definition of the NN used in this study and of efficient techniques for NN calibration can be found in Bertsekas (1995). The following descriptors are considered.

1. Slope. The slope relates the 85- and 37-GHz vertically polarized brightness temperatures (T_85V and T_37V) in a 7 (across track) by 3 (along track) pixel window centered on the pixel to be classified. This descriptor makes use of the observation that the 85- and 37-GHz brightness temperatures vary differently with respect to rain intensity. That is, a given increase in rain intensity may produce larger decrease in the 85-GHz temperature versus the 37 GHz. Consequently, this descriptor is expected to be less than one for raining pixels. For nonraining pixels, the 85-GHz brightness temperature values are expected to exhibit smaller variations in the window of 7 × 3 pixels when compared with the 37-GHz values, because the 85-GHz observations are sensitive only to scattering, which occurs when large ice particles and precipitation are present, whereas the 37-GHz observations are sensitive to both scattering and absorption, where the last can also be significant in nonprecipitating clouds. Consequently, in nonprecipitating areas (on the order of 100 km²) the variations in the 37-GHz temperature are expected to be larger than those in the 85-GHz temperature, which leads to a supraunitary slope.

2. Correlation. The correlation between T_85V and T_37V for the space window of descriptor 1 is a measure of confidence in descriptor 1. That is, a large correlation indicates variations in brightness temperature caused by precipitation rather than by ground emissivity variability. The first two descriptors were also used by Adler et al. (1993) but for somewhat different purposes.

3. Scattering index. This descriptor is defined by Grody (1991) as SI = 451.9 − 0.44T_19V − 1.775T_22V + 0.0575T²_22V − T_85V. The first four terms in the scattering index formula represent the estimation of the 85-GHz vertical brightness temperature as a function of the 19- and 22-GHz vertical brightness temperatures. This allows the identification of the scattering component in the actual 85-GHz vertical observation.

4. Ten-GHz mean temperature. The mean of T_10V over the space window defined for descriptor 1 is not relevant as a single descriptor, but it was found to be helpful in conjunction with the others.

5. Nineteen-GHz polarization signature. The difference between T_19V and T_19H has been used by Ferraro et al. (1998) in identifying arid and semiarid surfaces. Vector radiative transfer calculations (Haferman 2000) indicate that polarization differences greater than 7 K at 19-GHz frequency are likely caused by ground emissivity rather than rainfall.

6. The T_85V and T_37V linear combination. This descriptor is defined as LC = 124.28 − 0.36T_37V − 0.1T_85V. We determined LC by a linear least squares fit of VIHC versus T_37V and T_85V. Only raining pixels, that is, strictly positive values of VIHC, were considered in this formula fitting. As a consequence, raining pixels are expected preponderantly to yield positive LC values.

7. Eighty-five-GHz vertically polarized brightness temperature. This is a simple discriminator but is not always effective, because low 85-GHz temperatures may occur even in nonraining areas when ground emissivity is low.

8. Gradient of T_85V. This descriptor is defined as the difference between T_85V and the mean of T_85V in a four-point neighborhood.

9. Standard deviation of T_85V in an eight-point neighborhood. The last two descriptors (8 and 9) were introduced to exploit the observation that T_85V is more variable in rain than in nonrain pixels (Anagnostou and Kummerow 1997).

For calibration of the NN scheme, we used 20 TRMM orbits from 1997–98 over the continental United States. For each TMI multifrequency brightness temperature array, we determined the corresponding PR precipitation profile using a geometrical matching scheme. Namely, a TMI 85-GHz observation was considered to be coincident with a PR profile if the intersection between the TMI pointing vector and the surface of constant altitude 1 km above the freezing level fell inside a PR effective field of view. The matching yielded a database of about 150 000 coincident PR–TMI pixels. The 2A25 PR rainfall products were used to classify a matched pixel as rainy or nonrainy; the nine classification descriptors were calculated from the corresponding TMI data. The histograms of the different descriptors for rain and no-rain situations derived from the above matched dataset are shown in Fig. 2. As evident from this figure, none of the single descriptors may offer accurate rain/no-rain discrimination. This inaccuracy also has been noticed in other studies, and, more complex discriminations based on rules involving several descriptors consequently have been proposed (Ferraro et al. 1998). In this study, we attempt to reduce further the uncertainty by optimally combining all nine descriptors within an NN scheme. Because the best performance in terms of number of pixels accurately classified does not necessarily mean higher accuracy in rainfall estimation, we use a weighted sum of squared errors as the objective function in the NN training. That is, the squared error of the NN output is multiplied by VIHC of the pixel being classified. The sum of squared errors then becomes SSE = Σ^N_i=1VIHC_i(y_i − y^o_i)², where N is the training set size; y_i is the NN output for pixel i; y^o_i is the PR classification, that is, 0 for no rain and 1 for rain; and VIHC_i is the vertically integrated water content for pixel i. This weighting scheme penalizes errors in discrimination of pixels with low precipitation less than those pixels with high precipitation.

