A Wavelet-Based Technique for Identifying, Labeling, and Tracking of Ocean Eddies

Transcription

1 MARCH 2002 LUO AND JAMESON 381 A Wavelet-Based Technique for Identifying, Labeling, and Tracking of Ocean Eddies JINGJIA LUO Department of Earth and Planetary Physics, Graduate School of Science, University of Tokyo, Tokyo, Japan LELAND JAMESON University of California, Lawrence Livermore National Laboratory, Livermore, California (Manuscript received 1 June 2000, in final form 20 July 2001) ABSTRACT Wavelet analysis offers a new approach for viewing and analyzing various large datasets by dividing information according to scale and location. Here a new method is presented that is designed to characterize timeevolving structures in large datasets from computer simulations and from observational data. An example of the use of this method to identify, classify, label, and track eddylike structures in a time-evolving dataset is presented. The initial target application is satellite data from the TOPEX/Poseiden satellite. But, the technique can certainly be used in any large dataset that might contain time-evolving or stationary structures. 1. Introduction Wavelet transforms provide information about a function or dataset with respect to scale and location in contrast to Fourier transforms, which provide a oneparameter family of coefficients representing the global frequency content (see Chui 1992; Daubechies 1988, 1992; Erlebacher 1996; Meyer 1990; Strang 1996). Due to their ability to resolve scales, wavelets can capture and identify local features of various structures contained in a given dataset that might be entirely missed by another form of analysis. In a word, we are searching for an efficient basis set with which to represent data that are local in nature. Such approaches have been carried out by others and one can see Saito (1998) for the search for an optimal basis set, Kirby (2001) for a complete discussion of basis sets of appropriate for various sets of data, and Jameson (2000) for wavelet analysis applied to numerical resolution of Kelvin Rossby waves. The idea is that wavelets form an efficient basis set for localized information such as ocean eddies. Our current application of wavelet analysis is to time-evolving structures found in oceanography. These structures might come from computational data or they might be contained in data collected by satellites. In either case, we are interested in separating meaningful structures such as eddies and fronts from noise and following the evolution of these structures with time. Corresponding author address: Dr. Leland Jameson, University of California, Lawrence Livermore National Laboratory, P.O. Box 808, MS L-312, Livermore, CA ameson3@llnl.gov In order to achieve our goal of completely identifying obects and structures that might appear in observational data or computational data, we begin here by building a straightforward database of wavelet signatures and from these base signatures we will have an idea of how to treat more complicated obects. Our signatures will be limited to those given in the Daubechies wavelet basis. We choose an orthogonal basis in order to have the ability to clearly separate information by scale and location without the possibility of overlap, which can happen in the case of a nonorthogonal bases sets. After choosing the wavelet, then we must specify ways to characterize information in datasets. One very simple and practical means of identifying and labeling obects in a dataset is by measuring the amount of energy that is contained at each scale. Simply put, one must define a region around an obect of interest, perhaps an eddy, and in this region one finds the energy present at each wavelet scale. This energy by scale approach can help to uniquely identify each structure, or specifically, each eddy. Noise is the structure that one can certainly expect to encounter in almost all data sources. So, it is a very important structure to understand. Once it is thoroughly understood in the wavelet basis, then it can more reliably be dealt with by, say, cancellation or other means. Our next relevant structure is a front, or near discontinuity in the dataset. The wavelet signature of a front will be easier to detect and identify than that of eddies and noise. For the case of a front we will see a sharp ump in the wavelet coefficients at most scales. In fact, if the front is a true discontinuity then the ump in the 2002 American Meteorological Society

