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DiscussionsClifford algebra-based structure filtering analysis for geophysicalvector fields

Z. Yu1,2, W. Luo1, L. Yi 1, Y. Hu3, and L. Yuan1,2

1Key Laboratory of Virtual Geographic Environment, Ministry of Education, Nanjing Normal University,No.1 Wenyuan Road, Nanjing, China2Jiangsu Provincial Key Laboratory for Numerical Simulation of Large Scale Complex Systems, Nanjing Normal University,No.1 Wenyuan Road, Nanjing, China3Department of Computer Science and Technology, Nanjing Normal University, No.1 Wenyuan Road, Nanjing, China

Correspondence to:L. Yuan (yuanlinwang@njnu.edu.cn)

Received: 28 May 2013 – Revised: 24 June 2013 – Accepted: 25 June 2013 – Published: 31 July 2013

Abstract. A new Clifford algebra-based vector field filter-ing method, which combines amplitude similarity and direc-tion difference synchronously, is proposed. Firstly, a modi-fied correlation product is defined by combining the ampli-tude similarity and direction difference. Then, a structure fil-tering algorithm is constructed based on the modified corre-lation product. With custom template and thresholds appliedto the modulus and directional fields independently, our ap-proach can reveal not only the modulus similarities but alsothe classification of the angular distribution. Experiments onexploring the tempo-spatial evolution of the 2002–2003 ElNino from the global wind data field are used to test the al-gorithm. The results suggest that both the modulus similarityand directional information given by our approach can revealthe different stages and dominate factors of the process of theEl Nino evolution. Additional information such as the direc-tional stability of the El Nino can also be extracted. All theabove suggest our method can provide a new powerful andapplicable tool for geophysical vector field analysis.

1 Introduction

The vector field data not only have higher data dimen-sions, but also show complex structures and express moreattractive, meaningful and vivid information (Mendoza et al.,2010). Traditional vector field analysis methods based onvector algebra and calculus are mostly used for ideal fields.They are sensitive to the noise of data and their robust-ness need further tests (Tafti and Michael, 2011). Statistical

methods such as EOF/PCA (Empirical Orthogonal Function/Principal Components Analysis) divide the vector field signalinto two or more scalar signals, which will lead to the seg-mentary, ambiguous or even wrong modes (Paulus and Mars,2006; Yu et al., 2011). The development of pattern filteringmethod for feature extraction may be helpful for the process-ing and analysis of vector field data (Yuan et al., 2013).

Template matching is one of the most commonly usedtechnologies for feature filtering. Clifford convolution is ap-plied to compute the direction and similarities between vec-tor field and ideal vector field template (Ebling and Scheuer-mann, 2005), and it is also used in trajectory and eddy char-acteristics researches (Brassington et al., 2011). These stud-ies provide references for developing more powerful vectorfield filtering technologies. Aiming at filtering meaningfulgeophysical signals, the new-developed method should meetthe following criteria: (1) it should be noise-insensitive, andcan directly support complicated geological observation dataas the template; (2) both the amplitude similarities and direc-tions should be carefully integrated in the filtering process;(3) it should allow a threshold range of both amplitude sim-ilarities and angular difference; (4) it should have efficientperformance, and can support global scale analysis.

In this article, the Clifford convolution is introduced as thefoundation to construct the vector field data filtering algo-rithm. The vector field data are expressed by the Clifford al-gebra bases and split into the modulus fields and unit lengthdirection fields to separate the amplitude and direction infor-mation. The Clifford convolution is applied into the originalvector field and the template data. The scalar part and the

Published by Copernicus Publications on behalf of the European Geosciences Union & the American Geophysical Union.

564 Z. Yu et al.: Geophysical vector fields structure filtering analysis

angular part of the convolution result are discussed. Then,both the scalar and the angular part of the convolution arereplaced with the Normalized Cross Correlation (NCC) ofthe modulus fields and the angle estimated by a SVD (Singu-lar Value Decomposition)-based optimization model, respec-tively. With given thresholds on the modulus correlation andmean direction difference, the original field can be filteredwith both the spatial structure of modulus distribution andthe directions. The preliminary materials for constructing themethods are given in Sect. 2. In Sect. 3, we apply the methodto extract the spatio-temporal evolution of the El Nino fromglobal wind data. Finally, the conclusions are given in Sect. 4.

