Abstract

1. Introduction.

DMSP satellites have substantially increased the amount of real-time meteorological data that is acquired over the oceans, which are used subjectively by marine meteorologists to improve ocean surface weather map analyses, and objectively by numerical analysis systems to provide initial conditions for numerical weather prediction models. With three satellites in orbit (F11, F13 and F14) and with a swath width of about 1400 km for each of the satellites, high-resolution coverage is now available almost globally on a daily basis.

Empirical retrieval algorithms (or transfer functions) have been developed separately for various geophysical parameters such as surface wind speed (Goodberlet et al. 1989; Petty 1993) (see also Appendix A), columnar water vapor (Alishouse et al. 1990) and columnar liquid water (Weng and Grody 1994). The empirical retrieval algorithm is usually derived from a high-quality data set that collocates the satellite brightness temperatures with buoy- and/or radiosonde-measured geophysical variables in time and space. The physically based algorithms use a large amount of such empirical data for parametrizations (Wentz 1997). The collocated matchup data set requires a large data sample in order to represent a wide range of global meteorological events. High wind speed events have been fairly rare in most matchup data sets because winds speeds of gale force (> 17 m/s) or greater at a given time cover no more than 5 % of the global ocean surface.

Some of the initially developed retrieval algorithms are based on a simple statistical technique such as linear regression and, as a result, have limited retrieval capabilities. Careful validations and evaluations of the retrievals over a period of time are required to sample a wide range of meteorological conditions and to determine the conditions for which the transfer functions do not perform well. Such validations invariably show that the initial algorithms have serious limitations in providing good quality data over regions where weather conditions are rapidly changing. Hence the necessity to examine the possibility of making improvements to the retrieval algorithm arises (Gemmill et al. 1996).

The purpose of this paper is to address the application of improved SSM/I multi-parameter satellite retrievals that are now available at NCEP due to using the latest SSM/I neural network algorithm and data from three SSM/I sensors in operational weather analysis and forecasting. This algorithm provides detailed and accurate fields of meteorological variables over the oceans and the coverage is extensive because of the number of satellites that are currently in operation. These fields can be seen at: http://polar.ncep.noaa.gov/marine.meteorology/marine.winds/. The new neural network algorithm derives surface wind speed (W), columnar water vapor (V), columnar liquid water (L) and sea surface temperature (SST) simultaneously from SSM/I brightness temperatures. Although these parameters have already been retrieved separately by other techniques, it is the simultaneous retrieval by the neural networks (NNs) that is unique, allowing the information from one parameter to contribute to a better estimate of the other parameters. The parameters retrieved by the NN, when analyzed together, can provide information about synoptic weather patterns over the oceans (Gemmill and Krasnopolsky 1998) that is more comprehensive and internally consistent than that from a single parameter (see also Appendix A).

In Section 2 we briefly review recent works on improvement of SSM/I retrievals. In Section 3 we discuss possible approaches to interpreting the data fields derived from SSM/I and presents several examples which show that significant meteorological features like fronts, convective areas and areas with high probability of precipitation can be identified and observed in SSM/I fields retrieved by the new algorithm. In Appendix A we review prior works on SSM/I algorithm development. In Appendix B some background NN theory which is relevant for SSM/I algorithm development is introduced.

2. Work on Improvement of Accuracy of SSM/I Retrievals

                                                                                g = f (T)                                                                                     (1)

where f is usually called a transfer function. In the case of SSM/I the transfer function f is essentially nonlinear especially when the amount of moisture in the atmosphere is significant (Petty 1993, Stogryn et al. 1994). Each particular retrieval algorithm corresponds to a particular choice of geophysical parameters to retrieve (vector g), brightness temperatures to use (vector T), a mathematical (statistical) model for the transfer function f (see Appendix A for details), and a development data set.

