LPCE, UMR 6115 CNRS-University of Orléans, 3A, avenue de la Recherche Scientifique, Orléans cedex 2, France. Abstract

Transcription

1 Identification of the ionospheric footprint of magnetospheric boundaries using SuperDARN coherent HF radars R. André and T. Dudok de Wit LPCE, UMR 5 CNRS-University of Orléans, 3A, avenue de la Recherche Scientifique, 57 Orléans cedex 2, France September 3, 22 Abstract Several studies of SuperDARN radar data have revealed a systematic dependence of the Doppler backscatter versus latitude and magnetic local time. In particular, the Doppler spectral width is known to increase at the vicinity ofthe ionospheric footprint of the cusp, and in a region that shapes like the poleward boundary of the auroral oval. Following earlier work, we make use of a large radar data base consisting of a set of key parameters that characterize Doppler backscatter, and a multivariate statistical method, to reveal a strong connection between specific geophysical regions and their backscatter properties. This opens the interesting perspective of monitoring magnetospheric boundaries in real time. Corresponding author: T. Dudok de Wit, ddwit@cnrs-orleans.fr, tel '25'52'77, fax '3'2'3

2 Introduction The intimate connection between the characteristics of ionospheric convection and the conditions of the solar wind/magnetosphere/ionosphere coupling is a well known property of geospace. The Super Dual Auroral Radar Network (SuperDARN) has become an established tool for studying this connection, partly because it provides excellent spatial and temporal coverage of high latitude regions. Akey issue here is the reliable identification of magnetospheric boundaries from radar data (Rodger, 2). Over the last decade, there has been a dramatic increase in the number of studies devoted to that problem. Clear evidence has been found for a dependence of the radar backscatter characteristics on the geomagnetic latitude (Mlat) and the Magnetic Local Time (MLT) (Baker et al., 995; Rodger et al., 995; Pinnock et al., 999; André et al., 22; Hosokawa et al., 22; Moen et al., 22; Villain et al., 22; Woodfield et al., 22a). Most of the attention has focused on the Doppler spectral width, whose value is strongly enhanced at the ionospheric footprint of the cusp. Such an enhancement is most likely due to broadband wave activity in the Pc/Pc2 frequency band (André et al., 2a). Some studies suggest that the spectral width may also be a proxy for other regions, such as the separatrix between the central plasma sheet and the boundary plasma sheet (Lewis et al., 997; Dudeney et al., 99) or the open/closed field line boundary (Lester et al., 2). There are also indications, however, that the spectral width by itself is not always sufficient to detect boundaries (Woodfield et al., 22a,b). It has been shown, from a more statistical point of view, that the average spectral width coincides with a region that shapes like the poleward boundary of the auroral oval (Villain et al., 22) (hereafter referred to as paper-). The Probability Distribution Function (PDF) or histogram of the spectral width significantly changes with magnetic latitude (Baker et al., 995; Woodfield et al., 22c). In a recent statistical study (hereafter referred to as paper-2), André et al. (22) introduced a set of three parameters to characterize radar backscatters. Their study was motivated by the fact that the different physical processes which ultimately generate 2

3 irregularities should lead to different characteristics on the observed backscatters. The authors showed that the SuperDARN data could be categorized into different classes by applying a judicious set of thresholds on the three parameters. These thresholds supposedly separated different kinds of physical processes. When mapping down the resulting classes onto the Mlat-MLT space, different regions were obtained, which fitted quite well the footprint of some magnetospheric boundaries (possibly the plasmapause and the central plasma sheet). The main problem with the approach of paper-2 is that the thresholds were derived empirically by visualization of both simulation and experimental data. Moreover, the thresholding did not yield the same results in both hemispheres, as one would normally expect (Pinnock, 22). Overall, these different studies support the idea that the radar backscatter data contain enough statistical information to map down the location of some magnetospheric boundaries. As such they open interesting perspectives of using SuperDARN to observe boundaries in real-time and on a large scale. Wenow need an assessment study to demonstrate that, without any a priori knowledge, the radar data can be: ) systematically categorized in a way similar to paper-2, and 2) reveal some of the dominant physical processes that affect the backscatter. This is the main goal of the present article. A similar goal is pursued in the field of data mining, where classification methods are applied to extract features from large databases (Banday et al., 2). Since this study focuses on a novel way of analyzing radar data, the methodology will receive comparatively more attention than the physical implications. The derivation of the database and its main characteristics are described in section 2. The statistical analysis procedure, which is based on the singular value decomposition, is presented in section 3. In section we discuss the results, compare them with results of paper-2 and conclude. 3