2 Method

2.1 Multivector expression of vector fields inCl2,0

The most commonly used 2-D vector field expression isfounded on Euclidean space with Cartesian coordinates.Their expressions are highly dependent on the coordinates.The Clifford algebra provides the coordinate-free expressionand computation of vectors. With orthogonal Clifford algebrabasese1 ande2 in Clifford algebra spaceCl2,0, any vectorvin Cl2,0 space can be directly expressed as

v = v1e1 + v2e2,v ∈ R2⊂ Cl2,0. (1)

A discrete 2-D vector fieldF : � ⊂ R2→ R2 can be ex-

pressed with a set of vectors in the following form:

F : � ⊂ R2→ R2

= {xij }, i = 1, · · · ,M,j = 1, · · · ,N. (2)

The geometric product can be defined for two vectors. Forany given two vectorx = x1e1 + y1e2 andy = x2e1 + y2e2,their geometric product can be defined as

xy = x · y + x ∧ y. (3)

In Eq. (3), the part constructed by inner product is a scalar.However, the part constructed by outer product is a bi-vector,which hasi2 = e1e2 part. Multivector, one of the fundamen-tal tools of Clifford algebra, integrates different dimensionalsubspace spanned by the basis elements in a single structure.The hybrid expression and unified computation of differentdimensional subspace provide ultimate power of the Clif-ford algebra. InCl2,0 space, there are only four elements,the scalar,e1,e2 and e12 = e1e2. The scalar part, which isa constant, has a grade of 0. Both thee1 and e2 parts aregrade 1 vectors and thee12 part is a bi-vector of grade 2. Allthe four parts can construct multivector functions inCl2,0space with a general form ofA = a0 + a1e1 + a2e2 + a3e12.All the computations inCl2,0 are closed in this generalform. Different from the complex or tensorial approaches,the Clifford algebra approach encodes geometric objects ofall dimensions (including all subspace dimensions) as al-gebraic objects and allows measurements of length, areas

and volumes, and of dihedral angles across all dimensions(Hestenes and Sobcyk, 1984; De Bie et al., 2011). Cliffordalgebra allows coordinate free formulations and computation(Hestenes and Sobcyk, 1984).

2.2 The Clifford convolution

The definition of convolution is extended to vector field do-main, and provides a template matching technology for vec-tor field data (Ebling and Scheuermann, 2006). In discretizedform, the convolution is computed directly by the geometricproduct. The center point value of the convolution result isthe sum of the geometric product of all the correspondingvalues of the vector field and the vector template in the con-volution window. Since the geometric product is composedby the inner and outer product, the convolution between twodiscrete 2-D vector fieldsf (x) andg(x) can be expressed as

(f ∗ g)(x) =∑t

f (t)g(x − t)

=

∑t

f (t) · g(x − t)︸ ︷︷ ︸The inner product part

+

∑t

f (t) ∧ g(x − t)︸ ︷︷ ︸The outer product part

. (4)

In Eq. (4), the part constructed by the inner product is ascalar field, which indicates the angle differences betweenthe original vector field and the template data in the sam-pling windowst . However, the part constructed by the outerproduct is a bi-vector, which hasi2 part. In Ebling’s templatematching technology, the angle information is pre-processedseparately, and only the magnitude fields are used in the finalcorrelations computation (Ebling and Scheuermann, 2006).