Most previous SSM/I retrieval algorithms (Appendix A) retrieve one variable at a time. The original global algorithm for retrieving ocean surface wind speed from SSM/I was developed by Goodberlet et al. (1989) (GSW algorithm). This algorithm is based on linear regression and is primarily limited to low moisture conditions. Further, there were only a few wind speed observations in the high range (>18 m/s) available in the matchup data set used in the formulation of the algorithm, so the GSW algorithm could not be expected to perform well at retrieving high winds. Because of these limitations, wind speeds cannot be accurately determined with this algorithm in areas with significant levels of atmospheric moisture (e.g., in large parts of the tropics) and cannot be retrieved in the vicinity of storms and fronts. Petty (1993) introduced a nonlinear correction to the GSW algorithm (GSWP algorithm) which improves the accuracy of the wind speed retrievals in areas with higher amounts of the water vapor (in much of the tropics for example). Recently (October 1997) this version of the algorithm became operational via the shared data processing center at Fleet Numerical Meteorological and Oceanographic Center.

Several algorithms have been developed to retrieve columnar water vapor (Alishouse et al. 1990; Petty 1993) and columnar liquid water (Weng and Grody 1994; Weng et al. 1997). However, all these algorithms (including wind speed algorithms) have been developed independently using different data sets. They were formulated without taking into account co-dependency of these parameters and without accounting for the physical relationships among the parameters.

For the past five years, NCEP has concentrated on improving the accuracy of SSM/I satellite derived ocean surface wind speeds, columnar water vapor, and columnar liquid water for both marine meteorology applications and numerical weather prediction (NWP). A series of algorithms (see Fig. 1) has been formulated using neural networks (NNs), each one more complex and accurate than the previous one. A brief review of NN theory as related to our topic is presented in Appendix B. NNs were chosen because they have been highly successful in meteorological and oceanographic applications (Hsieh and Tang 1998) and in resolving complex non-linear relationships between the sensor output and the atmospheric variable of interest (Thiria et al. 1993, Stogryn et al. 1994, Jung et al. 1998). Hence, they were able to provide an effective method for dealing with high moisture conditions while deriving wind speeds. In 1994 , an initial NN algorithm (OMBNN1) was formulated (Krasnopolsky et al. 1995a), using the same matchup data base of SSM/I brightness temperatures (from the F8 satellite) with buoy wind speeds that was used to develop the GSW algorithm. The OMBNN1 algorithm used brightness temperature from four of the SSM/I channels to produce one output, wind speed. That initial study showed that OMBNN1 was capable of providing ocean surface wind speeds from SSM/I brightness temperatures with better accuracy, and in areas with higher levels of atmospheric moisture, than the GSW algorithm. But when the OMBNN1 algorithm was applied to global SSM/I data for operational use, the algorithm was unable to provide high wind speeds ( > 15 m/s) with acceptable accuracy. Many wind speed retrieval algorithms have problems with retrieving high wind speeds (Boutin and Etcheto 1996). This problem is usually attributed to the lack of high winds in the matchup data.

In order to improve accuracy in the high wind speed range, a new neural network, OMBNN2 (Krasnopolsky et al. 1995b), was developed by emphasizing the few high winds that were in the original data set, and including a bias correction. Also, the OMBNN2 algorithm used brightness temperatures from five of the SSM/I channels to produce one output, wind speed. The OMBNN2 algorithm did improve the ocean surface wind speeds compared to OMBNN1, but still could not produce higher wind speeds with the desired accuracy. Also, the OMBNN2 algorithm incorporates a bias correction which is sensor dependent. Nevertheless, wind speed fields produced by OMBNN2 were able to show high-resolution structure of the wind speed patterns over the ocean normally not observed in conventional surface data sets. The output from the OMBNN2 algorithm has been evaluated within the NCEP global prediction model (Yu et al. 1997) to determine its impact. The evaluation showed that the wind speed analyses and forecasts based on the OMBNN2 neural network ocean surface wind speed data were better than those based on the GSW algorithm when compared to buoy data.

More recently, a rather comprehensive SSM/I and buoy matchup data set was provided by the Naval Research Laboratory (NRL) for algorithm development. The NRL data set contains more data and has better coverage of high wind events than the previous data set used by GSW. Further, other high latitude SSM/I ocean weather ship matchup data sets were obtained from Bristol University (D Kilham, personal communication). The NN was retrained with the new wind speed data for one parameter (wind speed only) retrievals, but large errors at high wind speeds still occurred.