2 382 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 19 wavelet coefficients will be seen in all scales. Intuitively one would expect this since one way of viewing wavelet analysis is as a sequence of high-pass filters and a discontinuity will certainly have high-frequency energy at all scales. Our final task will be to follow and to track the various structures in our dataset as time evolves. 2. Wavelet analysis Possibly the most instructive way to think of wavelets is to compare with traditional analysis techniques such as Fourier analysis. With Fourier analysis we analyze discrete or continuous data using basis functions that are global, smooth, and periodic. This analysis yields a set of coefficients, say a k, which gives the amount of energy in the data at frequency k. Wavelet analysis, by contrast, analyzes data with basis functions that are local, slightly smooth, not periodic, and which vary with respect to scale and location. Wavelet analysis thereby produces a set of coefficients b,k which give the amount of energy in the data at scale and location k. Wavelet analysis can serve as a good complement to Fourier analysis. In fact, data that are efficiently analyzed with Fourier analysis often is not efficiently analyzed with wavelet analysis and the opposite situation also holds. For our purposes here we will confine our discussion to the so-called orthogonal wavelets and specifically the Daubechies family of wavelets. The orthogonality property leads to a clear indication when data deviate from a low-order polynomial, the importance of which will become clear when we discuss numerical methods. a. Defining the Daubechies wavelet To define Daubechies-based wavelets, see Daubechies (1988) and Erlebacher (1996), consider the two functions (x), the scaling function, and (x), the wavelet. The properties of the scaling function and wavelet are defined in terms of the low-pass filter coefficients h k and the high-pass filter coefficients g k. Here g k is actually the same filter as h k with the order reversed and with an alternating sign on the coefficients. More precise details on these high- and low-pass filters will be given in sections 2b and 2d after our current and more general introduction. First we define the dilation equation by L1 k k0 (x) 2 h (2x k), (1) which carries the name dilation equation since the independent variable x appears alone on the left-hand side but is multiplied by 2, or dilated, on the right-hand side. One also requires that the scaling function (x) be normalized: # (x) dx 1. The wavelet (x) is defined in terms of the scaling function, L1 k k0 (x) 2 g (2x k). (2) One builds an orthonormal basis from (x) and (x) by dilating and translating to get the following functions: /2 k(x) 2 (2 x k), and (3) /2 k(x) 2 (2 x k), (4) where, k Z. Here is the dilation parameter and k is the translation parameter. b. The spaces spanned by wavelets It is usual to let the spaces spanned by k (x) and k (x) over the parameter k, with fixed, be denoted by V and W, respectively. span V k Z k(x), (5) span k Z k W (x). (6) The scaling function subspaces form a nested set according to scale and the nesting is V1 V0 V1 (7) The wavelet subspaces W are the difference between the scaling function subspaces at different scales. The spaces V and W are related by V V1 W 1, (8) where the notation V 0 V 1 W 1 indicates that the vectors in V 1 are orthogonal to the vectors in W 1 and the space V 0 is simply decomposed into these two component subspaces. c. The high- and low-pass filters and orthogonality L1 L1 The coefficients H {h k} k0 and G {g k} k0 are related by g k (1) k h Lk for k 0,...,L 1. All wavelet properties, hence the definition of the wavelet being used, are specified through the parameters H and G. If one s data are defined on a continuous domain such as f(x) where x R is a real number, then one uses k(x) and k(x) to perform the wavelet analysis. If, on the other hand, one s data are defined on a discrete domain such as f(i) where i Z is an integer then the data are analyzed, or filtered, with the coefficients H and G. In either case, the scaling function (x) and its defining coefficients H detect localized low-frequency information; that is, they are low-pass filters, and the wavelet (x) and its defining coefficients G detect localized high-frequency information; that is, they are high-pass filters. Specifically, H and G are chosen so that dilations and translations of the wavelet, k (x), form an orthonormal basis L 2 (R) and so that (x) has M vanishing moments that determine the accuracy of the wavelet approximation. That is, scaling functions have the ability to represent polynomials up to a given order exactly and wavelets are, therefore, orthogonal to these same polynomials. This orthogonality is in terms of vanishing moments and therefore the number of vanishing