Any 2-D vector can be linearized as a modulus fieldmultiplied by a unit length vector. The assumption is thatfm(x) = |f (x)| andgm(x) = |g(x)| are the modulus of geo-physical vector fieldf (x) and template fieldg(x), and thenormalized directional vector fieldfd(x) = (f1(x),f2(x))

and gd(x) = (g1(x),g2(x)) with unit length, which meansf1(x)2

+ f2(x)2= 1 and g1(x)2

+ g2(x)2= 1. Then there

is a relation that f1(x)g1(x) + f2(x)g2(x) = cosθ(x)

and f1(x)g2(x) − f2(x)g1(x) = sinθ(x), where θ(x) isa function which states the angular differences betweenf (x) and g(x) at each corresponding locationx. By sep-arating the modulus field and the direction field, both theoriginal vector field and the template field can be rewrittenas f (x) = fm(t)fd(t) = |f (x)|(f1(x)e1 + f2(x)e2) andg(x) = gm(t)gt (t) = |g(x)|(g1(x)e1 + g2(x)e2). WithEq. (4), we have

(f ∗ g)(x) =∑t

f (t)g(x − t)

=∑t

(fm(t)fd(t))(gm(t)gt (t))

= |(f ∗ g)(x)|(cosθ(x) + i2sinθ(x))

= |(f ∗ g)(x)|ei2θ(x)

(5)

whereθ(x) = arctan(<(f ∗g)(x)>2(−i2)<(f ∗g)(x)>0

).

Nonlin. Processes Geophys., 20, 563–570, 2013 www.nonlin-processes-geophys.net/20/563/2013/

Z. Yu et al.: Geophysical vector fields structure filtering analysis 565

In Eq. (5), the convolution result of two vector field dataf (x) andg(x) can be separated as two parts: the|(f ∗ g)(x)|

part and theei2θ(x) part. The former part is a scalar field,which has a complicated structure and the latter part is afunction which mostly indicates the angular information be-tween the two vector field dataf (x) andg(x) (Ebling andScheuermann, 2005). Unfortunately, this angular informa-tion is not an exact expression of the angular difference be-tween any arbitrary vector field. Any arbitrary linear vectorfield can be decomposed as a linear combination with twosaddles, one source and one vortex field, but not all parts havethe same meaning of the direction information. AlthoughBu-jack et al.(2012) defines the total rotation of any arbitrarytwo-dimensional vector fields and provides an estimation al-gorithm, it is not very easy to apply the algorithm into geo-physical vector field data, especially when the amplitude ofthe vector field (i.e. the length of each vectors) is taken intoconsideration.

2.3 The modified correlation product

From Eq. (5), it is clear that the two parts are index of ampli-tude similarity and direction information respectively. There-fore, it will be possible to remodel the formula by replacingthe computation of both the amplitude similarity and angu-lar difference. For the scalar part, there are already lots ofefficient algorithms that can be used, thus we can directly re-place the scalar part with a numeric indicator that can expressthe interaction between the modulus to form a new signal.And for the angular part, the critical thing is how to estimatethe overall difference between the vector fields. As proposedin Bujack et al.(2012), the total rotation can be seen as ap-plying a rotorR between the original and template directionfield data. Therefore, an optimization method can be devel-oped to estimate the least square mean angular difference be-tween the two data. After that, we can then easily apply thethreshold for both amplitude and angular-based filtering.

Traditional convolution between scalar fields only producea value that indicates the relation between the two fields. Butthe result is not invariant with the change of amplitude andnot robust to noise. The NCC coefficient overcomes thesedifficulties by normalizing the results, yielding a cosine-likecorrelation coefficient. For two scalar fieldsf (x) andg(x),the NCC index is defined similarly to the correlation coeffi-cients as followsLewis (1995):

NCC(x) =1

n

∑t

[f (x − t) − f ][g(t) − g]√∑t

[f (x − t) − f ]2∑t

[g(t) − g]2, (6)

wheref , g are the means off (x) andg(x) in the convo-lution window, respectively, andn is the number of pointsin the window. The range of NCC is from 0 to 1, thus theNCC field is a scalar field that measures the similarities ofamplitudes of the two vector fields.

The direction differences between the two vector fieldscan be expressed by a rotor (Luo et al., 2012). For thetwo-direction fieldsfd(x) = f1(x)e1+f2(x)e2 andgd(x) =

g1(x)e1 + g2(x)e2, the adaptive rotor estimation based onthe SVD (Singular Value Decomposition) method can be ap-plied. A versor equation can be defined withfd(x) andgd(x)

with a rotorR:

fd(x) = Rgd(x)R−1+ ε(x). (7)

So for fd(x) = a1,a2, · · · ,ai andgd(x) = b1,b2, · · · ,bi ,the objective function can be defined as

minF(R) =

k∑i=1

(ai − RbiR−1)2

=

k∑i=1

(a2i + b2

i − 2 < aiRbiR−1>0).