Hence, a new NN architecture was formulated (Fig. 1) which takes into account the interdependence of physically-related atmospheric and oceanic parameters (wind speed, columnar water vapor, columnar liquid water and sea surface temperature). The new OMBNN3 algorithm (Krasnopolsky et al. 1998a,b, 1997, 1996) utilizes five SSM/I brightness temperature channels. It simultaneously produces all four parameters. This algorithm was trained to preserve proper physical relationships among these parameters. The algorithm has extended the range of wind speeds over which useful retrievals can be obtained. It not only improves the accuracy of the wind speed retrievals, especially at high wind speeds (without bias correction), but makes available three additional fields. Table 1 indicates the importance of the inclusion of water vapor, liquid water, and SST in retrieval algorithms on the accuracy of wind speed. The GSW is the original linear algorithm, and the GSWP algorithm contains the water vapor correction suggested by Petty (1993). The RMS error statistics of the OMBNN3 algorithm, which takes into account also the liquid water and SST influences, are lower than those of the GSWP algorithm over all wind speeds, and especially at wind speeds > 15 m/s.

Table 1. Comparison of bias, algorithm RMS error (sensor noise and matchup uncertainties are removed), total RMS error and high wind speed (W>15m/s) RMS error, for buoy wind speed vs. SSM/I wind speed, and for five SSM/I wind speed algorithms, with all errors in m/s. Errors calculated over more than 15,000 buoy/SSM/I matchups. Numbers outside the parentheses correspond to clear and in the parentheses to clear+cloudy conditions.

Algorithm	Bias	Algorithm RMSE	Total RMSE	W > 15 m/s RMSE
GSW1	-0.2 (-0.5)	1.4 (1.8)	1.8 (2.1)	(2.7)
GSWP2	-0.1 (-0.3)	1.3 (1.6)	1.7 (1.9)	(2.6)
GS3	0.5 (0.7)	1.4 (2.2)	1.8 (2.5)	(2.7)
Physically-Based4	0.1 (-0.1)	1.3 (1.8)	1.7 (2.1)	(2.6)
OMBNN35	-0.1 (-0.2)	1.0 (1.3)	1.5 (1.7)	(2.3)

¹Multiple Linear Regression (Goodberlet et al. 1989)
²Multiple Linear Regression with nonlinear water vapor correction (Petty 1993)
³Nonlinear Regression (Goodberlet and Swift 1992^)
4Inversion of a physically-based forward model (Wentz 1997)
⁵Neural Network (Krasnopolsky et al. 1996)

Fig. 2 shows the wind speed difference (buoy minus satellite wind speeds) characteristics (retrieval errors in wind speed) as functions of three other parameters: columnar water vapor, columnar liquid water and sea surface temperature for three algorithms GSW, GSWP and OMBNN3. Including nonlinear water vapor correction in GSWP reduced the bias and its dependence on the water vapor concentration (and partly on SST which is closely related to water vapor); however, it did not reduce its dependence on the liquid water concentration. This correction also did not significantly improve the standard deviation of the differences. The OMBNN3 algorithm, with its simultaneous multi-parameter retrievals, reduced the bias and its dependence on all three other parameters together with a significant improving the standard deviation of the differences.

NN retrievals for columnar water vapor and columnar liquid water are in good agreement with existing SSM/I algorithms. No attempt was made to verify these retrievals against observed data because of the lack of collocated observations. The details of the development of OMBNN3 and its validation have been documented by Krasnopolsky et al. (1998a,b, 1997, 1996). The accuracy of the SST retrievals is lower than the accuracy of high resolution AVHRR SST; however, the retrieved SSTs give secondary information that improves the accuracy of the wind speed retrievals, especially at high wind speeds. Because this algorithm is inherently nonlinear, it increases areal coverage in areas with significant levels of atmospheric moisture and under more active and critical weather systems such as storms and fronts.

Comparison studies of the impact of ocean surface wind retrievals on the Global Data Assimilation System (GDAS) at NCEP showed a more positive impact using retrievals from OMBNN2 algorithm than from the GSW algorithm (Yu et al. 1997). In the meanwhile, the OMBNN3 algorithm was developed, and the wind speed retrievals from the OMBNN3 algorithm were shown to be better than retrievals from the OMBNN2 algorithm, especially for high winds. Impact studies similar to those performed for OMBNN2 have been performed using the OMBNN3 algorithm and further positive impact was observed as compared to both GSW and OMBNN2 (Yu 1998). Based on these results the OMBNN3 ocean surface wind speed retrievals were incorporated into the operational GDAS at NCEP in April 1998.

