34;True" field Kriged gauge field
S- PROG
4.2 Extracting spatial structure from rainfall data
4.2.4 Empirical mode decomposition in a single dimension
The basic idea embodied in EMD analysis,is to allow for an adaptive and unsu- pervised representation of the intrinsic components of linear and non-linear sig- nals basedpurely on the propertiesobserved in the data without requiring thatthe signals exhibitstationarity. As Huang et al.(1998) point out in their abstract
"This decomposition method is adaptive and theref ore highly effi-
149
dent. Since the decomposition is based on the local characteris- tic time scale ofthe data, it is applicable to non-linear and non- stationary processes."
Few sequences of ob ervations ofnatural phenom ena are long enough to jus- tify the hypothesisofstationarity and frequently thephenomena are patently non- stationary. Thistacitly appliesin the measurem entof rainfall at a point or inspace- time because sequences of rain are interspersed with dry period (intermittency) and during the rainin g periods, the variability of the intensity due to mixture sof rainfall type (stratiform, convective, frontal) confound the homo geneity defini- tion. The EMD algorithmcopes with stationarity (or the lack of it) by ignoring the concept,embracing non-stat ionarity as a practical reality. For a fullerdiscus- sion of the genesisof these ideas, see theintroduction of Huang et al.(1998), who alsoheuristically demonstratethe implicit orthogonalityof thesequences of IMFs defined by theEMD algorithm.
In the application of theEMD algorithm, thepossiblynon-linear signal, which mayexhibit varying amplitudeand localfrequencymodulation ,islinearly decom- posedintoa finite numberof (zero mean)frequency andamplitude modulated sig- nals,as well as a residual function which exhibits asingle extremum,is a mono- tonic trend or is simply constant. Although EMD is a relatively new data analysis technique, its power and simplicity have encouraged its application in a myriad of fields. It isbeyond the scopeofthisthesisto giveanexhaustive review of the applications, however afewinteresting examplesarecitedhere togive the reader a feeling for the broad scope of applications. Chiew et al. (2005) examin e the one-dimensional EMD of severalannualstreamflow time series to search for sig- nificant trendsin thedata,usingbootstrappingtotestthe statisticalsignificance of identified trends.The techniquehas beenused extensively in theanalysisof ocean wave data (Huang et al., 1999;Hwang et al., 1999) as well as in the analysis of polar ice cover (Gloersen and Huang, 2003). EMD has also been applied in the analysis ofseismologica l data by Zhang etal. (2003) and has even been used as an aid in diagnosingheart rate fluctuations(Balocchietal.,2004).
highestlocal frequency from the data ("detail" in the Wavelet context or the result of a high-pass filter in Fourier analysis) , leaving the remainder as a "residual"
("approximation" in Wavelet analysis). Successive applications of the algorithm onthe sequence of residuals produce a complete frequency decomposition of the data. The final residual is a constant, a monotone trend or a curve which only has a single extremum.
The EMD of a one-dimensional dataset z(k )is obtained using the following procedure:
1. Setro(k)=z(k) and seti=1.
2. Identify allofthe extrema(maxima andminima) inTj-l(k).
3. Compute a maximal envelope, maxj_l(k), by interpolating between the maxima found instep 2. Similarly compute the minimal envelope,min.i.,(k).
Cubicsplines(assuggested by Huang et aI., 1998)appear to be the mostap- propriate interpolation method for deriving these envelopes in one dimen- sion (Flandrin et al., 2004).
4. Compute the mean value function of the maximal and minimal envelopes
. (k) Imaxi-l (k)+mini-dk»)
rn,_l 2 '
5. The estimate of the IMF is computed from IMFj(k )=Tj_l (k )- mj_l (k ).
Each IMF is (by definition) supposed to oscillate about a zero mean and in practice it is necessary to perform a"sifting" process by iterating steps 2-5 (settingTi - l=IM Fi before each iteration)until this is achieved.
6. Once the IMFj has a mean value that is suffic ientlyclose to zero over the length of the data (defined by a topping criterion within some predefined tolerance e:) the residual rj(k) =rj_l(k)- IMFj(k) is computed. Alterna- tively the sifting procedure can be stopped when the difference in the stan- dard deviation ofsuccess ive estimatesof IMFjfalls below a critical thresh- old (Huang et al., 1998).
151
} i#Y0;:;;~
111IMFi~
2nd IMF}VWV\NY\M/;;v;JV;d
RnldlUllT..nd
.2 L - - - - ' _ - ' - _ - ' - _ - - ' - _ - ' - _ . . L - _ - ' - - _ ' - - - - ' _ - '
Fig. 4.22: EMD based signal eparation; all IMFs are plotted to the same vertical scale. Top panel is the combined signal; lower 3 panels are the decomposition which recaptures, almost exactly, the original components.