4 Data selection SuperDARN is an international network of HF radars that primarily measure the convection patterns in the ionospheric auroral zone and in the polar cusp (Greenwald et al., 995). In the northern hemisphere, 9 radars permanently measure the signal that is backscattered by field-aligned irregularities occurring in the radar beam. Each radar consists of an array of log-periodic antennae, from which beams are formed through electric phase delay switching. The statistical analysis has been performed on a database covering three years (instead of 2 in paper-2) of measurements made during the winter time, in the northern hemisphere and with radars. The winter time was chosen for it has the highest data rate. Care was taken to select only data from the F-region and to remove spurious data. All the observations are mapped onto a grid defined by Mlat and MLT. The grid resolution is 3 min in time and deg in latitude (ranging from 5 deg to 9 deg). The main output of the radars is the complex AutoCorrelation Function (ACF) of the backscattered signal. This ACF is routinely fitted with a model that provides the power, the line-of-sight Doppler velocity of the irregularities and the Doppler spectral width (Villain et al., 97). See Fig. for an example. Following paper-2, we introduce three key parameters. The first one is the Doppler spectral width. Various processes, such as microscopic plasma turbulence, Pc wave activity, and large-scale vortices affect the spectral width, but their signatures differ. The latter two generate multicomponent spectra, whereas turbulence generally preserves one single maximum in the spectrum. One of the manifestations of multi-component spectra is a modulation of the ACF, which degrades the quality of the model fit. Therefore, we include the standard deviation of the ACF phase fit and the standard deviation of the ACF power fit as indicators of the existence of multiple components in the Doppler spectrum. It can be argued that this set of three parameters is not necessarily the best one for describing the statistical properties of the data set. Finding a proper set of parameters is indeed a problem on its own. Yet, as we shall see below, the chosen set is sufficiently complete to allow a clear separation between regions with different physical properties.

5 Our database contains 2: samples of each parameter, which are stored in 2 grid cells. Figure 2 shows that the cells do not contain equal numbers of samples. Two reasons for this are the finite coverage of the radars and the backscatter dependence versus Mlat and MLT. Cells that contain few samples (especially at low Mlat) often tend to be dominated by a few intense geomagnetic events. We discard such cells because they are not representative of the average conditions. Figure 2 also displays the mean value of the three key parameters, for each cell. As expected, the Doppler spectral width is strongly enhanced in the vicinity of the cusp, where it exceeds 3 m=s, and in a region that roughly coincides the poleward limit of the auroral oval. The average position of the latter is obtained from a model (Holzworth and Meng, 975), for moderate magnetic activity. The observed enhancement of the spectral width, the phase error and the power error are in agreement with previous studies (Lester et al., 2; Dudeney et al., 99; Villain et al., 22), which are compared in paper-2. The main difference appears in the morning side, where larger averages are observed. We believe that this bias is caused by a few intense geomagnetic events. Statistical analysis procedure The starting point for our analysis is the PDF of the three key parameters, computed in each grid cell. Statistical studies traditionally start with an investigation of moments of the PDF such as the mean (displayed in Fig. 2), the standard deviation, etc. Although such quantities (which have already been discussed in paper- and paper-2, and agree with our results) are useful for getting first insight, they are certainly not as informative as the PDF itself. Each PDF is estimated with 2 bins and accordingly each grid cell is described by the values contained in the 3 2 = bins. When two cells have similar PDFs, then the radar backscatter properties are statistically the same, and so these two cells are likely to be affected by the same physical processes. The idea thus consists in comparing the 5