3. Interpretation of SSM/I Neural Network Derived Data for Weather Analysis

The ocean surface wind speed data have the most direct use in marine weather analysis and weather forecasting. Although these data provide wind speed only, the extensive coverage of the three satellites depicts high-resolution wind speed patterns across synoptic weather systems. These data can be used directly to improve ocean surface wind analyses, and indirectly to improve sea level pressure analyses.

The columnar water vapor and columnar liquid water are the vertically integrated values through the entire atmosphere. The columnar water vapor is also known as "total precipitable water", which is the depth of water that would fall on the ocean if all the water vapor were condensed and precipitated. Columnar water vapor is an air mass characteristic closely related to synoptic scale features. Its primary source is the warmer waters of the tropics, and it is advected to higher latitudes by storms and low- and mid-level jet streams. As a result, regions with large gradients of columnar water vapor have been shown to be good objective indicators of the position of an ocean surface front (Katsaros et al. 1989) .

The liquid water resides in clouds, and is more directly related to regions of precipitation and to active weather systems such as storms and fronts (McMurdie and Katsaros, 1996). Large liquid water amounts are generally associated with strong convective activity (cumulus clouds) and turbulent surface weather conditions, whereas small amounts of liquid water are associated with near neutral or stable regions (stratiform clouds) and constant or steady surface weather conditions.

OMBNN3 has been extensively evaluated at NCEP using real-time data from F10, F11, F13 and F14 SSM/I instruments. Simultaneous retrievals of wind speed, columnar water vapor and columnar liquid water fields using OMBNN3 were examined to reveal significant information concerning weather patterns over the ocean. Two examples of such a consideration are presented below.

a. Some Examples

Each example includes a series of plots (Figs 4 and 6): (a) SSM/I ocean surface winds speed, (b) SSM/I columnar water vapor, (c) SSM/I columnar liquid water, (d) ocean surface wind

data from buoys and ship, (e) ERS2 scatterometer wind vector data⁽³⁾, and (f) the sea level pressure analysis available from the NCEP Global Data Analysis System. The plots of satellite data are within a +/- 3 hour time window about the analysis time. The SSM/I data are a composite from three DMSP satellites (F11, F13 and F14), which together provide almost complete and extensive regional coverage. The two cases represent synoptic weather patterns that were well analyzed from other real-time data sources, so there are no discrepancies between the data sets in terms of the meteorology. Their purpose is to show that the variables retrieved from the SSM/I through the NN algorithm provided information consistent with the actual weather situation. These examples demonstrate that neural networks have the capability to retrieve useful meteorological information from SSM/I brightness temperatures.

1) EXAMPLE 1. EASTERN NORTH PACIFIC OCEAN, 12 MARCH 1998, 06UTC.

Figure 3. Surface marine weather map for eastern North             Figur e 4. (a) SSM/I derived ocean surface wind speed data, (b) SSM/I derived columnar
  Pacific Ocean, 12 March 1998, 06 UTC                                   liquid water data, (c) SSM/I columnar water vapor data, (d) buoy and ship wind data,
                                                                                                  (e) ERS2 scatterometer wind vector data and (f) sea level pressure analysis from global analysis
                                                                                                  system over eastern North Pacific Ocean, 12 March 1998, 06UTC. Each panel covers an area
                                                                                                  from 15 to 65 N and from 165 to 115 W.

The sea level pressure analysis from the global model at 06 UTC shows little difference from the MPC analysis. The MPC analysis has the storm slightly deeper by 4 mb than the global model analysis (Fig. 4f).

The SSM/I wind field (Fig. 4a) shows the storm to be fairly circular and moderate in size. The yellow region shows the outer limit of the 20 knot winds, and the orange region shows the gale force region southwest of the center and 40 knot winds near the center. Due to high moisture content that makes retrievals impossible, and due to an occasional bad scan line, there are areas without wind speed data. The northward moving occluded front is associated with a band of high wind speeds (30 knots). Just ahead of the eastward moving cold front there is a wind band with winds up to 25-30 knots, but the strongest winds are masked due to high moisture contamination (possible rain). The weak low south of Alaska has winds near 30 knots along the coast. The storm further west is already generating 40-45 knot winds ahead of the occlusion.