3 MARCH 2002 LUO AND JAMESON 383 moments determines the accuracy of the approximation. In other words, (x) will satisfy k m klm k(x) l (x) dx, (9) where kl is the Kronecker delta function, and the accuracy is specified by requiring that (x) (x) 0 satisfy 0 m (x)x dx 0, (10) for m 0,..., M 1. Under the conditions of the previous two equations, for any function f(x) L 2 (R) there exists a set {d k } such that where k f (x) d k(x), (11) Z k Z dk f (x) k(x) dx. (12) d. Quadrature mirror filters and the Haar wavelet The two sets of coefficients H and G are known as quadrature mirror filters. For Daubechies wavelets the number of coefficients in H and G, or the length of the filters H and G, denoted by L, is related to the number of vanishing moments M by 2M L. For example, the famous Haar wavelet is found by defining H as h 0 h 1 1. For this filter, H, the solution to the dilation equation (1), (x), is the box function: (x) 1 for x [0, 1] and (x) 0 otherwise. The Haar function is very useful as a learning tool, but because of its low order of approximation accuracy and lack of differentiability it is of limited use as a basis set. The coefficients H needed to define compactly supported wavelets with a higher degree of regularity can be found in Daubechies (1988). As is expected, the regularity increases with the support of the wavelet. Note that the usual notation to denote a Daubechies-based wavelet defined by coefficients H of length L is D L D 2M. However, one can also find examples where the Daubechies wavelet is denoted by a subscript that is the number of vanishing moments D M. We will use the first and most common convention for notation. e. Setting a largest and smallest scale In a continuous wavelet expansion, functions will arbitrarily small-scale structures can be represented. In practice, however, there is a limit to how small the smallest structure can be depending on, for example, the numerical grid resolution or the sampling frequency in a signal-processing scenario. Hence, on a computer an approximation, using Eq. (8), would be constructed in a finite space such as V W W W V, J J with the approximation being with J J J V0 k k k k k Z 1 k Z P f(x) s (x) d (x), (13) J J k k k k d f (x) (x) dx, s f (x) (x) dx utilizing orthogonality. Within this expansion, the scale 0 is arbitrarily chosen as the finest scale required, and scale J would be the scale at which a kind of local J average, k (x), provides sufficient large-scale information; that is, the first term in Eq. (13) provides the local mean around which the function oscillates. One must also limit the range of the location parameter, k. Assuming periodicity of f(x) implies periodicity on all wavelet coefficients, sk and dk, with respect to k. For the nonperiodic case, since k is directly related to the location, a limit is imposed on the values of k when the location being addressed extends beyond the boundaries of the domain. f. Implementation on a computer The wavelet decomposition matrix is the matrix embodiment of the dilation equation, Eq. (1), defining the scaling function and the accompanying equation, Eq. (2), defining the wavelet. The following two recurrence relations for the coefficients, sk and dk, in Eq. (13) are given as L 1 k n n2k2 n1 L 1 k n n2k2 n1 s hs and (14) d gs (15) as obtained from Eqs. (1) and (2), and we recall that h n refers to the chosen filter while we have g n (1) n h Ln. Denote the decomposition matrix embodied by these,1 two equations, assuming periodicity, by P N where the matrix subscript denotes the size of the square matrix while the superscripts indicate that P is decomposing from scaling function coefficients at scale to scaling function and wavelet function coefficients at scale,1 1; that is, maps s onto s 1 and d 1 : P N [ ] d s 1,1 P N : [s ], (16) 1 where we by s refer to the vector containing the coefficients at scale. Note that the vectors at scale 1 are half as long as the vectors at scale.

4 384 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 19 FIG. 1. Contour plots for a Gaussian eddy and the same eddy with computer-generated noise superimposed. g. Wavelet analysis in two dimensions Performing the wavelet analysis of a given field in higher dimensions is a straightforward extension of the ideas from one dimension. Once can simply perform the wavelet analysis in a tensor product approach dimension by dimension. In fact, the regions of the domain which can be felt, as discussed above, carry over directly dimension by dimension. The tensor product approach works as follows: Recall that in one dimension a decomposition of the finest scale subspace yields V0 V1 W 1. So, if one takes the tensor product of this one-dimensional analysis with another one-dimensional analysis then one obtains V0 V0 (V1 W 1) (V1 W 1), (17) which yields V0 V0 (V1 V 1) (V1 W 1) One can think of the subspace V1 V1 (W V ) (W W ). (18) as representing the horizontal and vertical average, lowpass filtering, of the information contained in V 0 V 0. The subspace V1 W1 represents a horizontal low-pass filtering process and a vertical high-pass filtering process. Such a subspace would capture horizontal edges. On the other hand, the subspace W1 V1 would represent a horizontal high-pass filtering process and a vertical low-pass filtering process. One would be able to detect vertical edges in such a subspace. Finally, when the high-pass filter is applied in both the vertical and horizontal directions one arrives at the subspace, W1 W 1. It is this subspace that we are primarily interested in since it will detect variation in both the horizontal and vertical directions. Note that as in the one-dimensional case we certainly will perform many levels of wavelet decompositions, and the subspaces that we are primarily interested in will be W W, for 1, 2, 3, 4... Generally, decompositions up to 4 will be sufficient. 3. Defining wavelet signatures In this section we document the wavelet signatures, see below for definition of signatures, of various structures that will commonly appear in the wavelet analysis of various fields associated with oceanography. Examples of such structures would be noise, eddies, fronts, etc. In order to achieve our goal of completely identifying obects and structures that might appear in observational data or computational data, we begin here by building a straightforward database of wavelet signatures. For now our signatures will be limited to those given in the Daubechies wavelet basis. a. Various ways to form a wavelet signature Our first task is a brief discussion of the various ways that one could form a signature using a wavelet basis. In a word, wavelets give us information with respect to scale and location. One could imagine a very large num-