7. Ifthe residual Ti(k) is a constant or trend then stop; else increment i and return to step 2.
Figure 4.22 shows the EMD of a composite data series (shown in the first panel) that is the summation of a sine wave, a triangular waveform and a slowly varying trend. The compact representation obtained by EMD extracts (almost per- fectly - except near the ends) the three separate data series (shown in panels 2 to 4) that make up the composite signal, without resorting to Fourier or Wavelet tech- niques with restrictive assumptions about the form of the underlying oscillatory modes (in the form of predefined basi functions). Figure 4.23 show the analysis of the same data, using Wavelet decomposition. Here a fifth order Daubechie wavelet basis wa (arbitrarily) cho en for illustration purposes; this choice of ba- sis function may not be optimal for detrending but serves to demonstrate a typical decomposition. Seven levels of decomposition were required before the trend became apparent; this decomposition is clearly far less compact and physically meaningful than the EMD results in this case.
12e.ga
Fig. 4.23: Wavelet based signal separation - The "data" are the same as in fig- ure 4.22, the vertical scale has been compressed for a compact presentation. An arbitrarily chosen db5 wavelet decomposition basis has been used for illustrative purposes.
A similar decomposition analysis can be carried out using Fourier techniques.
Figures 4.24 and 4.25 show the result of decomposing the data using a finite Fourier series approximation (see equation 4.5). Figure 4.24 shows the first 5 harmonics; while figure 4.25 shows the series reconstruction by accumulating the lower harmonics up to m. Computing the Euler-Fourier coefficients provides a compact approximation of the original signal (useful for data compression) but fails to extract physically meaningful information. The ability to determine mean- ingful structural information is clearly important in a nowcasting context, which cannot be bound by the periodicity assumption implicit in Fourier methods.
153
Fig. 4.24: Fourier based signal separation, the first 5 of 75 compon ents - The dashed lines show thesine component and the solid lines thecosinecomponent.
; f : : M o' : : : : 1
~E : : : : m ,' : : : : j
~ ~
. :
Fig. 4.25: Reconstruction ofthe signalfrom the sine and cosinecomponents, m representsthe number of Euler-Fouriercoefficients used ineach reconstruction.
In two dimensions the EMD process is conceptually the same as for a single dimension, except that the curve fitting exercise becomes one of surface fitting and the identification of extrema becomes (slightly) more compl icated. Very lit- tle work appears to have been done which applies the EMD technique to two- dimension al data. Han etal. (2002) use EMD in one dimension along four dif- ferent direction tosmooth Synthetic Aperture Radar (SAR) images and remove speckle. Nuneset al.(2003) develop atechnique,whichtheyterm "Bidimensional Empirical Mode Decomposition " (BEMD) in the context of texture analysis in image data wherethey demon strate severalexamples of intrinsic modeextraction from image data. Linderhed (2002, 2004) examined the use of EMD in two di- mension sfor imagecompression. Both oftheseimplementation sare verysimilar to what is proposedhere,with the exception thatin therainfall field context there are raining and non-raining areas on the same image. The 2D EMD provides a truly two-dimensional analysis of the intrinsic oscillatory modes inherent in the data. Two-dimension al Fourier and Wavelet analyses are really application s of their one-dimension al counterparts in a number of principal directions. Fourier analysis concentrates on orthogonal "East-West" and "North-South" directions (e.g. Presset al., 1992). Wavelet analysis can,in general , consider any direction of the wavelet relativeto the data,however atypical 2D Wavelet analysisexam- ines only horizontal, vertica l and diagon al orthonormal wavelet basis functions (Daubechies, 1992, pp. 313; Kumar and Foufoula-Georgiou, 1993). In contrast, EMD producesa fully two-dimensional decomposition ofthe data,based purely onspatial relation ship sbetween the extrema, independent oftheorientationofthe coordinatesystem inwhichthe dataare viewed.
Description ofthealgorith m
The algorithm follows intuitively from the one-dim ensional case and may be briefly summarizedas follows:
155
1. Locate the extrema in the 20 space including maximal and minimal plateaus.
2. Generate the bounding envelopes using appropriate surface fitting tech- niques. Conical Multiquadrics are suggested (for reasons explained later).
