34;True" field Kriged gauge field
S- PROG
4.3 Case Study - Application of two dimensional EMD to rainfall data
4.3.2 Results
An analysis of over 800 individual radar scans, embodying mixtures of various ratios of Strati form and Convective rainfall types, was carried out to determine the effectiveness of the20EMO algorithm in eparating the high wavenumber spatial components from the low wavenumber components of the original rainfall data.
Working on the basis that the average characteristics of the data over a range of spatial scales summarized by the power spectrum is intuitively useful, the (radially averaged) power spectra of(i) the original data, (ii) the first IMS and(iii} the first residual of each image were examined and compared. Figure4.29shows a typical result; the power spectrum of the residual shows a very close correspondence with that of the original data at large wavelengths while it contains far less power at the shorter wavelengths (note the logarithmic scale on both axes of the figure). In contrast, the spectrum of the first IMS has very little power relative to the data's spectrum at high wavelengths but shows a strong correspondence at the lowest wavelengths. Figure4.29clearly indicates how the20EMO technique moves the bulk of the high frequency components in the original data into the first IMS and leaves the high power, lower frequencies in the first residual. The decomposition behaves as a low pass spatial filter, without presupposing the shape of the filter function. Figure 4.30 shows a time average of this behaviour by plotting the mean values at each wavelength of the three spectra over five consecutive radar scans (beginning with the data used to produce figure4.29).The radar scans are captured at approximately five-minute intervals. It is interesting to observe that the average of the spectra of the first IMS is flat (constant mean) for wavelengths longer than
10 km, suggesting nearly white noise over this range.
The temporal persistence exhibited at the spatial scales represented in each of the three sequences of: (i) the data, (ii) the first IMS and(iii) the first residual was examined by considering the persistence of their temporally consecutive power spectra. The notion of "spectral persistence" was used to determine how variable the spatial structure (at a particular spatial scale) is in time and hence to give an indication of the temporal predictive capability at each spatial scale. A summa- rized example of the analysis of a sequence of 5 radar rainfall images is presented
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Fig. 4.29: Comparisonof individual radially-averaged powerspectra of the radar rainfalldata (of figure 4.19)with its EMD comp onents: the firstIMS and the first residual. Note the logarithmic scale on both axe .
in figures 4.31,4.32 and 4.33 where a"matrix" of scatter plots is shown in each case. Scatter-plot s of the pairs of power values at each discrete wavelength for five consecutive spectra (with the I: I line indicated) are shown for the original data (Figure 4.31),thefirstIMS(Figure 4.32),and the firstresidual(Figure 4.33).
The rowsand columns of thescatter-plot matricesare labelled from TotoT4 and indicate separate radarscans betweentime T=Oand timeT=4.Each block in the scatter-plot matrix represent s a scatter-plot of the power at each wavelength for the spectrum computed atT,versus that of thespectrum computed atTj• Clearly the plotson the "matrix" diagonal each compare aspectrum to itselfand a perfect 1:I relationship is observed in this case. For the off-diagonal plots, the degree ofscatter amongst the data points indicates the degree ofsimilarity between the spectra at individual wavelength sat increasing time lagswith a largescatter indi- cating a weak similarity. The trends shown here are typical of the data analysed and show how the first (high average wavenumber) IMS has a temporally inco- herentspatialstructure, while the first (low average wavenumber)residual shows
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Fig.4.30 : The sameas figure4.29 but for themean of indiv idual power spec trafor five consec utive, radar scans- Beginning with the spec tra show n in figure 4.29.
Note the logarithmic scaleon both axes.
a temporally consistent structure. The behaviou r show n in Figures 4.29 - 4.33 sugges ts that the high frequency IMS components in spatial rainfall data do not contain much predictivecapabil ity. Thisobservation supports the sugges tions of Seed (2003) and Turner etal.(2004) who propose to increase the degree of spa- tial smoothing andgive more credi bility to the information contai nedin the lower freq uencycomponentsas forecast lead times increase.
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Fig. 4.31: Spectral persistence scatter plotsof the original data for asequence of rainfall fields and those at successive intervals. This is constructed by plotting the values of power for each field at corresponding wavelengths coaxially. For example points A and B at the 10 km wavelength are plotted against each other and appear ringed in the upper right diagram.
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Fig. 4.32: Spectral persistence scatter plots of the sequence of 1st IMS of each pair of rainfallfieldsTo,'." T4 •
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Fig. 4.33: Spectral persistence scatter plots of the equence of 1st Residual of each pair of rainfall fieldsTo.. . .,T4•
167
ChapterSummary: This chapter began with a briefoverview of some of the nowcasting techniques and model in the literature. This was followed by a dis- cussion oftheOptical Flowmethod as a technique for determining the advection of rainfall fields measured by radar or satellite. This techn ique is a core com- ponent of the S-PROG and SBM stochastic nowcasting models, as well as the accumulation scheme presentedinsection 3.3.
The SBM rainfall simulation model (Pegram and Clothier, 1999) was de- scribed and extended to provide a nowcasting implementation. The differences between the simulation and nowcasting modes were highlighted. In addition, a well known nowcasting model called S-PROG (Seed, 200 I) was described and it' smain feature sdiscussed.
With the nowcastingmodels introducedsome comparitiveinvestigationswere carried outusing obser ved radar reflectivity data from two South African radars.
It turns out that S-PROG's strategy of decomposing the observed rainfall fields according to pre-specified spatial scales gave it a performance advantage when compared with SBM in nowcastingmode for two mean field errorstatistics.
With the results of the nowca t comparisons in mind, a new technique for analysing the spatial scaling structure ofrain fall fields was presented. The tech- nique is a two dimensional extension of Empirical Mode Decomposition for the analysisof non-linear and non-homogeneoustime series. An EMD analysisintwo dimension slinearly decomposesthe spatially distributed rainfalldata into aset of Intrinsic Mode Surfaces, which are approximately mutually orthogon al (Huang et al., 1998)and sum back to theorigin aldata. Each IMS contains an oscill atory mode inherent in thedata atadifferent (narrow) rangeofspatialfrequencie . The EMD analysis success ively extracts the IMS with the highest local wavenumbers in a recursiveway, which is effectively aset ofsuccess ivelow-pass spatialfilters based entirely on the properties exhibited by the data. The utility of the EMD technique for signal separation has been demonstrated in both one and two di- mension sand applied to the analysis of a large set of 800 radarrainfall imagesin South Africa. The 2D EMD technique is proposed here in the context of rainfall nowcastingtoseparate the lesspersistenthigh wavenumbercomponentsfrom the
add little structural information to nowcasting algorithms. The scale separation achieved by 20 EMO has been analysed using radially averaged power spectra tosummarize the spatialstructure of the dataand filter outputs. In addition these power spectra have alsobeen used to examine the temporal persistenceof the spa- tial structure exhibited by the first IMS and residual. The results presented here support other work in the Hydrometeorologicalliterature,which suggests that the low frequency spatial components in rainfall data are mostuseful in a nowcasting context.