6 PDFs and determining whether they can be classified into different categories. Data classification can be carried out in many different ways, see for example Chatfield (995). We shall focus on one simple method of doing unsupervised classification. Let us consider each grid cell as a single point in a -dimensional phase space. The coordinates of this cell are the values contained in the bins. Our task then consists in finding clusters of points that are closely spaced. Note that the data are very redundant, for it is likely that the same physical driving mechanisms may affect more than one bin at a time. For example, the value of bin k should not differ much from that in bin k +, regardless of the cell. We can take advantage of this redundancy to reduce the dimensionality of the data by projection it on a lower-dimensional subspace. A quantatively rigorous method for reducing dimensionality is the Singular Value Decomposition (SVD) (Golub and van Loan, 995), which is closely connected to principal component analysis (Chatfield, 995). Let us look for a simple transformation that transforms the -dimensional axis system into a new one, without distorting the cloud of points. This can only be achieved by rotating and by translating the points. Consider the rotation that transforms the data in such a way that their variance or scatter is maximized along some axes. Figure 3 illustrates this for a two-dimensional phase space. By tilting axes and 2 by about 5 deg, we can almost entirely describe the scatter of the points with one axis only (axis ') instead of two. In this example, the variance along axis ' will be much larger than that along axis 2'. A one-dimensional projection suffices here to describe the salient features of the data. In higher dimensions, this problem amounts to fitting the points with plane surfaces. The SVD is a technique that provides a new orthonormal frame in such a way that the variance is maximized along some axes. If p(x; k) denotes the number of events in bin k for cell x, then the SVD provides a unique decomposition X p(x; k) = V i f i (x) g i (k) ; () i= with the constraint that the functions f i (x) and g i (k) be orthonormal hf i (x)f j (x)i = hg i (k)g j (k)i = >< >: if i = j if i = j ; (2)

7 where brackets denote averaging over x of k. The profile g i (k) has the same dimension as a PDF: it tells us how to express the value along axis i in the new frame as a linear combination of the old axes (or bins). The location of the corresponding data points along that new axis is given by f i (x): it tells us how g i (k) isweighted in space. Finally, V 2 i is the abovementioned variance. By convention V V2 ::: V are sorted in decreasing order, and so the prime axes of interest are the low index ones. The orthogonality constraint of the SVD arguably does not have a sound physical justification. Various alternatives exist, some of which could provide deeper insight into our data. Let us therefore emphasize that the SVD is not a physical model, but a statistical technique to extract the salient features from a multivariate data set. The physics is expressed by the linear combination of the low-index axes. Result of the decomposition The data are most easily stored in a 2 rectangular matrix, in which eachrow corresponds to a cell and each column to a bin. Using the SVD, this matrix can be directly decomposed into a product of three matrices, which respectively contain the spatial structures f i (x), the square root V i of the variances and the profiles g i (k). Some preprocessing is needed to get meaningful results. First, equal weight should be given to all the cells. We dothisby normalizing the area of all PDFs to. Second, we avoid spurious events by flagging out cells that contain less than 5 events. Finally, since we are interested in comparing different PDFs, we remove from each bin its value μp(k) averaged over all grid cells, to concentrate on deviations only. This operation is equivalent to translating the cloud of points to the origin of the phase space. If this were not done, then the first axis of the SVD would just give this average itself. The SVD of the data thus actually gives X p(x; k) =μp(k)+ V i f i (x) g i (k) (3) i= The distribution of the variances V 2 i, which is displayed in Fig., is the key to the interpretation of a SVD. The occurrence of a few outstanding values, followed by a 7

8 flat plateau, suggests that the prominent features of the data are captured by a few axes only. If the points had been randomly distributed in phase space, then a plateau with approximately constant variance would have resulted. By projecting the data on 3 axes, we can describe over 9 % of the signal variance. With axes, this figure exceeds 99 %. All the PDFs can thus be described by a linear combination of 3 to independent parameters only, depending on the sought accuracy. Consequently, the statistical properties of the radar data are governed by just a few driving mechanisms only. The spatial structure f i (x) associated with the dominant axes is shown in Fig. 5 for axes, 2, 3, and. Let us recall that finding regions with similar statistical properties is equivalent to finding groups of grid cells with similar values of f i (x). The most conspicuous result in Fig. 5 is the identification by the SVD of several such groups. These groups, far from being scattered in space, are assembled in large continuous regions. We must stress that no information about the spatial location of the cells enters the SVD in any way, and so the emergence of continuous regions is really a consequence of the underlying statistical properties, and not of the analysis procedure. Clearly, regions with the same value of f i (x) are affected by the same physics and therefore should somehow be connected to the footprint of specific magnetospheric regions. Several regions can readily be identified. Large positive values of f(x) coincidewith the location of the polar cusp (75 to deg latitude, betweenand2mlt). This region is very similar to the location of the class 'm' in paper-2. Strongly negative values are found at low latitudes, where the magnetic field lines are closed. This region is also very similar to the location of the class 'S' in paper-2. Large positive values of f2(x) coincide with the poleward boundary of the auroral oval, and the low latitude boundary layer (night sector). The third axis f3(x) describes a dawn-dusk asymmetry. The fourth and subsequent axes reveal less structured regions, which can be interpreted as small-scale corrections to the preceding ones. The sixth axis unexpectedly captures a few grid cells around 2 MLT. We found out that these cells are corrupted by hours of data coming from one single radar that had interfer-