The NN performance is tested on the larger set (i.e., 664) of TRMM orbits from 1999. The validation results are given in Table 1 in the form of a contingency array. The integers in the table represent the number of the analyzed pixels sorted according to the PR and NN rain/no-rain discriminations. In the parentheses, the percentages of pixels relative to the total number of pixels are specified. Based on these results, we calculated the observed agreement P_A, which is the ratio of correctly classified pixels to the total number of pixels. The observed agreement is 97.29%, which denotes an overall error of 2.71% (18.89% in rain- and 1.77% in no-rain pixels). An index often used in assessing the performance of classification schemes is the kappa coefficient. This coefficient evaluates the algorithm performance by eliminating the contribution of chance in P_A. It is defined as (Carletta 1996)

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where P_E is the proportion of cases for which the classification and reality would agree by chance. Here, P_E is estimated as the probability of both PR and NN indicating rain plus the probability of both PR and NN indicating no rain. That is, for Table 1, P_E = 0.0541 × 0.0604 + 0.9459 × 0.9396 = 0.892, resulting in K = 0.75. Analyzing the K value relative to its range (from −1, in case of complete disagreement, via 0, when there is no agreement above that expected by chance, to +1, in case of perfect agreement), one may notice satisfactory agreement of NN discrimination with the PR observations. In section 5, we present implications of the rain discrimination accuracy on rain estimation.

Rain-regime classification

A strategy similar to that used in rain/no-rain discrimination was devised for the rain-regime classification. A set of descriptors was determined and fed to an NN, which was trained to predict the regime type (stratiform vs convective rain). These descriptors, which are similar to the ones proposed in studies of Anagnostou and Kummerow (1997) and Hong et al. (1999), include the following.

1. The minimum 85-GHz vertically polarized brightness temperature T_85V in a 7 (across track) by 3 (along track) pixel window centered on the pixel to be classified.

2. The 85-GHz vertically polarized brightness temperature. The first two descriptors describe the likelihood that the current pixel belongs to a convective core. This is because the 85-GHz brightness temperature is a strong indicator of rain intensity.

3. The standard deviation of T_85V. This descriptor exploits the observation that the convective rain is more variable in a small area (on the order of a few pixels) than is the stratiform rain.

4. The scattering index as described in section 4a.

5. The T_85V and T_37V linear combination presented in section 4a. Descriptors 4 and 5 give a measure of the rain intensity and consequently of the likelihood of convective rainfall.

6. Gradient of T_85V as described in section 4a. This descriptor was considered for reasons similar to those in the case of descriptor 3.

The calibration/validation setup for this classification scheme is the same as in section 4a. For each matched TMI–PR pixel, the 2A23 PR rain classification product was used as reference. Histograms of the proposed descriptors for both regimes are shown in Fig. 3. It is apparent that there is much overlap in these histograms, which justifies the use of a complex classification approach, such as NN, instead of simple threshold-based decisions. The validation results are given as a contingency array in Table 2. The observed agreement calculated from the table is 0.58 (classification error is 42%); the K coefficient is about 0.2. This performance is slightly lower than that in Hong et al. (1999), which was, however, demonstrated for overocean-only PM data. Note also that, although the classification may not seem to be significantly more skillful than the chance classification (K = 0.2), its effect on rain estimation can be important. This is because the NN scheme groups into the two rain types pixels that are statistically similar, which reduces the variability of the predictors used in the rain estimation algorithms. The positive effect of this fact will be presented in section 5. An application of the combined NN-based rain/no-rain discrimination and C/S rain-regime classification schemes is illustrated in Fig. 4, which shows good agreement between the PR and TMI predictions. It is apparent that, with such satisfactory performance, TMI offers reliable information of the rain development over a broader area, which is valuable in filling up the sampling gaps of PR measurements. We visually inspected other TMI C/S classification cases and found that for organized widespread events, the technique is able to detect properly the major convective cores.