The liquid water (Fig. 4b) shows a wrap-around pattern along the cold front and then along the occlusion into the center. The greatest liquid water values quite likely are associated with rain areas precede the occlusion and cold front. Drier air is indicated behind the fronts. The water vapor (Fig. 4c) shows the distinct pattern associated with the air masses. The large water vapor gradient zone recognized as a quantitative parameter for the location of oceanic fronts by Katsoras et al. (1989) clearly depicts the location of the cold front. The water vapor shows a strong flow of moist air moving from north of Hawaii to the U.S. northwest coast.

The in-situ buoy and ship wind data (Fig. 4d) are plotted four times a day at the standard synoptic times of 00, 06, 12 and 18 UTC. The satellite data are taken within three hours of the surface ship and buoy data. Although the winds show the circulation associated with a storm, the intensity and location of the storm center can be not determined from the ship and buoy data alone. Determination of those values is aided by the SSM/I derived data, and where there are in-situ surface wind reports, they corroborate the values of the SSM/I derived wind speed data. Likewise, the ERS2 scatterometer wind data (Fig. 4e) and SSM/I wind data are in close agreement.

2) EXAMPLE 2. WESTERN NORTH ATLANTIC OCEAN, 25 FEBRUARY 1998, 12UTC

Figure 5. Surface marine weather map for western                          Figure 6. (a) SSM/I derived ocean surface wind speed data, (b) SSM/I derived columnar
North Atlantic Ocean, 25 February 1998, 12 UTC.                         liquid water data, (c) SSM/I columnar water vapor data, (d) ship and buoy wind data,
                                                                                                       (e) ERS2 scatterometer wind vector data and (f) sea level pressure analysis from global 
                                                                                                       analysis system over western North Atlantic Ocean, 25 February 1998, 12 UTC.
                                                                                                       Each panel covers an area from 15 to 65 N and from 82 to 32 W.
                                                                                                           

The SSM/I wind field (Fig. 6a) shows the storm to be fairly circular. The yellow region shows the outer limit of the 20 knot winds, and the orange region shows the gale force winds around the center of the system with speeds to 45 knots. The SSM/I data indicate 50 knots winds for the Greenland storm. SSM/I data also indicate a band of 35 knot winds on the western side of the weak system in the central Atlantic.

The liquid water values (Fig. 6b) are greatest to the southwest of the storm center, east of Cape Hatteras and south of Cape Cod, associated with a trough line crossing the area east of the Gulf Stream.

The water vapor (Fig. 6c) shows the distinct air mass pattern, but does not identify the storm system very well. However, the associated fronts are clearly identified, especially by the large water vapor gradient across the southwest portion of the figure. The water vapor shows a strong flow of moist air moving from the Caribbean north into the eastern portion of the storm.

The surface wind data reports (Fig. 6d) and the corresponding satellite wind data are in close agreement. Note the region near 42 N between 50 W - 60 W, where the surface wind speed data approach 50 knots. The satellite wind speeds in that region are only slightly lower, about 45 knots. Also, note the ship report of 60 knots east of Greenland. In that area the satellite data indicate wind speeds of 50 knots. Likewise, the ERS2 scatterometer winds (Fig. 6e) and SSM/I winds are in close agreement. The wind speeds and pattern are similar.

4. Summary

The OMBNN3 algorithm retrieves wind speed, columnar water vapor, and columnar liquid water signals contained in the SSM/I brightness temperatures, with accuracies which are operationally useful. Also, multi-parameter retrievals preserve the correct physical relationships among the retrieved parameters.

The algorithm generates high wind speeds (>15 m/s) in areas where such winds are well supported by other data and are expected from sea level pressure analyses.

The algorithm generates columnar water vapor patterns which are able to delineate and characterize air masses. Low values are associated with air masses originating in high latitudes that are cold and dry. High values are associated with air originating in tropical areas that are warm and moist. High gradients of the columnar water vapor are related to the position of ocean surface fronts.

The algorithm generates columnar liquid water patterns which are related to regions of water vapor convergence, resulting in clouds, which are closely associated with cyclones and active frontal location.

These data can be used to improve the interpretation of the weather at the ocean surface. These fields can be seen at: http://polar.ncep.noaa.gov/marine.meteorology/marine.winds/ every day. The OMBNN3 algorithm was incorporated into the GDAS at NCEP on April 7, 1998.

