A Neural Networks–Based Fusion Technique to Estimate Half-Hourly Rainfall Estimates at 0.1° Resolution from Satellite Passive Microwave and Infrared Data

Satellite rainfall retrievals

Two main approaches have been used to obtain satellite rainfall retrievals. Some review articles have analyzed the capabilities and performance of visible/infrared (VIS/IR) algorithms (e.g., Levizzani et al. 2001). However, there is only an indirect relationship between cloud-top temperatures and actual rainfall. VIS/IR- based algorithms seem to work best for large spatial scales or long time series mainly because of a statistical smoothing effect. Another reason is the relationship between cold-cloud duration and long-term measured rainfall for certain climatic typologies. Despite reasonably good correlations, the physics of the rainfall retrieval problem and the dynamics of the frontal systems are overlooked altogether.

Conversely, PMW techniques provide more direct observation of the rainfall since it is the raindrops themselves that modify the upwelling radiation measured by the satellite. Nonprecipitating clouds are almost transparent at frequencies below 40 GHz, but at the same wavelength hydrometeors interfere with the background radiation. Since radiation scattering, absorption, and emission are related to the size and number of raindrops, a direct physical relationship—although complicated and affected by several impairments—can be established. However PMW techniques have poor spatial and temporal resolutions limited by technical requirements of the sensors and their low earth orbit (LEO) platforms. Spatial resolution cannot be improved because of the antenna diffraction limit related to the centimeter wavelength of the radiation itself, and temporal resolution can only be enhanced by using a constellation of satellites in low orbit. On the other hand, VIS/IR images are available at higher spatial and temporal resolutions. New sensors such as the new Meteosat Second-Generation (MSG) Spinning Enhanced Visible and Infrared Imager (SEVIRI) will provide 15-min, 12-band measurements with a resolution at subsatellite point varying from 1.4 (high-resolution VIS) to 4.8 km (IR) (Schmetz et al. 2002).

Data fusion approaches intend to take advantage of the strengths of both data types while avoiding their weaknesses as much as possible. The ultimate goal is to find a more direct measurement of the rainfall than that inferred from IR cloud-top brightness temperatures, while providing a spatial resolution one order of magnitude better than that of the PMW sensors at a much better temporal sampling.

A complete discussion on the advantages and disadvantages of VIS/IR and PMW rainfall retrieval algorithms can be found in Levizzani et al. (2001). PMW algorithms are either statistical–empirical algorithms or physical, radiative transfer equation–model based. Whereas the ultimate goal of rainfall estimation might be to propose a physical model, this task is hindered by two facts. First, because climate is a time-dependent phenomenon, changes must already be known to establish a deterministic physical model. Even more important, the nonlinear behavior of the variables involved prevents a deterministic modeling approach. Despite the importance of microphysical studies some statistical assumptions are required even for small-scale estimates around 1 km. Note that considering statistical methods versus deterministic ones does not mean lack of scientific content. On the contrary, it can be argued that when a large number of particles or variables are involved, the statistical approach is the only viable methodology. Jaynes' arguments on statistical mechanics (Jaynes 1990) can be effectively applied to the problem of rainfall estimation.

Retrieval validation issues

Results are validated over a Mediterranean region, a climatic scenario substantially different from that of the Tropics since neither the diurnal rainfall cycle nor the precipitation processes are comparable, and the accumulated rainfall amounts are completely different. The lower rainfall rates in the Mediterranean as compared with the convectively active Tropics imply less chance to encompass errors through spatial or temporal averaging. The general dominance of the frontal systems over convective processes represents a challenge for rainfall estimation since the ability to discriminate rainfall areas within satellite imagery is limited by the complexity of the intermixed phenomena (convective and stratiform) and the generally lower rain rates.

Monthly accumulated products are not necessarily superior with respect to instantaneous estimates. On the contrary, a loss of any intermediate data can result in a very biased estimation. For example, while a couple of lost satellite scenes does not represent a problem in the high-rainfall areas of the intertropical convergence zone (ITCZ), these could be precisely the images that reveal the weekly rainfall in a given cell over a lower- rainfall area in the Mediterranean basin. Consequently validation data and satellite imagery used in the fusion algorithms must be as continuous as possible.

