Most of the common techniques for estimating conditional probability densities are inappropriate for applications involving periodic variables. In this paper we apply two novel techniques to the problem of extracting the distribution of wind vector directions from radar scatterometer data gathered by a remote-sensing satellite

Bishop, Christopher M.

Nabney, Ian T.

English

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Modelling Wind Direction from Satellite Scatterometer DataIan T Nabney and Christopher M BishopNeural Computing Research GroupAston University, Birmingham B4 7ET, United Kingdom.Email: bishopcm@aston.ac.uk nabneyit@aston.ac.ukTel: +44 (121) 359 3621 x4685Fax: +44 (121) 333 6215Neural Computing Research Group Report: NCRG/95/011Available from http://www.ncrg.aston.ac.uk/Abstract: Most of the common techniques for estimating conditional probability densitiesare inappropriate for applications involving periodic variables. In this paper we apply two noveltechniques to the problem of extracting the distribution of wind vector directions from radarscatterometer data gathered by a remote-sensing satellite.IntroductionIn an earlier paper (Nabney and Bishop, 1995), we have introduced three novel techniques formodelling conditional distributions for periodic variables, and investigated their performanceusing synthetic data. In this paper we show how these techniques can be applied to the problemof determining the wind direction from radar scatterometer data gathered by the ERS-1 remotesensing satellite.Extraction of the wind vector from the radar signals represents an inverse problem which istypically multi-valued. A conventional neural network approach to this problem, based on aleast squares estimate of the direction, would predict directions which were given by conditionalaverages. Since the average of several valid wind directions is not itself a valid solution, such anapproach would clearly fail. In this paper we show how to extract the complete distribution ofwind directions (conditioned on the backscattered power and incidence angle) and hence avoidsuch problems.Application to Radar Scatterometer DataThe European Remote Sensing satellite ERS-1 is equipped with three C-band radar antennaewhich measure the total backscattered power along three directions relative to the satellitetrack, as shown in Figure 1. When the satellite passes over the ocean, the strengths of thebackscattered signals are related to the surface ripples of the water (on a scale of a few cen-timetres) which in turn are determined by the low level winds. The value 0 is the ratio oftransmitted to backscattered power signal. Noise is introduced into the signal by the eects ofwave breaking, modulation by long waves and rain.Extraction of the wind vector from the radar signals represents an inverse problem which istypically multi-valued. Although determining the wind speed is relatively straightforward, thedata gives rise to `aliases' for the wind direction. For example, a wind direction of  will giverise to similar radar signals to a wind direction of  + , and there may be further aliasesat other angles. A conventional neural network approach to this problem, based on a least500 km785 kmforebeamsatellitetracksatellitemidbeamaftbeamFigure 1: Schematic illustration of the ERS-1 satellite showing the footprints of the three radarscatterometers.squares estimate of the direction, would predict directions which were given by conditionalaverages. Since the average of several valid wind directions is not itself a valid solution, suchan approach would clearly fail. In this paper we show how to extract the complete distributionof wind directions (conditioned on the three 0 values and incidence angle) and hence avoidsuch problems.Earlier neural network studies (see, for example, (Thiria et al., 1993)) were based on simulateddata. In this paper, the direction was modelled with a target variable discretised into 36bins. During testing, the estimated direction was calculated by interpolating over the 3 binssurrounding the histogram peak for a given input vector. To improve the disambiguation,inputs were taken from a 3  3 grid of cells around the cell of interest. Our approach is tomodel the conditional density purely locally (i.e. based on the values from a single cell) andthen to combine these density functions (for example, using div-curl splines, see (Amodei andBenbourhim, 1991)) using global information about wind eld properties. Here we report onthe rst step in the process.A large dataset of ERS-1 measurements, spanning a wide range of meteorological conditions,has been assembled by the European Space Agency in collaboration with the UK MeteorologicalOce. Labelling of the dataset was performed using wind vectors from the Met Oce NumericalWeather Prediction model. These values were interpolated from the inherently coarse-grainedmodel to regions coincident with the scatterometer cells. This was the best estimate of the windvectors that was available in sucient quantities for developing a neural