3. Compute the mean surface function as the average value of the upper and lower envelopes.
4. Determine the first estimate of an IMS by subtracting the mean surface from the data.
5. Iterate until the IMS mean surface function is close to zero everywhere.
6. Estimate the IMS and Residual.
7. Ifthe Residual is a constant or a monotone trend, then stop; else return to step 2.
Surface fitting for extremal envelope generation
The generation of maximal and minimal envelopes is of key importance to a suc- cessful 20 EMD implementation and is the most computationally intensive task.
The problem is a familiar one of collocating a smooth surface to randomly scat- tered data points in two-dimensions. There are several options available to achieve this. Ultimately the fitting procedure reduces to computing the unknown value of the surface at a point Si=(Xi Yi), by some linear (or nonlinear) weighting of the known data. In general, a basis function determines the influence of each known data point based on its spatial position relative to the unknown point Si' Nunes et al. (2003) use radial basis functions while Linderhed u es bi-cubic splines (Lin- derhed, 2002) and later (Linderhed, 2004) chooses the more suitable option of Thin Plate Splines. The choice here is radial basis functions (technically,conical Multiquadrics),which are identical to Kriging with a purely linear erni-variograrn model (Borga and Vizzaccaro, 1997). It could perhaps be argued that it would be more appropriate to fit a serni-variogram model to the maxima and minima, but this would be over-elaborate and presumptuous, as the extrema are only related by
the idea is to let the data do the talking and conical Multiquadrics assume the least structure of any linear surface fitting algorithm.
The Ordinary Kriging estimate Zi at any point i based on n observed data points is
n
Zi = l::AkZk
k=l
where Zkare the observations and Akare weights as ociated with each observa- tion and the target point. The mean is assumed unknown and the weights Akare constrained to sum to unity. The vector of weight A is obtained by solving the linear system in equation 4.6
(4.6)
where 1 is a vector of semivariogram values, in this application simply defined by the linear distance basis function i ( Sij ) =
ISijl
with Sij the distance between point i and thej=l,2,""" !n.observation locations.r
is the matrix of distances between the observations, u is a vector of n ones and J1. is a Lagrange multiplier ensuring that the Kriging weights A sum to unity, as required. The solution of equation 4.6 is obtained using Singular Value Decomposition (SVD), to ensure that a stable solution is assured (when the matrix is ill conditioned). This is achieved by trun- cating singular and near-singular components. Although SVD is computationally less efficient than (for example) LV decomposition as a means of solving a dense linear system, it's use is preferred here because of robustness in the face of the near-singular Kriging systems which are frequently encountered in gridded data applications (Wesson and Pegram,2004).A more efficient choice of interpolation technique would be useful and more work could be done in this regard, however care is required. Moving-neighbourhood Kriging (a possible alternative to reduce the number of control points) can pro- duce unwanted discontinuities in regions that are data sparse (Chiles and Delfiner,
157
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.
' .; ' . .' , ,, ,
Signa l NOMcorrupted..lg~1
5 5
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40 40
40 40
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hllMS 1I.tRe .idual
5 5
.1 .1
40 40
40 40
o 0 o0
Fig.4.26 : Exampl eof EMO used for noiseremo valon a 20sine wave. The bulk ofthe additivewhite noiseinthecorruptedsignal iswellcaptured by thefirstIMS.
1999, pp 20 I), such discontinuities would be amplified through the EMO sifting process. In addition, the particul ar choice of Ordinary Kriging as a method of generating the bounding envelopes was (partially) directed by the propert y that the estimatesdecay asymptoticallyto the meanof the observed extrema.
Simple two-dimensional EMD
In this section, application s of the 20 EMO technique are presented. As an ar- tificially constructedexample figure4.26 shows the successful remo val of noise added toa synthetically generated two-dim ensional sinesignal. The noise (with it's high local spatial frequ ency) is almostcompletely describ ed by the firstIMS leaving a residual,which isclosely representativeof theunderlying signal.
Turningto amorerealistic example, figure 4.19showedan instantaneousradar rain fall field with dimensions lOOx200 km. A complete EMO of this field is shown infigure 4.27 using adirect applicationofthe 20 EMO processdescrib ed in this section; note the change inscaleofthe individual IMS. The final residu al
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4. "'t '.
v •
~
. ...
3rdlMS
.
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....e.
r -t/'
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o
-5
Fig. 4.27: Naive EMD of the observed rainfall field shown in figure 4.19 - note the change in scale of the rain rates in the IMS.
(with a single extremum) gives a clear indication of the position of the largest con- vective raincell evident in the field. Co-incidently itturns out that this particular raincell is in fact the feature which persists for the longesttime in the observed rainfallsequence.