A statistically based approach is used to fuse the PMW and IR data so that the strengths of each individual dataset are maintained and their weaknesses are at least partially corrected for (cf. Marzano et al. 1999). The use of gauge data was intended for the validation at several temporal and spatial scales. However, caution must be used while comparing these data. The large spatial variability of the rainfall rates implies that areal estimations from point measurements are difficult to estimate. Interpolation procedures always contain an unrealistic smoothing effect, even if the least possible bias is involved. Moreover, the relationship between rainfall rate and rainfall duration makes it difficult to covert the satellite instantaneous snapshots into accumulated 30- min rainfall. Intensity–duration–frequency (IDF) curves provide only a certain degree of accuracy and should be applied to large time and spatial scales. Therefore the bias associated with the 0–30-min transformation must be considered when the quality of validation results is assessed.

Passive microwave data

Based on the Electronically Scanning Microwave Radiometer (ESMR) aboard Nimbus-5 and -6, spacecraft that compose the Defense Meteorological Satellite Program (DMSP) carry aboard the enhanced SSM/I system. The first DMSP satellite (F8) with an SSM/I was launched in 1987. The latest DMSP spacecraft (F15) was launched in December 1999 and became operational in February 2000. The SSM/I sensor measures the earth's microwave emission and has a near-circular, sun- synchronous, near-polar orbit at an altitude of 860 km with an inclination of 98.8° and an orbital period of 102 min. This provides complete coverage of the earth, excluding two 2.4° circular sectors centered on the Poles. Spatial sampling resolution is 12.5 km pixel⁻¹ at 85 GHz and 25 km pixel⁻¹ at the lower frequencies. Calibration is done once each sensor scan (period 1.899 s) with a cold (3.1 K) and a warm (300 K) target. SSM/I data are obtained from the National Aeronautics and Space Administration's (NASA) Marshall Space Flight Center (MSFC).

Many rainfall-rate retrieval algorithms have been proposed (cf. Ferraro 1997; Smith et al. 1998). Here, a frequency difference algorithm described in (Ebert 1996) is used. Its physical basis is that the 85-GHz channel will be more greatly affected than the lower- frequency, 19-GHz channel because of scattering from precipitation-sized particles. Rain rates (RR) are generated through a lookup table derived from coincident measurements with TRMM PR. This empirical method aims to minimize the effects of surface temperature, emissivity, and atmospheric effects. Vertical polarization channels are used over the ocean, with the horizontal polarization used over land areas. Estimates over coastal regions are produced by a polarization-corrected temperature algorithm (see Kidd 1998).

Infrared data

Since the VIS images cannot be properly used during nighttime, only IR can provide a complete daily series. Globally merged, full-resolution (∼4 km), radiometrically corrected IR data from the Climate Prediction Center/National Centers for Environmental Prediction (NCEP)/National Weather Service (NWS) are used. These data are formed from the ∼11-μm IR channels aboard the Geostationary Meteorological Satellite-5 (GMS-5), GOES-8, GOES-10, Meteosat-7, and Meteosat-5 geostationary satellites. Details are provided by Janowiak et al. (2001). Global images are resampled to 0.036° resolution for the area of interest. This merged product and the location of the area in the Meteosat domain ensures a valid image every 30 min.

Rain gauge data

Gauge data were provided by the 89 agro-meteorological stations of the European Union (EU) Regional Government of Andalusia, Spain. Figure 1 shows the validation area with the spatial distribution of rain gauge data. Both the area covered (35°–39°N, 7.5°–1.5°W) and the 30-min temporal sampling of the measurements are deemed appropriate for validation purposes.

The interest in this region is twofold. First, the available data, which come from automatic stations, are quality controlled through preprocessing and comparison with independent measurements from the national rain gauge network. Complete station series may be omitted if deemed unreliable, thus ensuring the best possible reliability of the measured values. The second reason of interest is the location of the region, allowing a different kind of comparison than those available from past PMW validation campaigns such as TRMM's, operating in areas where the diurnal cycle is pronounced. The Andalusia climate can be summarized as Mediterranean: The average annual temperature is 16.8°C, while rainfall is 630 mm. However, the rainfall oscillation is pronounced: from 200 to 2000 mm with very variable instantaneous rainfall rates. From a physical point of view, the region can be divided into three main sectors: 1) the mountain areas including the Sierra Nevada (SE), with elevations up to 3478 m, and the Sierra de Grazalema- Ronda; 2) the Gualdalquivir river valley crossing the region SW–NE, and 3) the plains in the east. Each of these areas presents very different rainfall regimes with peak rainfall amounts in the SE.