9 ence problems. Beyond the tenth axis, the spatial correlation between adjacent cells practically drops to zero, and so no significant information is likely to be conveyed anymore. Since the new axes are merely a rotation of the old ones, we can readily determine what kind of PDF is associated with each axis, and with each region. Indeed, each g i (k) profile (see Eq. 3) indicates by howmuch axis i departs from the average PDF μp(k) for bin k. The PDF of each cell is thus nothing but linear combination of its average μp(k) with the corrections g i,weighted by the spatial profiles f i. The corrections are typically an order of magnitude smaller than μp(k), which ensures the positiveness of the PDF. Figure displays the corrections g;g2 and g3 associated with the three dominant axes. In the cusp region, for example, the PDF is the weighted sum of μp with g (and subsequent terms), whereas at low latitudes it is the sum of μp minus g. The different axes can now beinterpreted as follows: ffl Axis : all the probability densities are affected in the same way, since the tails of the distributions are enhanced at the expense of small values. Regions in Fig. 5 with large positive excursions, such as the cusp, thus exhibit multi-component Doppler spectra with a large spectral width. These characteristics are in full agreement with previous findings (Baker et al., 995; André etal., 2b, 22) and are the consequence of the large level of wave activity in the cusp. The opposite happens at low latitude, where the spectra are narrow. ffl Axis 2: for the spectral width and the phase error, bins corresponding to moderately high values only (bins 5 to ) are enhanced. The opposite occurs for the the power. Some spectral broadening thus takes place, and there is a mild tendency for the spectra to have multiple components. ffl Axis 3 is the weakest of the three. The tail of the spectral width is depleted, in contrast to the tail of the phase error, which is slightly enhanced. We tentatively attribute the occurrence of narrower spectra in the afternoon sector (as compared 9

10 to the night and morning sector) to the solar irradiance, which is known to improve the radar backscatter and produce narrower spectra. Axis is by far the strongest one (its variance exceeds 9 %), so its characteristics really correspond to a dominant feature of the data. Axes 2 and 3 are significantly weaker, which means that the parameters of interest may either be the axes themselves, or a linear combination thereof. The sum of axes 2 and 3 (= unchanged spectral width but enhanced phase error) indeed coincides very well with class `T' of paper-2. The main conclusion is that we can ) identify the footprint of several magnetospheric boundaries and 2) determine the ionospheric signature associated with these boundaries, simply on the basis of a statistical analysis of the radar backscatter data. This is of course just a starting point for further studies, and so we would like to make some comments. First, there remains an important issue to determine how the regions with differentstatistical properties map onto known geophysical boundaries. Although some regions can readily be identified in Fig. 5, a quantitative comparison with known (e.g. Tsyganenko) models is at order. Second, since our database encompasses quite different conditions of the solar wind and the magnetosphere, the location of the boundaries is necessarily smeared out. We checked this effect by repeating the analysis for subsets that correspond to specific solar wind conditions. The transitions between the regions are sharper as expected. Third, it is interesting to see what would happen if one or two only instead of three key parameters are analyzed. The phase error and the power error have similar meanings, so they should be partly redundant. The SVD analysis indeed confirms that one of the two parameters can be left out without much affecting the outcome. The different regions just do not stand out as clearly, probably because there is less information to classify the cells. The situation is slightly better if the power error is skipped and not the phase error. If the SVD is applied to the spectral width only,oronthecontrary, if the spectral width is left out, then the results completely differ. The variances are still strongly ordered,