5 MARCH 2002 LUO AND JAMESON 385 ber of ways to use this information to form a uniquely identifying signature. We will explore only a few of these ways. b. Local variance or energy as a signature One possible way to find a wavelet signature is to find the amount of energy present at the various wavelet scales in a given region. This could be accomplished by summing up the square of the wavelet coefficients at the various scales in the following manner: 1 2 K,K,K C, (19) where will be the wavelet-detected variation at scale and C will be a corresponding scaling constant. Here is defined as k1k1n k2k2n k3k3n (2n 1) k1k1n k2k2n k3k3n [d (k, k, k )]. (20) As above, the parameter n will define a box in three dimensions about which the summation of the wavelet coefficients occurs. The reason that one would have a different n for each scale is that the wavelets at the larger scales, higher values of, will cover larger portions of the domain. One can see this from the above discussion of regions of influence. Therefore, one would expect that the values of n will decrease as increases. This will, of course, depend on the size of the region that one wished to use in finding their estimate of the energy at a given wavelet scale. The point (K 1, K 2, K 3 ) will be a wavelet translation index at the scale corresponding to a point in the physical space. The point is roughly centered at the location of the physical space where one needs an estimate of variance or energy. c. The wavelet subspaces for signature measurement We begin by observing the energy signature at various wavelet scales using the D4 wavelet. Recall from our earlier discussion that a wavelet decomposition in two dimensions provides us with four subspaces, V V, V W, W V, and W W. For now, we will give the energy contained in the subspaces W W, for 1, 2, 3,.... This energy will simply be the summation of the squares of the wavelet coefficients at scale in both the horizontal and vertical directions. These energies will give us a kind of wavelet noise energy spectrum at various scales. Note that in detecting the global energy in a domain one is not using the key strength of wavelet analysis, that is, the ability to separate structures by location. That is, one key difference between wavelet analysis and Fourier analysis is the information on location or the parameter k in the coefficient. In taking a global sum, this information d k FIG. 2. (a) The energy profile of noise with most of the energy in the smaller scales. (b) The energy profile of the single Gaussian eddy with noise added. is not used. However, this global analysis is an essential first step in building our wavelet signature database. d. Wavelet spectra and a bounding box Note that the manner in which one uses the information from wavelet analysis is far from unique and that in some manner one must combine the obtained information into some kind of meaningful and easily utilized information. For our purposes we have chosen to combine the wavelet information by scale and this has worked quite effectively. Further, we have a bounding box defined around each eddy for practical convenience. Without such a clearly defined bounding box one can encounter practical problems such as deciphering where one eddy ends and where the next eddy begins. Of course, deciding the beginning and end of an eddy becomes at the most precise level somewhat arbitrary, one still must have a precise manner to separate information. For our purposes we have not found the bounding box to be a limitation. If, however, situ-

6 386 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 19 FIG. 3. The contour diagram of two eddies of different strength and size. ations are encountered in the future where the bounding box is a limitation, then appropriate modifications can be incorporated. 4. A catalog of wavelet signatures In this section we conduct testing of the technique on relatively simple datasets that consist of Gaussian structures called Gaussian eddies in this study. Of course, one could spend a great deal of time building models of eddies more realistic than ours, but as one will see, our Gaussian eddies are sufficient for testing our method, which will be proven by effectiveness when applied to real TOPEX/Poseidon (T/P) datasets. a. One Gaussian eddy with and without noise In this subsection we begin with our simplest test case, that of a single Gaussian eddy first without noise and then with computer-generated noise. In Fig. 1 we have two contour plots of sea surface height which indicates the size and strength of the eddy. The left plot is of the pure eddy and the right plot we show the eddy with computer-generated noise superimposed on it. This eddy is very simple but a good place to begin our catalog. In Fig. 2a we see the wavelet energy spectrum of the noise alone, and then in Fig. 2b the eddy alone and finally the eddy plus the noise. The horizontal axis represents a measure of energy in dimensionless units. The vertical axis is the wavelet scale FIG. 4. The wavelet energy profiles of two eddies of different strength and size. with the smaller numbers representing the smaller scales. Essentially, there are two features that should be noted: first, the energy level of the noise is much lower than the energy level of the eddy. Of course, since the noise is computer generated, we could have made it as strong as we like. However, we tried to make it as close as possible to noise levels found in real datasets. Second, note the structure of the energy spectrum for the noise and the eddy. One can see that the noise spectrum has most of its energy in the smallest scales and the energy level decreases as the scale size goes up. On the other hand, the energy spectrum of the eddy is quite different. The eddy has more and more energy with increasing scale until it reaches a scale 6 at which point the energy decreases. The values of the energy at scale and the structure of the energy spectrum work effectively as a wavelet signature. b. Two Gaussian eddies without noise From now on we will leave out noise since we have a basic understanding of it in the wavelet basis and since the energy levels of noise are extremely small.