Spatial interpolation procedures

Several interpolation methods were applied to the rain gauge point data to derive areal estimates. For cumulative values several methods have been proposed (e.g., Sivapalan and Blöschl 1998). Instantaneous values are usually estimated through interpolation procedures as described in a special issue of the Journal of Geographic Information and Decision Analysis (1998, Vol. 2, no. 1–2). Distance weight-averaged, bilinear and quadratic interpolation, various kriging models, and maximum entropy interpolation are compared in the search of the least-biased method, using different grid intervals. Eventually, maximum entropy interpolation method is used for 0.1° and 0.5° interpolations.

The maximum entropy method (MEM) was developed by Jaynes (Jaynes 1963, 1990). His Shannon's- based definition of information entropy (Shannon 1948) provides a criterion for setting up probability distributions without spurious assumptions. Epistemologically, the MEM presents some advantages in science, as has been reinforced by many researchers (in particular Shore and Johnson 1980). This method has been widely applied in physics, as when reconstructing the major concepts of the classical statistical mechanics from a purely Shannon information definition start. Several methods have been derived from Jaynes' ingenious findings and have been applied in signal processing (Skilling and Bryan 1984), image restoration (Noll 1997), and satellite imagery data fusion (Tapiador and Casanova 2002). Christakos (1991) presented a method to apply the formalism in geospatial interpolation. This procedure is followed here in the Lee and Ellis (1997) development.

The MEM provides an objective procedure to find the least-biased probabilistic distribution of a univariate or multivariate random function. It can be demonstrated that this distribution extracts the maximum knowledge out of the problem (maximum entropy principle). The formalism is as follows. The information contained in a probabilistic distribution or random variable R(x, y) is consistently measured by their entropy function:

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The MEM states that the most unbiased probabilistic distribution yields from the maximization of S subject to the set of r + 1 constraints:

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where g_k(x, y) are linear moments with respect to the R(x, y) vector with expected values c_k. We apply here the discrete formalism without loss of generality. However, some discussion remains on the continuous formalism general-case theoretical base.

To maximize the entropy S subject to the constraints, a variational approach is normally used, solving the system through Lagrangian multipliers. In its simplest, nontrivial two-constraints form,

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Then solving (for x) yields

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This results in a nonlinear system of equations that must be solved by numerical methods. Alternatively, a dual problem can be defined minimizing the a posteriori uncertainty through a maximum entropy estimate z*. After some calculations, the necessary condition for a minimum results

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Because the Shannon's information function is convex, this condition is also sufficient. In the general case, this results in a set of equations that need to be solved through numerical methods. Nevertheless, if the prior information (the constraints) consists of the two first moments of the distribution, the joint distribution characterized by the maximum entropy principle is then Gaussian, as was shown by Kapur and Kesavan (1992). Thus

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with Σ the covariance matrix and a_ij the ijth value of the Σ⁻¹ matrix (Lee and Ellis 1997). This allows a practical implementation of the MEM. Equation (6) is then solved, which yields

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This can then be calculated easily using a simple kriging algorithm (Lee 1998). It can be proved that the second moments are also equivalent. Empirical tests between the actual calculation of the MEM and this approach result in high correlation values (>0.99).

Neural network model analysis

Several NN models were tested in the preliminary stage of the present work. The selection of the model was not a blind process but a conscious and analytical quest. Following Anders and Korn (1999), a statistical analysis was performed with the data in order to find the most appropriate structure for the net through hypothesis testing, information criteria, and cross-validation methods. In particular, the information criteria model selection was followed once the most appropriate architecture was chosen, that is, a multilayer perceptron (MLP) NN. Model selection can be guided by previous statistical modeling, which provides a conscious NN architecture selection. The procedure followed here was to use the network information criterion (NIC; Murata et al. 1994) with a previous transfer function linear transformation through Taylor series expansion. NIC is defined as

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with 𝗔 ≡ −E[∇² logL_i], 𝗕 ≡ E[∇ logL_i∇ logL^′_i], n the number of observations, ŵ the set of parameters, ∇ the gradient, ∇² the Hessian, and log L the estimated maximum log likelihood. This criterion is not justifiable, however, for overparameterized nets whose limiting distributions are mixed Gaussian. However, a Taylor series expansion of an additional hidden unit transfer function reduces the hidden unit to linear terms and makes it possible to use the NIC.

The method is as follows. First, the number of hidden units is decided. Then, the Taylor expansion for the additional unit and the principal components are calculated and regressed with the residuals of the models. The next step is to calculate the NIC, accepting the additional component if it improves the overall performances. In a second stage, this fully connected net is pruned, extracting subsets and comparing with the NIC estimator again. After this process, Monte Carlo simulations show that the results increase the classification capabilities.