11 but no clear-cut regions can be identified anymore. More importantly, the regions are not collocated anymore with known magnetospheric boundaries. We conclude that both the Doppler spectral width and the phase fit error (or alternatively, the power error) are needed to properly distinguish magnetospheric boundaries from HF radar data. This is corroborated by recent results bywoodfield et al. (22a,b), showing that a high spectral width only is not a reliable means for locating the boundary between open and closed field lines. Conclusion The main result of this study is the identification of large-scale regions in which the statistics of the SuperDARN radar backscatter share similar properties. These regions coincide with the footprint of known magnetospheric boundaries such as the cusp, the auroral oval, the low-latitude boundary layer, etc. The novelty of this approach lies in the rigorous classification of the different regions on the sole basis of the statistical properties of the radar data, without prior physical information. The technique which does this is the Singular Value Decomposition (SVD). These results open the interesting perspective of performing a real-time classification, which would enable the location of the boundaries to be tracked with a time resolution of a few minutes. Indeed, since we know what are statistical properties of the different regions, it is straightforward to set up an automatic classification scheme with predetermined classes. This idea had already been advocated in (André et al., 22) by empirical thresholding, but the SVD now puts it on a firm basis. Our approach, however, should be considered a first step toward a more elaborate methodology, and several improvements are at order. The first one would be to study the dependence of the location of the different regions versus seasonal and solar wind conditions. A comparison with results obtained in the southern hemisphere is also in progress. Finally, wehave nowmoved from the SVD toward more performing classification schemes (such ask-means clustering and self-organized maps), which are conceptually closer to the idea of identifying clusters in phase space. Preliminary re-

12 sults so far confirm the existence of the regions that have been identified here and in paper-2 (André et al., 22). Acknowledgements We gratefully acknowledge discussions with colleagues from the SuperDARN network, and thank J.-P. Villain for making comments on the manuscript. References André, R., Pinnock, M., and Rodger, A., 2. Identification of the low altitudecusp by SuperDARN radars: a physical explanation for the empirically derived signature. J. Geoph. Res. 5, André, R., Pinnock, M., Rodger, A., Villain, J.-P., and Hanuise, C., 2. On the factors conditioning the Doppler spectral width determined from SuperDARN HF radars. Int. J. Geomagn. Aeron. 2, 77. André, R., Pinnock, M., Villain, J.-P., and Hanuise, C., 22. Influence of magnetospheric processes on winter HF radar spectra characteristics. Annales Geoph., in press. Baker, K., Greenwald, R., Ruohoniemi, J., Dudeney, J., Pinnock, M., Newell, P., Greenspan, M., and Meng, C.-I., 99. Simultaneous HF-radar and DMSP observations of the cusp. Geoph. Res. Lett., Baker, K., Dudeney, J., Greenwald, R., Pinnock, M., Newell, P., Rodger, A., Mattin, N., and Meng, C.-I., 995. HF radar signatures of the cusp and low-latitude boundary layer. J. Geoph. Res., Banday, A. J., Zaroubi, S., and Bartelmann, M. (eds.), 2. Mining the sky, Springer Verlag, Berlin. 2

13 Chatfield, C., and Collins, A. J., 995. Introduction to multivariate statistics, Chapman & Hall, London. Dudeney, J., Rodger, A., Freeman, M., Pickett, J., Scudder, J., Sofko, G., and Lester, M., 99. The ionospheric response to IMF B y changes. Geoph. Res. Lett., 25, 2 2. Golub, G. H., and van Loan, C. F., 995. Matrix computations, Johns Hopkins Univ. Press, Baltimore. Greenwald, R., Baker, K. Dudeney, J., Pinnock, M., Jones, T., Thomas, E., Villain, J.-P., Cerisier, J.-C., Senior, C., Hanuise, C., Hunsucker, R., Sofko, G., Koehler, J., Nielsen, E., Pellinen, R., Walker, A., Sato, N., and Yamagashi, H., 995. DARN/SuperDARN: a global view of the dynamics of high latitude regions. Space Sci. Rev. 7, Holzworth, R., and Meng, C.-I., 975. Mathematical representation of the auroral oval. Geoph. Res. Lett. 2, Hosokawa, K., Woodfield, E. E., Lester, M., Milan, S. E., Sato, N., Yukimatu, A. S., and Iyemori, T., 22. Statistical characteristics of Doppler spectral width as observed by the conjugate SuperDARN radars. Ann. Geophys. 2, Lester, M., Milan, S., Besser, V., and Smith, R., 2. A case study of HF radar spectra and 3. nm auroral emission in the pre-midnight sector. Ann. Geophys. 9, Lewis, R. V., Freeman, M. P., Rodger, A. S., Reeves, G. D., and Milling, D. K., 997. The electric field response to the growth phase and expansion phase onset of a small isolated substorm. Ann. Geophys. 5, Moen, J., Walker, I. K., Kersley, L., and Milan, S. E., On the generation of cusp HF backscatter irregularities. J. Geophys. Res., in press. Pinnock, M., Rodger, A., Baker, K., Lu, G., and Hairston, M., 999. Conjugate observations of the day-side reconnection electric field: a GEM boundary layer campaign. Ann. Geoph. 7,