network solution. Itis suitable for predicting a background wind direction, but may be less appropriate in regionswhere there are smaller scale features, such as frontal zones and other areas of high gradientsin wind speed or direction.The data that was selected for the experiments reported in this paper was collected from lowpressure (cyclonic) and high pressure (anti-cyclonic) circulations. These conditions, rather thancases that were homogeneous or with a simple gradient in speed or direction, were chosen toprovide a more challenging task to test the modelling techniques. Ten windelds from eachof the two categories were used: each windeld contains 19  19 = 361 cells each of whichrepresents an area of approximately 50  50km1. This gives a total of 7220 patterns, althoughthe data for some of the cells was missing. When the data was split into three subsets, eachcontained 1963 patterns.1Given that the total footprint width is 500 km, this implies that the cells overlap to some extentDensity Estimation for Periodic VariablesA commonly used technique for unconditional density estimation is based on mixture modelsof the formp(t) =mXi=1ii(t)(1)where i are called mixing coecients, and the kernel functions i(t) are typically chosen to beGaussians. In order to turn this into a model for conditional density estimation, we simply makethe mixing coecients and any parameters in the kernels into functions of the input vector x.This can be achieved by setting these parameters from the outputs of a neural network whichtakes x as input. This technique underlies the `mixture of experts' model (Jacobs et al., 1991)and has also been considered by a number of other authors ((Bishop, 1994); (?)). In this paperwe use a standard multi-layer perceptron with a single hidden layer of sigmoidal units and anoutput layer of linear units.In this section we briey show how we have extended this technique to provide three distinctmethods for modelling a conditional density p(jx) of a periodic variable . For further details,see (Nabney and Bishop, 1995). For all three models, the error function E which is used is givenby the negative logarithm of the likelihood function for the data with respect to the densityfunction given by the network/mixture model combination.Mixture of Wrapped Normal DensitiesThe rst technique involves a transformation from a Euclidean variable  2 ( 1;1) to theperiodic variable  2 [0; 2) of the form  =  mod 2. This induces a transformation ofdensity functions p with domain IR to functions ~p with domain [0; 2) as follows:~p(jx) =1XN= 1p( + N2jx)(2)It is clear by construction that the function ~p is a density function (since its integral is 1) andthat it also satises the periodicity requirement ~p( + 2jx) = ~p(jx).Various choices for the component density functions which make up the mixtureep(jx) arepossible, but here we shall restrict attention to functions which are Gaussian of the formi(tjx) = 1(2)1=2i(x) exp( kt   i(x)k22i(x)2)(3)where t 2 IR. The transformed density function ei is known as the `wrapped normal' distribu-tion (Kotz and Johnson, 1992).The density function p(jx) is modelled using a combination of a neural network and a mixturemodel as described above.Mixtures of Circular Normal DensitiesThe second novel approach is also based on a mixture of kernel functions, but in this case thekernels themselves are periodic, thereby ensuring that the overall conditional density functionis periodic. The distribution we use has the formp() = 12I0(m) expfm cos(    )g(4)which is known as a circular normal or von Mises distribution (Mardia, 1972). The normal-ization coecient is expressed in terms of the zeroth order modied Bessel function of therst kind, I0(m), and the parameter m is analogous to the inverse variance parameter in aconventional normal distribution. The parameter  corresponds to the peak of the densityfunction.With this choice of kernel function, we again use a neural network to determine the parametersi(x), i(x) and mi(x) in a mixture model to generate a periodic conditional density function.Expansion in Fixed KernelsThe third and nal technique introduced in this paper involves a conditional density modelconsisting of a xed set of periodic kernels, again given by circular normal functions as inequation (4). In this case the mixing proportions alone are determined by the outputs of aneural network (through a softmax activation function) and the centres  i and width parametersmi are xed. We selected a uniform distribution of centres, and mi = m for each kernel, wherethe value for m was chosen to give moderate overlap between the component functions.Clearly a major drawback of xed kernel methods is that the number of kernels must growexponentially with the dimensionality of the output space. However, for a single output variable,as in this application, they can be regarded as practical techniques.Density ModellingAfter selection of the wind eld datasets, we then trained neural networks to model the winddirection distribution. The inputs used were the three values of 0 