(8)

With a constraint condition ofRR−1= 1, the most suit-

able rotorR can be solved with a SVD model:

R∗= [[V]] [[ U]]T (9)

whereR∗ is the optional rotor estimated.[[U]] , [[V]] are es-timated by the SVD of[[fd]] , and[[fd]] = [[U]] [[ S]] [[ V]]T

=

k∑i=1

[[bi ]] [[ ai ]]T .

Rotor can be written as an exponential form ofR = e−Iθ/2

(Dorst et al., 2007), whereθ and I are the rotation angleand rotation plane. In the grade two fields space, the rota-tion plane is exactly thee12 plan, so the exponential equationcan also be reformed as

R = e−θ2 i2 = cos

θ

2− i2sin

θ

2, (10)

then theei2θ(x) part of Eq. (5) can be expressed as

ei2θ(x)= e−

−2θ(x)2 i2 = R(−2θ(x)). (11)

Since the convolution of Eq. (5) is a multvector, by re-placing the scalar part of Eq. (5) with the NCC coefficientsand the estimated mean rotated angleθ∗(x), we can combineamplitude correlation and direction in a single equation. Tomake this equation meaningful and computable, we definethe modified correlation product� to express the similaritiesbetweenf (x) andg(x) as follows:

(f � g)(x)

= NCC(x)︸ ︷︷ ︸Scalar Correlation

[cos(θ∗(x)) + i2sin(θ∗(x))]︸ ︷︷ ︸Angular Vector Field

= NCC(x)R(−2θ∗(x)).

(12)

In Eq. (12), the result of the modified correlation productsuggests a structure similarity scalar field is multiplied by amultivector function of the angular relations. Thus, the mod-ified correlation product can process both the magnitudesand angles synchronously in a signal equation. The physicalmeanings of both magnitudes and angular correlations arealso maintained.

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566 Z. Yu et al.: Geophysical vector fields structure filtering analysis

f(x)

g(x)

fm(x)

fd(x)

fm(x)

gm(x)

fd(x)

gd(x)

NCC(x)

θ*(x)

fout(x)gm(x)

gd(x)

Structure

threshold

Filtered data,

, tN

Structure

threshold , tθ

Fig. 1.The procedure of the template-based structure filtering.

2.4 Template-based filtering of vector field inCl2,0

According to the discussion above, we can design thetemplate-based filtering algorithm (Fig. 1). The original vec-tor field data are first split into the modulus field and the unitlength direction field. Since the modulus field is scalar field,we use the NCC computation algorithm proposed byYoo andHan(2009). For the unit length direction field, the optimizedangle difference field is first estimated by the SVD method.Then, we can apply a certain threshold on the NCC similar-ity NCC(x) and the angleθ∗(x) (e.g. a amplitude correlationmore than 0.6 and the angular difference is less than 45◦)to extract the features according to the relations between theoriginal and template data.

3 Experiments

3.1 Data

Global wind filled and the El Nino Modoki event are studied.The 0.5◦ Grided Global Average QuikSCAT Surface WindVelocity Field data during May 2002 to March 2003 are usedas the original data, and the Nino 3.4 region of the compositeEl Nino Modoki spatial pattern (Wang and Xin, 2013), whichaverages the NCEP-NCAR reanalysis wind velocity vectordata during the seven El Nino Modoki events, are used as thetemplate. Since the composed wind field data (2.5◦

× 2.5◦)have a lower spatial resolution than the QuikSCAT wind ve-locity field data (0.5◦×0.5◦), the composite data are first lin-early interpolated in the 0.5◦ × 0.5◦ grid.