To determine the MLP choice, several models were previously tested. Also, some adaptive resonance theory (ART) nets (Carpenter et al. 1989) were tested because of their fast learning ability. ART1 (adapted to the input dataset), ART2, ARTMAP, distributed ARTMAP, and fuzzy ARTMAP were generated, trained using the 1800 UTC 10 December 2001 SSM/I image and tested with a different SSM/I–IR pair (0630 UTC 10 December 2001).

ART architectures develop the Grossberg (1969) adaptative resonance theory. The idea behind this paradigm is to develop a procedure to solve the false encoding problem, using two fields of neurons. One F^x field acts as “sensorial cells” and the other F^y acts as “high level” processing units. The inclusion of a “trigger” ability to inhibit or disinhibit false patterns represents an improvement over the MLP approach, which is unable to distinguish false patterns that are actually used in the training process. ART nets present good behavior in near-real-time processes. However, this feature is not the major reason for interest in using the ART paradigm in rainfall data fusion—rather it is its ability to identify spatial patterns. This was the primary use of ART models in their original image processing– related applications.

Because of the ART model sensitivity to the order in which the data are presented to the net, a voting procedure is required. Carpenter et al. (1997) themselves acknowledge the need for this additional feature in their model when applied to remote sensing data. It is empirically shown in this study that for one particular case no less than 30 different arrangements were needed to obtain any results.

Several approaches have been described to overcome this problem (Carpenter et al. 1997; Dagher et al. 1999). The algorithm suggested by Dagher et al. (1999) is applied to select the pattern order presentation within a rational framework, reaching stationary values around a large value of 1000 orderings, slowing down the learning process. Nevertheless, this procedure provides a rational, straightforward, easy-to-compute method to present the patterns to the net in the training phase. The algorithm consists of four stages. First, for each training pattern x = (x₁, … , x_k, x_k+1, … , x_n),

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is computed. Second, the training pattern that maximizes Eq. (10) (say x₀) to the NN is presented, and then x₀ is removed from the training data. The next c patterns are chosen as follows. The Euclidean distance between x₀ and the training patterns is calculated, and the minimum x_j is selected. This is iterated using x₀ = x_j until j = c, where c is the number of classes desired plus one. This generates a set of c I_c cluster centers, whose maximum value is computed. The corresponding training pattern is the next to be presented to the net. Once used, the pattern is removed and the calculation of the cluster centers step iterates.

The third stage consists of the selection for the next n − c parameters computing the Euclidean distance of the remaining patterns to the cluster centers and selecting the minimum of these distance patterns as the next input. Then, the pattern is removed and the process iterated until all the input patterns have been used.

It can be proved that the generated ordering is independent of the one associated with the original training set (Dagher et al. 1999). Experimental work of the authors also proves that the generalization performance is increased. Table 1 shows the results of this selection process. Note that, despite good behavior in the training phase, ART models fail when applied to new patterns.

Architecture selection

The chosen architecture is a simple MLP feed-forward model with 11 input neurons, 23 and 7 units in two hidden layers, and a unique output neuron. From a mathematical point of view it has been argued that no more than two hidden layers are required for any kind of data, and only one if the function to be modeled is revealed as continuous (Cybenko 1989). However, practical issues such as training speed can suggest other decisions. In this study, after the model selection an empirical test confirms that no further improvement was obtained by increasing the number of hidden layers. Sigmoidal functions are used in all the neurons but with different parameterizations. Also, a binary output in 100 classes has proved to be worse than the chosen one for increasing the efficiency of the net.

Inputs selection

The inputs of the net have been selected bearing in mind the precipitation processes in the area of study. Only the IR band was used, enabling generation of daytime and nighttime estimates: further research must be done using more IR bands to help in identifying cirrus, cloud, fog, and mist, which are sometimes mistaken for rain areas.

The first six inputs consist of the current and the five previous brightness temperature values for a particular pixel. This is expected to model not only the convective and orographic processes but also the frontal ones in the southern Mediterranean area. Each rainfall process generates a very different signature or sequence of digital numbers in the same pixel, but it was expected that the training process would be able to recognize them when associated with a certain amount of rain and additional spatial information. The next two inputs are the average over a 3 × 3 and 5 × 5 neighborhood of the pixel. This minimizes the geolocation and synchrony problems of both datasets. Second, it provides valuable texture information about the cloud cover (Bellerby et al. 2000). Standard deviations of the current value with the 3 × 3 and 5 × 5 windows were taken as the next two inputs. Last, the number of pixels whose associated temperature is lower than 225 K in the 3 × 3 slot was also included. This value is intended to provide information about the heaviest rain areas. A previous empirical test showed that the 225-K threshold was more appropriate to this area than the more usual 235-K value recommended in the literature. This variability of the threshold has been pointed out by several authors. Todd et al. (1995) and Todd and Washington (1999) showed that optimum IR threshold values (calculated by comparison with rain gauges) over East Africa are highly variable in time and space, as a result in local-scale rainfall/cloud characteristics. Such optimized IR threshold fields were shown to identify rainfall events more accurately than a fixed 235-K threshold. Xu et al. (1999) also demonstrated the value of local calibration of IR thresholds (using PMW satellite on a monthly basis) over the Japanese islands.