Estimation of vertically integrated hydrometeor content

We investigated VIHC estimation from TMI observations, based on algorithms of varying complexity and error characteristics. A common feature of the investigated algorithms is the use of multilinear regression to relate their predictors to VIHC. The investigated algorithms are described below.

The fifth algorithm uses the Goddard Scattering Technique (GSCAT), which was developed for rainfall estimation over land, to define the descriptor (Adler et al. 1994):

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The P_GSCAT is related to VIHC using a linear regression similar to those of the previous algorithms [e.g., Eq. (10)].

Resampling experiment

A Monte Carlo statistical resampling procedure is deployed to assess the performance of the proposed VIHC estimation algorithms. The procedure involves the following steps. A fraction p of matched TMI–PR pixels, varying from 20% to 60%, is selected at random from the database for estimation of the algorithms' linear regression coefficients. The derived relationships are applied to the remaining fraction of matched TMI–PR pixels to calculate the algorithms' prediction efficiency and bias. The efficiency is defined as

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where N is the number of pixels considered for evaluation, VIHC_i is the reference value (determined from the PR retrieval), and VIHC^e_i is the value estimated by the investigated algorithms. Similar, the bias is defined as

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For a certain fraction p, the random selection of the calibration/validation dataset, parameter estimation, and performance evaluation are repeated 300 times to ensure statistical significance.

The pixels selected for the calibration/validation exercise are the ones indicated by the PR as rainy, which according to Table 1 represent a total of 100 599. Three scenarios are selected for this analysis. In the first scenario, distinct regression coefficients are determined for the convective and stratiform rain regimes classified by our NN scheme. In the second scenario, the regression coefficients are determined for C/S rain regimes provided by the PR 2A23 products. In the third scenario, no classification is considered.

Resampling results

The results for a 50% calibration set size are shown in Fig. 5 in terms of mean and standard deviation of the performance scores. Results suggest that absorption channel observations are informative, even over land, and that a multispectral estimation is preferable to single-frequency retrievals. This is evident for the MLR algorithm, which yields noticeably better results than the GSCAT. However, the EV and AEV algorithms, which also use multiple frequency information, do not yield better results than GSCAT. This result indicates that not only the calibration but also the functional form of the algorithm used in the retrieval is important. It is also apparent from Fig. 5 that the C/S classification has a considerable impact on the VIHC estimation. In the case of linear algorithms (EV, AEV, and GSCAT), there is an improvement of about 16% in the coefficient of efficiency induced by the NN classification. Note that for linear algorithms our NN classification leads to better performance in comparison with that of PR classification. This is because the NN classification can be statistically more consistent with the structure of these empirical algorithms than can the PR classification. For the nonlinear algorithms (MLR and SI), the PR classification yields better results. In terms of bias, the algorithms are similar and it may be stated that, for long-term applications, the algorithms induce no bias. Note that the NN classification makes the performance of four of the algorithms almost similar. This performance similarity, which is somewhat unexpected given their markedly different formulations, suggests that the information derived from overland TMI observations might cause an objective indeterminacy (no unique solution may be associated with a set of observations). Nevertheless, in this validation exercise we could show that the MLR formulation can give improved accuracy with about 19% higher relative efficiency than the rest of the algorithms.

To investigate the influence of the training size on the results (and, implicitly, the results' statistical significance), we calculated the performance scores distinctively for each rain regime and various calibration sizes. The results are illustrated in Fig. 6. One may notice negligible dependence of the performance results on the calibration set size. The standard deviation slightly increases with respect to the set size, which is most likely an effect of the decrease of the validation set size and implicitly of the numerators in Eqs. (13) and (14). It is noted from Fig. 6 that the coefficient of efficiency in the stratiform regime is consistently higher than in the convective regime for all algorithms. It is also noted that the better performance of the MLR algorithm shown in Fig. 5 is caused by its superior performance in the convective regime.