7 MARCH 2002 LUO AND JAMESON 387 Table 2. Location of centers and averaging regions for three eddies at time 2. Eddy label A B C Eddy center 36.75N, E 34.75N, E 32.25N, E Neighborhood of eddy for averaging N, E N, E N, E d. The wavelet signature of a front Our final feature of interest is that of a front or discontinuity in the dataset. Locating discontinuities represents no challenge at all for wavelets and we will not spend much time on it. In a word, wavelets detect information at various scales and locations. Discontinuities have large amounts of energy at all scales and at one location and is straightforward to identify. In fact, if the front is a true discontinuity then the ump in the wavelet coefficients will be seen in all scales. Intuitively one would expect this since one way of viewing wavelet analysis is as a sequence of high-pass filters and a discontinuity will certainly have high-frequency energy at all scales. FIG. 5. The contour diagram of four eddies of different strength and size. The eddies are labeled A through D from the bottom to the top of the domain. The second entry in our catalog is a domain that contains two eddies, one significantly larger than the others, see Fig. 3 for the contour diagrams. In Fig. 4, we can see the wavelet energy spectrum for each of these two eddies. The top plot shows the energy profile for the smaller eddy and the bottom plot shows the energy profile for the larger eddy. One can see that these two eddies have significantly different energy profiles especially the energy levels at each wavelet scale. c. Four Gaussian eddies without noise The final entry in our catalog is an example with four eddies in the domain (see Fig. 5). Note that in this case the salient point is the ability of the technique to automatically find and separate these eddies. This is a key feature of the technique and the software implementation. In Fig. 6 we see the wavelet energy profiles for the four eddies. The energy levels at each scale for each eddy are quite distinct and there is no danger of ambiguity. 5. Identify, label, and track eddies in TOPEX/ Poseidon data This section is the culmination of the paper. Here we apply the previous developed method to real satellite data and illustrate that we can actually identify, label, and track real eddies. a. Eddies in a wavelet basis First of all, notice a simple wavelet decomposition of one snapshot of T/P data. In Fig. 7 we have plotted simply the magnitude of the wavelet coefficients. One can see that in the wavelet basis the eddies become easily observed. Our next challenge will be to identify, label, and track such eddies. b. Tracking real eddies The software implementation of this method easily finds and counts eddies in the domain. Here we choose three eddies from a full set of eddies and show that we can track the eddies and give a history of the evolution of their energy at various wavelet scales. We label our Table 1. Location of centers and averaging regions for three eddies at time 1. Table 3. Location of centers and averaging regions for three eddies at time 3. Eddy label Eddy center Neighborhood of eddy for averaging Eddy label Eddy center Neighborhood of eddy for averaging A B C 36.75N, E 35.25N, E 31.75N, E N, E N, E N, E A B C 36.75N, E 34.75N, E 32.25N, E N, E N, E N, E

8 388 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 19 FIG. 6. The wavelet energy profiles of four eddies of different strength and size. three eddies A, B, and C; they have been selected in the first snapshot for tracking (see Fig. 8). The second and third snapshots illustrate that we can maintain a lock on the proper eddies without our scheme umping to other eddies. In Fig. 9 we give the wavelet energy profiles of the three eddies in the three different snapshots. The energy profiles are the amounts of energy at the wavelet scales from 1 to 6. For such energies to be found, we must define boxes around each of the eddies that contain most of the eddy energy. The bounds for these boxes are given in Tables 1 3. FIG. 7. The magnitude of D4 wavelet coefficients after one wavelet decomposition. One can see that the D4 wavelet clearly identifies the eddies and eddy centers. c. A dense field of eddies In a real application we would strive to analyze data from T/P that are essentially dense with eddies. That is,