Acknowledgments

                            W_GSW = 147.9 + 1.0969 T19V - 0.4555 T22V - 1.76 T37V + 0.786 T37H                                (A.1)
 

The GS algorithm developed by Goodberlet and Swift (1992) attempts to improve the performance of the GSW algorithm, using nonlinear regression with a nonlinear approximation of function f of the following form:

where W_GSW is given by (A.1). Since the nature of the nonlinearity of the SSM/I TF is not known precisely, application of this nonlinear regression may not improve the results. Because (A.2) has a singularity at = 1 or equivalently at |T37V - T37H| = 30.7 K, when BTs fall close to this pole, this algorithm generates spurious high wind speeds. The authors do not recommend using this algorithm when |T37V - T37H| < 40. K (Goodberlet and Swift 1992); however, this limitation is not based on physical principles, but rather it is caused by an improper choice of nonlinear regression function.

The nonlinear approximation introduced by Petty (1993) is another type of regression algorithm. Nonlinear functions introduced in the linear regression in this case represent the nonlinear behavior of the transfer function (1) much better:

                                                                   (A.3)

Here again, W_GSW is given by (A.1), -2.13 is a bias correction, and 0.2198 V - 0.4008×10^-2V² is a nonlinear function which corrects the one-parameter linear TF (A.1) for water vapor.

Single-parameter NN algorithms have been introduced as an alternative to nonlinear regression (e.g., A.2) because they can model the nonlinear behavior of TFs without specifying a particular type of nonlinearity a priori. The NN algorithms developed by Stogryn et al. (1994) and Krasnopolsky et al. (1995a) (OMBNN1) have identical architectures which are shown in Fig. 1. The OMBNN1 algorithm is represented by expression (B.3, Appendix B) where n = 4 (inputs - T19V, T22V, T37V, T37H), m =1 (output - W) and k = 2 (hidden nodes). An improved for high wind speeds single-parameter NN algorithm was developed by Krasnopolsky et al. (1995b), the OMBNN2 algorithm, has an architecture shown in Fig. 1. A new method of NN training which enhances learning at high wind speeds by using a weighting schema which is inversely proportional to the wind speed probability distribution was used. The OMBNN2 algorithm is represented by expression (B.3) where n = 5 (inputs - T19V, T22V, T37V, T37H, T85V), m =1 (output - W) and k = 2 (hidden nodes).

Single-parameter algorithms discussed above retrieve a single geophysical parameter g_i (e.g., surface wind speed), using SSM/I BTs, without regard to any information about other geophysical parameters which are related to, or correlated with, g_i. In this case, the signatures (contributions to BT) of these related parameters, if they are not properly taken into account, act as additional noise (pseudo noise) in the BT signal. As a result, first, useful information about related geophysical parameters contained in these signatures is lost, and second, this pseudo noise causes additional errors in the parameter of primary interest, g_i (e.g., wind speed). If the data set used for algorithm development spans the full dynamical range of observed wind speeds, water vapor, etc., these additional errors will not contribute to the bias, but only to the scatter. However, for high values of wind speed, atmospheric moisture, etc., where the data are sparse, a single-parameter algorithm will produce a significant bias (and a higher scatter) for the estimated parameter g_i (see Fig. 2). This error is the primary source of error in wind speed retrievals at high wind speeds. This type of error can be minimized using simultaneous multi-parameter retrievals. A multi-parameter NN algorithm OMBNN3 (Krasnopolsky et al. 1998a,b, 1997, 1996) has an architecture shown in Fig. 1 and retrieves simultaneously four geophysical parameters: wind speed, columnar water vapor, columnar liquid water, and SST. Multi-parameter retrievals performed by this algorithm reduces error in the wind speed which is due to co-variability of related geophysical parameters (see Fig. 2). The OMBNN3 algorithm is represented by expression (B.3, Appendix B) where n = 5 (inputs - T19V, T19H, T22V, T37V, T37H), m =4 (outputs - W, V, L, SST) and k = 12 (hidden nodes).