Table 3 contains values of coefficients a₁, a₂, and a₃ described in section 4c. These values are obtained using a randomly chosen calibration set with size equal to 50% of the total dataset size. The resulting VIHC is in kilograms per meter squared, and the units of a₁, a₂, and a₃ are kilograms per meter squared per U unit, where U represents the units of corresponding descriptors. These units are kelvin for eigenvalues EV1 and EV2, kelvin for temperatures, and millimeters per hour for P_SI and P_GSCAT.

The same calibration/validation exercise is applied to rainfall estimation by replacing the VIHC variable with the rainfall rate. We noticed that the correlation between brightness temperatures and vertically averaged rainfall rate increases with respect to the depth of columns considered in averaging. Consequently, we chose to evaluate relationships between the TMI observations and the average rainfall rate in columns extending from ground to the freezing level. The TMI–rainfall rate algorithms examined have the functional forms of the VIHC algorithms. The reference rainfall rates used for algorithm calibration and validation are based on the TRMM 2A25 rainfall products, which are very similar to our estimates. The results for 50% calibration set size and using the NN C/S classification are shown in Fig. 7, which illustrates a significant difference between the performance of VIHC estimation and that of rainfall rate estimation. The coefficient of efficiency is about 29% lower in the case of rainfall rate estimation. This suggests that the VIHC is a variable more strongly related to the brightness temperatures than is the mean rainfall rate. An additional argument for considering the VIHC variable is that it can be more efficiently converted to rainfall accumulations than can instantaneous surface rainfall fluxes (Tsintikidis et al. 1999).

Effect of rain/no-rain discrimination on rain estimation

In addition to the performance scores presented in section 4, we conducted further evaluation to illustrate the impact of the NN rain discrimination scheme on rainfall estimation accuracy. Namely, we evaluated the MLR algorithm's estimation error (with respect to PR) for the following three scenarios: 1) both TMI-based discrimination and PR observations indicate rain, 2) the TMI discrimination predicts no rain while the PR detects rain, and 3) the TMI discrimination predicts rain while the PR detects no rain. Error histograms were determined distinctly for the three scenarios and for three different TMI rain discrimination schemes. One discrimination scheme is the NN procedure developed herein: the other two are adapted from Ferraro et al. (1998). In one of them, referred to as DS1, the raining pixels satisfy conditions T_22V − T_85V ≥ 3 K and T_85H < 253 K, and all other pixels are considered to be nonraining pixels. DS1 is the best discrimination scheme that we could formulate by nonautomatically considering combinations and variations of the schemes in Ferraro et al. (1998). In the second scheme, referred to as DS2, only condition T_85H < 253 K must be satisfied by the raining pixels. The error histograms are presented in Fig. 8. The relative frequency in the histograms is defined as the ratio of pixels falling in a bin to the total number of PR rain pixels. The dash- and dot–dash-line histograms, which correspond to scenarios 2 and 3, show the effect of rain discrimination error on estimation accuracy. It is apparent from Fig. 8 that the NN scheme yields much smaller errors than do DS1 and DS2. Exact figures are given in Table 4, which contains the relative error of total false rain and the relative error of total missed rain. The relative error of total false rain is determined as the ratio of the total amount of rain estimated from TMI, given that the discrimination erroneously predicted rain, to the total PR rain. Similar, the error of total missed rain is defined as the ratio of the total amount of rain estimated from TMI, given that the discrimination erroneously predicted no rain, to the total PR rain. Table 4 also shows the efficiency of the MLR algorithm, determined as specified in Eq. (13), applied with each of the three discrimination schemes. It is apparent that efficiency decreases relative to the no-discrimination-error case by about 9% for the first scheme, 18% for the second, and 30% for third. These numbers may be determined by comparison with the MLR efficiency value plotted in Fig. 5. It is concluded from these results that the application of the NN discrimination scheme is justified by the improvement of the overall rain estimation accuracy.