3.2 The directional filtering with parallel template

Clifford convolution with a parallel template can extract thedirection information of the vector field. A 11× 11 paral-lel templateT (x) is generated withT (x) = sin[θ(x)]e1 +

cos[θ(x)]e2 by assigning each point of a constant 2-D vector.The north direction is set asθ(x) = 0◦. Therefore, the tem-plate is directly northward. The directional part of the modi-fied correlation product indicates the direction of the angulardifference from the north. We also compute the direction an-gle with the formula ofθ(x) = arctan(y/x) for comparison,wherex andy are the longitudinal and latitudinal projectionof the normalized direction vector in each point (Fig. 2).

The overall spatial patterns of the angle value calculatedfrom both the convolution and the original data are very sim-ilar. This proves the correctness of our method. Furthermore,the highlighted area suggest our method will produce lesssingularities than the direct computation. The zonal distri-bution of the global wind (e.g. Western wind zone, South-east trade wind zone etc.) can be clearly revealed, and fromeast to central Pacific (10◦ S∼10◦ N, 120◦ E∼180◦), there areclearly boundaries of eastern and western wind. This area isalso the birth place of El Nino event and a place that hasdrastic changes and large influence on global climate.

3.3 The El Nino global impaction pattern extraction

A time slice (September 2002) of the original wind field, thetemplate data, the NCC coefficient and the directional clas-sified angular coefficients are depicted in Fig. 3. The areawith higher NCC coefficient may be strongly affected by thisEl Nino event. The high impact regions are located in theEquatorial Pacific between 30◦ S∼30◦ N and the east coastalregions of the main continents (Fig. 3c). The value ofθ∗(x)

reveals the angular relationship between the original data andthe El Nino Modoki template, which indicate the directionalstructure change. For better expression, we classify the angu-lar differences as eight directions. The direction classificationof the filtered wind field is expressed in Fig. 3d.

The directional pattern of the wind field in September2002 and the template clearly show distinct spatial distribu-tion patterns in different areas. There are more eastward re-gions (in cold color) than westward regions (in warm color)in the whole global ocean. However, significant westward re-gions exist in the Pacific and Indian oceans, which suggeststhe closed linkage with ENSO. The most obvious westwardregion is located at East Pacific, east coast of South Amer-ican, Africa, Asia, Australia and the west coast of Europe.The September 2002 is the forming stage of the El Nino eventand the ENSO influences in these places are much stronger.The boundaries between the eastward and westward regionsin central and east Pacific also have very similar structure tothat of the weak El Nino composed by SST anomalies (Wangand Fiedler, 2006). Yu et al.(2012) suggests its SST anomalycenter located near the coast of South America.

Since the spatial signal can be better separated from thefrequency domain, testing the frequency characteristics ofthe modified correlation product is helpful. The 2-D Fourierspectrums of the original data, the template and the filteredfield are computed and depicted in Fig. 4, which suggeststhat the modified correlation product can extract well andfilter significant patterns and can also reveal the directionaldistribution. From Fig. 4, the spectrums also suggest clearlythe filtering characteristics. In the original vector field, thereare clearly periodical patterns, which have been filtered inthe filtered field. The spectrum structures of the filtered fieldare more similar to the template field. For the directionalpatterns, there is no clear direction in original vector field.

Nonlin. Processes Geophys., 20, 563–570, 2013 www.nonlin-processes-geophys.net/20/563/2013/

Z. Yu et al.: Geophysical vector fields structure filtering analysis 567

2 43 5 6m/s

0°90°S

60°E 120°E 180° 60°W120°W 0°

6

60°S

30°S

0

30°N

60°N

90°N

130°E 170°W160°E5°S

15°N

35°N

170°W 110°W140°W45°S

25°S

5°S

0 1Angle (rad)

6m/s

0°90°S

60°E 120°E 180° 60°W120°W 0°

60°S

30°S

0

30°N

60°N

90°N

130°E 170°W160°E5°S

15°N

35°N

170°W 110°W140°W45°S

25°S

5°S

Angle (rad)

(a)

(b)

2 43 5 60 1

A1

A2

B2

B1

Fig. 2.The direction pattern comparison.(a) is the angle value calculated by optional rotor estimation, and(b) is calculated with the originalvector data.