Neural networks (NNs) are well-suited for a very broad class of continuous approximations and mappings. Neural networks consist of layers of uniform processing elements, nodes, units, or neurons. The neurons and layers are connected according to a specific architecture or topology. Fig. B.1 shows a simple architecture which is sufficient for any continuous nonlinear mapping, a multilayer perceptron. The number of input neurons, n, in the input layer is equal to the dimension of input vector X (T in our particular case). The number of output neurons, m, in the output layer is equal to the dimension of the output vector Y (g in our particular case). A multilayer perceptron always has at least one hidden layer with k neurons in it. A typical neuron (processing element) usually has several inputs (components of vector X), one output, z_j, and consists of two parts, a linear part and a nonlinear part. The linear part formes the inner product of the input vector X with a weight vector  _j (which is one column of the weight matrix _ji ), and may also add a bias term, B_j. This linear transformation of the input vector X feeds into the nonlinear part of the neuron as the argument of an activation function. For the activation function, it is sufficient that it be a Tauber-Wiener (nonpolynomial, continuous, bounded) function (Chen and Chen 1995a,b). Here we use a standard activation function - the hyperbolic tangent. Then, the neuron output, z_j , can be written as,

                                                                         (B.1)

The neuron is a nonlinear element because its output z_j is a nonlinear function of its inputs X.

From the discussion above it is clear that NN generally perform a continuous (and nonlinear) mapping of an input vector Xⁿ (n is the dimension of the input vector or the number of inputs) onto an output vector Y ^m(m is the dimension of the output vector or the number of outputs). Symbolically, this mapping can be written as,
 

                                            Y = f_NN(X )                                                                                      (B.2)
 

where f_NN denotes this neural network mapping (the NN input/output relation).

For the topology shown in Fig. B.1 for a NN with k neurons in one hidden layer, and using (B.1) for each neuron in the hidden and output layers, (B.2) can be written explicitly as,

where the matrix _ji and the vector B_j represent weights and biases in the neurons of the hidden layer; _qj in R^k×m and the _q in R^m represent weights and biases in the neurons of the output layer; and a_q and b_q are scaling parameters. It can be seen from (B.3) that any component (y_q) of the NN's output vector Y is a complicated nonlinear function of all components of the NN's input vector X. It has been shown (e.g., Chen and Chen 1995a,b, Funahashi 1989) that a NN with one hidden layer (e.g., NN (B.3)), can approximate any continuous mapping defined on compact sets in Rⁿ.

For each particular problem, n and m are determined by the dimensions of the input and output vectors X and Y. The number of hidden neurons, k, in each particular case should be determined taking into account the complexity of the problem. The more complicated the mapping, the more hidden neurons are required. Unfortunately, there is no universal rule that applies. Usually k is determined by experience and experiment. In general, if k is too large, the NN will reproduce noise as well as the desired signal. Conversely, if k is too small, the NN is unable to reproduce the desired signal accurately. After these topological parameters are defined, the weights and biases can be found, using a procedure which is called NN training. A number of methods have been developed for NN training (e.g., Beale and Jackson 1990, Chen 1996). Here we use a simplified version of the steepest (or gradient) descent method known as the back-propagation training algorithm.

Because the dimension of the output vector Y may obviously be greater than one, NNs are well suited for modeling multi-parameter transfer functions (1). All components of the output vector Y are produced from the same input vector X. They are related through common hidden neurons; however, each particular component of the output vector Y is produced by a separate output neuron which is unique.

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1. OMB Contribution Number: 165

2. Science Applications International Corporation/General Sciences Corporation, Laurel, MD 20707

3. These ERS scatterometer wind vectors have been reprocessed at NCEP from the real-time ERS backscatter and geometrical data records, in order to improve the wind direction selection.  For more information, see: Peters C, V Gerald, P Woiceshyn and W Gemmill: 1994. Operational Reprocessed ERS-1 Scatterometer Winds:  A Documentation, OPC Cont. No. 96, NMC, Camp Springs, MD, 20233, 12pp.
Gemmill W, P. Woiceshyn, C Peters and V Gerald, 1994: A preliminary Evaluation of Scatterometer Wind Transfer  Functions for ERS-1 Data, OPC Cont. No. 97, NMC, Camp Springs, MD, 20233, 35pp.

4. Here we switch from the scientific units of m/s to knots which is used by the U. S. maritime industry and back; 1 m/s 2 knots.