9 MARCH 2002 LUO AND JAMESON 389 FIG. 8. Sea surface height contours showing eddies in three snapshots of T/P data at 10-day intervals. Three eddies have been identified, labeled (A, B, and C), and tracked. FIG. 9. The wavelet energy profiles of the three eddies labeled A (solid line), B (dash line), and C (dotted line) at three different times at 10-day intervals. in the entire two-dimensional ocean surface we would find ocean eddies at all locations and our technique would need to be robust in order to be able to handle such a challenging environment. In Fig. 10 we analyze such a dense field of eddies. We label and order the eddies from weakest to strongest, see Table 4 and we give the coordinates of a bounding box. Note that such a complex eddy field presents very little for the algorithm. One should note that if the eddy is near to the boundary of the domain, we elected not to attempt to label and measure the wavelet energy of such an eddy since it would be difficult to get a clean measure of the eddy s energy without corruption from the boundary. 6. Conclusions In this manuscript we have shown how wavelet analysis can be used to identify various structures that might FIG. 10. Fifteen eddies identified and labeled according to wavelet energy.

10 390 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 19 Table 4. Fifteen eddies ranked according to wavelet energy at the finest scale. Columns 2 and 3 give the location of the eddy in plot 10. Eddy label Eddy center 37.25N, E 31.75N, E 34.75N, E 35.25N, E 36.25N, E 37.75N, E 36.25N, E 32.75N, E 37.25N, E 31.75N, E 34.75N, E 34.25N, E 34.75N, E 34.75N, E 36.75N, E Neighborhood of eddy for averaging N, E N, E N, E N, E N, E N, E N, E N, E N, E N, E N, E N, E N, E N, E N, E occur in a dataset. Such a dataset might come from observational data or perhaps from numerical simulations. We have tried to present a general approach to structure identification using wavelet analysis, but in the end we focused on one structure, the eddy, that seems to occur in many fields. Understanding and tracking eddies can provide valuable information about various physical processes especially mixing. We consider our technique to be nothing more than a first attempt at eddy tracking and in the future we expect to refine our method and to test it on more challenging datasets. The key advantage of the approach presented here is that it can easily be automated so that very large datasets can be analyzed without human input. Further, the scale information provided by wavelet analysis offers a very straightforward manner for a wavelet signature, which provides essentially a unique identifying mark to help to keep eddies separated. Also, the time evolution of the wavelet energy provides information on the transfer of turbulent energy in the eddies across scales. In short, the information provided by wavelet analysis cannot easily be obtained from other methods such as Fourier analysis or direct observation of contour plots. Acknowledgments. The two authors would like to express their appreciation to Professor Toshio Yamagata for his support and for bringing the two of us together for a successful collaboration. This work was performed under the auspices of the U.S. Department of Energy by the University of California Lawrence Livermore National Laboratory under Contract W-7405-Eng-48. REFERENCES Chui, C., 1992: Wavelets: A Tutorial in Theory and Applications. Vol. 2. Academic Press, Daubechies, I., 1992: Ten Lectures on Wavelets. Society for Industrial and Applied Mathematics, 357 pp., 1998: Orthonormal basis of compactly supported wavelets. Commun. Pure Appl. Math., 41, Erlebacher, G., M. Y. Hussaini, and L. Jameson, 1996: Wavelets: Theory and Applications. Oxford, 510 pp. Jameson, L., and T. Miyama, 2000: Wavelet analysis and ocean modeling: A dynamically adaptive numerical method WOFD- AHO. Mon. Wea. Rev., 128, Kirby, M., 2001: Geometic Data Analysis: An Empirical Approach to Dimensionality Reduction and the Study of Patterns. John Wiley and Sons. Meyer, Y., 1990: Ondelettes et Operators. Hermann, 215 pp. Saito, N., 1998: Least statistically-dependent basis and its application to image modeling. Wavelet Applications in Signal and Image Processing VI, A. F. Laine, M. A. Unser, and A. Aldroubi, Eds., The International Society for Optical Engineering, Strang, G., and T. Nguyen, 1996: Wavelets and Filter Banks. Wellesley-Cambridge Press, 490 pp.