6m/s

0°90°S

120°E 120°W 0°

45°S

0

45°N

90°N

Speed Speed170°W 150°W

4°S

2°S

0

2°N

4°N

0 4.0 6.0 8.0 10.02.02.0 6.0 8.0 10.00

Niño3.4

130°W

6m/s

6m/s

6m/s

(a) (b)

140°W 80°W110°W30°S

10°S

10°N

145°E 155°W175°E10°N

30°N

50°N

0.2 0.4 0.6

0°90°S

60°E 120°E 180° 120°W 0°

0.8

60°S

30°S

0

30°N

60°N90°N

1.00.0NCC

140°W 80°W110°W

60°W

145°E 175°E 10°N

30°N

10°N

50°N

30°S

10°S

10°N

0° 60°E 120°E 180° 120°W

1 2 3 4 5 6 7 8Angle classes

60°W

(d)

0°

(c)

90°S

60°S

30°S

0

30°N

60°N90°N

Fig. 3. The modified correlation product of September 2002.(a) is the original data,(b) is the template, and the NCC coefficient result anddirection classification result can be seen in(c) and(d).

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568 Z. Yu et al.: Geophysical vector fields structure filtering analysis

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

|F{ux}|2+|F{ux}|2

k y(c

ell-1

)

kx(cell-1)

−3

−2

−1

0

1

2

3

k y(c

ell-1

)−3

−2

−1

0

1

2

3

k y(c

ell-1

)

−3 −2 −1 0 1 2 3

kx(cell-1)

−3 −2 −1 0 1 2 3

kx(cell-1)

|F{ux}|2+|F{ux}|2 |F{ux}|2+|F{ux}|2

Fig. 4. The 2-D spectrum of original data (left), the template (mid-dle) and the filtered field (right). The spectrum is the squared sumof the 2-D Fourier spectrum of the vectors projected in the two di-rections.

However, the direction distribution is clear and strengthenedin the spectrum of the filtered field. These characteristics sug-gest the modified correlation product can be used for pat-tern extraction which integrated both the amplitude and di-rectional part of the two vector fields.

3.4 The evolution of high El Nino impact area

Since both the NCC and the angular filtering can provideclear signals from vector field with template-based filtering,we can explore the tempo-spatial evolution characteristics ofthe 2002/03 El Nino event. The NCC coefficients changingwith time indicate the amplitude similarity structure evolu-tion, which may mainly suggest the strength of the El Ninoevent. And changing in the directional field classification in-dicates the directional effect evolution. In addition, the NCCthreshold of NCC> 0.6 is chosen to select the region highlyrelated to or affected by the El Nino event. In the regionof high NCC (Fig. 5a), the angular fields are divided withineight directional classes for better visualization (Fig. 5b). Thedistributions of the directions as well as the NCC coefficientsare also expressed with the windrose plot, in which, the fre-quencies of the number of grid in each direction are countedand the associated NCC coefficients are expressed with thecolorbar (Fig. 6).

The NCC coefficients suggest the structure change fromMay 2002 to March 2003. This El Nino event is strongly as-sociated with active phases of the MJO (Madden–Julian Os-cillation), an eastward propagating pattern affecting surfacewinds over the warm pool (Zhang, 2005). The characteris-tics of the MJO and its effects on surface wind can also bepartially revealed in the data of NCC filtering. Several con-vective high NCC regions in the Pacific Ocean follow theeastward propagation of MJO. In May and June 2002, twoof the highest NCC centers in the Pacific region appear in(10◦ N, 170◦ E) and (15◦ S, 130◦ W). But in July 2002 thesetwo centers moved to (15◦ N, 175◦ E) and (25◦ S, 115◦ W).The whole high NCC regions in Pacific also totally show aclearly northeastward trend.