14 Pinnock, M., 22. Personal communication. Rodger, A., Mende, S., Rosenberg, T., and Baker, K., 995. Simultaneous optical and HF radar observations of the ionospheric cusp. Geoph. Res. Lett. 22, Rodger, A. S., 2. Ground-based imaging of magnetospheric boundaries. Adv. Space Res. 25, 7. Villain, J.-P., Greenwald, R., Baker, K., and Ruohoniemi, J., 97. HF radar observations of E-region plasma irregularities produced by oblique electron streaming. J. Geophys. Res. 9, Villain, J.-P., André, R., Hanuise, C., and Grésillon, D., 99. Observation of the high latitude ionosphere by HF radars: interpretation in terms of collective wave scattering and characterization of turbulence. J. Atmos. Terr. Phys. 5, Villain, J.-P., André, R., Pinnock, M., and Hanuise, C., 22. Statistical study on Spectral width from SuperDARN coherent HF radars, Ann. Geophys., in press. Woodfield, E. E., Davies, J. A., Eglitis, P., and Lester, M., 22a. A case study of HF radar spectral width in the post midnight magnetic local time sector and its relationship to the polar cap boundary, Ann. Geophys. 2, Woodfield, E. E., Davies, J. A., Lester, M., Yeoman, T. K., Eglitis, P., and Lockwood, M., 22b. Nightside studies of coherent HF radar spectral width behaviour. Ann. Geophys., in press. Woodfield, E. E., Hosokawa, K., Milan, S. E., Sato, S. Lester, M., 22c. An interhemispheric, statistical study of nightside spectral width distributions from coherent HF scatter radars. Ann. Geophys., in press. Yeoman, T. K., Hanlon, P. G., and McWilliams, K. A., 22. A statistical study of the location and motion of the HF radar cusp. Ann. Geophys. 2,

15 Figures Figure : Example of an ACF observed by the Stokkseyri radar on the th of February 99 at 22: UT in geomagnetic coordinates (MLT, Mlat). Dots correspond to observations and the line to the best fit by the ACF model. The captions on the left (a and c) show the phase of the complex ACF and the captions on the right (b and d) the modulus or power. The upper row corresponds to a single component spectrum with a relatively large Doppler spectral width, and the bottom row toamultivalued spectrum with a large width. 5

16 number of samples 2 5 DOP average PHA average 2 POW average Figure 2: Number of samples per grid cell (top left, with logarithmic scale), mean value of the Doppler spectral width (DOP, expressed in m/s), the phase fit error (PHA) and the power fit error (POW), in Mlat/MLT coordinates. The average position of the poleward boundary of the auroral oval is indicated by the dashed curve. Regions in blank either are not covered by the radars or have insufficient statistics to obtain meaningful averages. By convention, the Sun is at noon. Figure 3: Example showing data points in a two-dimensional phase space, and the rotation into a new, more appropriate axis system. The points are almost distributed along a line, and so a single axis (') almost suffices to locate them.

17 2 variance [%] 2 2 axis # Figure : Distribution of the variances V 2 i associated with the P axes of the new frame. The variances are normalized with respect to the total variance ( i= V 2 i ) of the data. The dominant axes are the low order ones. 7

18 axis variance = 9% 2 axis 2 variance = % axis 3 variance =.7% 2 axis variance =.35% Figure 5: Spatial distribution f i (x) associated with axes,2,3 and. Each distribution is represented as a 2D surface in geomagnetic coordinates. The maximum amplitude is arbitrarily set to. The average position of the poleward boundary of the auroral oval is indicated by the dashed curve. Cells with no radar coverage or insufficient statistics are left blank.

19 axis DOP axis 2 DOP axis 3 DOP PDF.5 PDF.5 PDF axis PHA axis 2 PHA axis 3 PHA PDF.5 PDF.5 PDF axis POW axis 2 POW axis 3 POW PDF.5 PDF.5 PDF bin number bin number bin number Figure : Probability distribution function (PDF) of the Doppler spectral width (DOP), the phase fit error (PHA) and the power fit error (POW), for axes to 3. The thick line is μp(k), the PDF averaged over all cells. The thin lines are the deviations g i (k) from the average PDF. In each cell, the PDF is a linear combination of the μp(k) and the deviates g i (k). The average PDF has been arbitrarily normalized to a maximum amplitude of but the ratio between PDFs of the same axis is conserved. 9