for the aft-beam, mid-beamand fore-beam, and the sine of the incidence angle of the mid-beam, since this angle stronglyinuences the reected signal received by the scatterometer. The 0 inputs were scaled tohave zero mean and unit variance, while the fourth input value was passed to the networkunchanged. The target value was expressed as an angle clockwise from the satellite's forwardpath and converted to radians. A conjugate gradient algorithm and `early stopping' were usedto train the networks.Table 1 gives a summary of the preliminary results obtained with each of the three methods.As expected, the complexity of the domain meant that there were many diculties with localoptima. In fact, over 75% of the training runs ended with the network trapped in a localminimum of the error surface. It was found that the validation error for methods 1 and 2 waslowest for models with eight centres, even though fewer centres are actually required to modelthe conditional density function well. This is also demonstrated by the graph in gure 2 whichshows the conditional distribution of wind directions given by this network at a typical datapoint from the test set, and which is clearly bi-modal. This graph also shows how the xedkernel model gives rise to a conditional density with a dierent alignment.Figure 2 also shows that the density given by the xed kernel model has four peaks. Notethat the extra peaks are not just caused by the variance parameter being too small, since thereare kernel centres close to the minima between the two pairs of adjacent local maxima. TheMethod Centres Hidden Validation TestUnits Error Error1 4 20 2474.6 2446.22 6 20 2308.0 2337.93 36 24 2028.9 1908.9Table 1: Results on satellite data.Method 1: Mixture of wrapped normal functions.Method 2: Mixture of adaptive circular normal functions.Method 3: Mixture of xed kernel functions.0.000.501.001.502.002.50Method 1Method 2Method 3  2 02 x0.500.000.501.001.500.5 10Figure 2: Plots of the conditional distribution p(jx) obtained using all three methods. (a) and(b) show linear and polar plots of the distributions for a given input vector from the test set.The dominant alias at  is evident. In both plots, the solid curve represents method 1, thedashed curve represents method 2, and the curve with circles represents method 3.dierences between the shapes of these distributions, despite the relatively close agreementof the data likelihood values, suggests that more training data is required to obtain betterestimates of the density functions.DiscussionIn this paper we have a new approach to modelling the wind direction based on satellitescatterometer data. It involves modelling the conditional density distribution of the directionat a local level, and we have shown how some novel neural network methods for doing this canbe successfully applied. A conventional neural network approach, involving the minimizationof a sum-of-squares error function, would perform poorly on this problem since the requiredmapping is multi-valued.An advantage of the density modelling approach is that it enables a better understanding of thedirection ambiguities to be formed. There is the suggestion that there are other aliases thanat an angle of  in the distribution. In addition, it provides the most complete informationfor the next stage of processing, which is to `de-alias' the wind directions by combining localinformation to determine the most probable overall wind eld.AckowledgementsWe are grateful to the European Space Agency and the UK Meteorological Oce for makingavailable the ERS-1 data. The contributions of Claire Legleye in the early stages of the projectare also gratefully acknowledged. We would also like to thank Iain Strachan and Ian Kirk ofAEA Technology for a number of useful discussions relating to the interpretation of this data.ReferencesAmodei, L. and Benbourhim, M. N. (1991). A vector spline approximation. Journal of ApproximationTheory, 67:51{79.Bishop, C. M. (1994). Mixture density networks. Technical Report NCRG/4288, Neural ComputingResearch Group, Aston University, U.K.Jacobs, R. A., Jordan, M. I., Nowlan, S. J., and Hinton, G. E. (1991). Adaptive mixtures of localexperts. Neural Computation, 3:79{87.Kotz, S. and Johnson, N. L., editors (1992). Encyclopedia of Statistical Sciences, pages 381{386.Wiley.Mardia, K. V. (1972). Statistics of Directional Data. Academic Press, London.Nabney, I. T. and Bishop, C. M. (1995). Modelling conditional probability distributions for periodicvariables. In 4th International Conference on Articial Neural Networks, pages 177{182. IEE.Thiria, S., Mejia, C., Badran, F., and Crepon, M. (1993). A neural network approach for modelingnonlinear transfer functions: Application for wind retrieval from spaceborne scatterometer data.Journal of Geophysical Research, 98(C12):22827{22841.

Modelling wind direction from satellite scatterometer data

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Modelling wind direction from satellite scatterometer data

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