From August to October in 2002, the MJO related convec-tion is growing weak. However, the original isolated peaks

in the south ocean and the north Pacific are becoming con-nected, which suggests a global strengthening impact of theEl Nino. Episodic forcing associated with the MJO helpedto “kickstart” the 2002/03 El Nino (McPhaden, 2004), butthen large-scale ocean–atmosphere feedbacks became thedominant forces that amplified and sustained the El Nino.In the maximum development of this event in October–December 2002, the area of the high NCC region also growsrapidly. In October 2002, in the Pacific region there are al-ready two continuous zones that cross from 150◦ E∼90◦ W.The slope of the El nino impact zone is even flatter. A strongsimilarity burst in the Pacific in November–December 2002witnessed the amplification of the westerly wind bursts as-sociated with the El Nino. And from January 2003, the areaof the high NCC region declines. The termination of this ElNino event starts.

3.5 The angular evolution of the El Nino event

By classifying the direction distributions in the region of highNCC with the angular filtering results (Fig. 6), the highestNCC regions locate in Pacific, and the major direction pat-tern is northwestward. Compared to the shape and structurechange of the high NCC region, the northwestward directionpropagation is stable over time. In the boreal summer 2002,the MJO was dominated by a strong eastward component,probably driven by the abnormally high SSTs in the centralPacific (McPhaden, 2004). And in July 2002, the directionalfiltering result suggests the eastward becomes strong, whichcan be well compared with the observation results.

The center where the El Nino event developed is also a keyproblem for different patterns of the El Nino events. Existingliteratures heavily argue the differences among the east Pa-cific (EP) El Nino and central Pacific (CP) El Nino. In ourangular direction classification, the central Pacific is locatedat the boundaries of the eastward and westward propagation.Other indicators also suggest the El Nino event was charac-terized by very high sea surface temperatures (SSTs) in thecentral Pacific (Kug et al., 2010). These further support thatthe central Pacific is the source of this El Nino event evo-lution. From the directional classification evolution we canalso find the main propagation direction is almost stable overtime (Fig. 7), although the original direction of the wind fieldchanges continuously throughout time. In most time slices,the majority direction is the Northwest. The only exceptionis the July 2002, and there are opposite El Nino propagationdirections on the east coast of China compared with that inNorth American. The influence of the heat and water trans-portation situation is also opposite (Cook et al., 2007).

4 Conclusions

The Clifford convolution is modified and applied to vectorfield analysis. By defining the modified correlation product

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Z. Yu et al.: Geophysical vector fields structure filtering analysis 569

0° 90°E 180° 90°W 0°0° 90°E 180° 90°W 0°

(a) (b)

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Fig. 5.The filtered results of modified correlation product with NCC> 0.6 during May 2002 to March 2003.(a) is the NCC filtering results,and(b) is the direction classification filtering results.

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Fig. 6.The direction and NCC distribution of high NCC (> 0.6) fil-tering results. The direction of the windrose indicates the evaluateddirection calculated by the modified correlation product, and the ve-locity of the windrose suggests the magnitude of NCC.

with the NCC and the SVD-based mean angle difference es-timation method, both the amplitude and directional relationscan be analyzed synchronously. The structure filtering algo-rithm can reveal both the modulus similarities as well as theangular distribution. The method can give smooth and con-tinuous result, which may be helpful for vector data segmen-tation or classification. Future improvements, such as intro-ducing more comprehensive correlation indexes as well asmore stable and accurate amplitude correlation and angularestimation model, will greatly extend the application area.

With the El Nino template, the tempo-spatial evolution ofthe global El Nino impact and propagation can be revealed.It suggests the close linkage between the 2002/03 El Nino isfirst forced by the MJO event and then the large-scale ocean–atmosphere feedbacks becomes dominant. Although the am-plitude similarities vary in both temporal and spatial area, thedirectional propagation of El Nino is stable, especially for theareas with high amplitude similarities. This result may leadfurther studies on exploring the implied mechanisms abouthow the wind power and direction change interact with theEl Nino evolution.

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570 Z. Yu et al.: Geophysical vector fields structure filtering analysis

Acknowledgements.This work was supported by the NSCF Project(Grant No. 41201377, 41231173) and NCET program (Grant No.NCET-12-0735).

Edited by: H. J. FernandoReviewed by: two anonymous referees

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