The time of the day input was considered as invalid early, since the NN can become coupled with the value itself. The reasonable performances of models using this input can be attributable to strong daily rainfall cycles such as in the Tropics where the net could learn the daily convective succession oblivious of the other inputs. Table 2 summarizes the inputs used. The output corresponds to the estimated SSM/I instantaneous rainfall rate (in tenths of millimeters). All absolute values were normalized to [0, 1].

Operative data processing

The procedure followed to generate the 30-min estimates is to train the net with every valid SSM/I overpass. The temporal sampling of these satellites allows typically twice-daily imagery of the area under study. This means that at least two nets are generated, trained, and applied each day, but because of the region's climate this produces various scenarios. In this Mediterranean area, only a single coincident image containing a rainfall pair is usually found over a long period of time. In this case, the closest nets are used to generate the estimates. This represents an additional uncertainty that is reflected in the final results.

The learning rates in the training phase were set to 0.1 for the first 10 000 iterations and to 0.05 for the next 10 000. Several selected cases showed that the mean-square error (MSE) of the net was always less than 0.002 within the 20 000-iterations limit. An independent validation dataset was also employed to ensure the generalization properties of the trained net. The overtraining effect was avoided by using these complementary data.

Inversion procedure

An inversion procedure was followed to analyze the influence of the inputs on the final result. Input selection cannot be parameterized in the same way as the NN model is. The physics of the problem relies on a rational choice of the factors that can have an a priori relationship with the instantaneous rainfall rate. Nevertheless, once the net is trained some algorithms have been proposed to uncover the almost black-box procedure that the training process represents. The NN inversion procedures seek to find one or more input values that produce a desired output response for a fixed set of synaptic weights (Jensen et al. 1999). The basic idea is to analyze the kind of input vector that generates a specific output. This could be useful both for the input selection process and for the analysis of the process being modeled.

The inversion of neural networks seeks to find the input vector that corresponds with a desired output. Once the training dynamic process has been completed, an NN is nothing but a matrix of weights that best fits the inputs to the outputs. If we consider an NN as a function f that gives a desired output y from some inputs x,

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where H and I denote the hidden and input layers, respectively, and g is the transfer function. Once the NN has been trained, the matrix 𝗪 becomes fixed. To retrieve an estimated input from a given output o*, a cost error function must be minimized:

exfxo^α(12)

In general, the solution is a set of k-uple on hypersurfaces, so that there are many solutions to a given output. In some NN applications this can be a problem, but in our case this is in agreement with our hypotheses on rainfall processes.

Several approaches have been proposed to solve the optimization problem. Here the Williams (1986) and Linden and Kindermann (1989) (WLK) method has been used. This method uses the gradient descent as does the back-propagation algorithm (Linderman and Linden 1990). Given an initial x, values are updated following

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where η is the back-propagation learning rate parameter, i is the iteration, x_k is the kth component of x, and e is the error function. To the described net, we solve in reverse order

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where the derivative delta δ_j is

δ_jo_jo_jo^*_j(15)

if the neuron belongs to the output layer O, and

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if it belongs to the input layer I or the hidden one H.

In this work, a previous, more complex model was built, containing two more IR images, the 3 × 3 and 5 × 5 averages, and standard deviations for the eight IR images and the coldest pixels in the 5 × 5 window. The inversion procedures demonstrated the redundancy of these inputs in this particular case. Moreover, the 5 × 5 windows were interpreted as a source of noise in the training phase for a problem in which contradictory exemplars are common. Note that despite the high correlation between the 3 × 3 and 5 × 5 t₀ inputs (Table 3), these were judged necessary by the inversion algorithm.

A good example of the difficulties in rainfall merging techniques and the utility of using NN is the correlation estimation by using some models. As can be seen in Table 4, none of the analyzed models are capable of offering any significant correlation between the IR-derived inputs and rainfall measured by the SSM/I.