Tensor-guided ﬁtting of subducting slab depths

CWP-726

Tensor-guided fitting of subducting slab depths

Farhad Bazargani1, Dave Hale

1& Gavin Hayes

2

1 Center for Wave Phenomena, Colorado School of Mines, Golden, CO 80401, USA2 U.S. Geological Survey, National Earthquake Information Center, USA

(a) (b) (c)

Figure 1. Scattered depth samples (a) from the subducting slab in South America with tensor-guided interpolation (b) of

depths accounting for the curvature of the earth’s surface, and tensor-guided fitting (c) of depths accounting for both that

curvature and estimated slab strikes. White ellipses represent models of spatial correlation. The solid black lines represent the

west coast of South America.

ABSTRACTEarthquakes and active-source seismic surveys provide estimates of depths ofsubducting slabs, but only at scattered locations. Constructing a useful 3Dmodel of slab geometry involves fitting the depth estimates with a uniformlysampled function of space. The method used to fit the data should account forthe curvature of the earth’s surface as well as data uncertainties.In addition to estimates of depths from earthquake locations, focal mechanismsof subduction zone earthquakes also provide estimates of the strikes of thesubducting slab on which they occur. We use estimated strike directions andthe earth’s curved surface geometry to infer a model for spatial correlationthat guides a blended neighbor interpolation of slab depths. The interpolationmethod is then modified to account for the uncertainties associated with thedepth estimates.

Key words: data fitting, interpolation, subducting slab, tensor-guided, uncer-tainty

262 F. Bazargani, D.Hale & G. Hayes

1 INTRODUCTION

Accurate knowledge of the geometry of the interfaceof a subducting slab has important applications ina variety of seismological analyses such as tsunamimodeling and propagation (Wang and He, 2008), seis-mic hazard assessment (e.g. Global Earthquake Model,http://globalquakemodel.org), tectonic modeling andplate reconstruction, and earthquake source inversions(Hayes and Wald, 2009).

One way of acquiring information about the ge-ometry of a subduction interface is through studyingthe locations and source mechanisms of the earthquakesthat occur on or within the subducting slab. However,earthquakes associated with the subduction process areunevenly distributed. The spatial distribution of suchearthquakes is dense in seismogenic sections of the sub-duction zone and sparse in other areas where seismic-ity rates are low or nonexistent. For example, shallowsections of subducting slabs (above 80 km) are mostlyaseismic. In such areas, in the absence of earthquakes,estimates of slab depths derived from active-source seis-mic or bathymetry surveys can facilitate a subductiongeometry model with higher resolution.

Even after combining earthquake data withbathymetry and active-source seismic surveys, slabdepths are sampled only at scattered locations. Thissituation is illustrated in Figure 1a for the subductingNazca slab beneath western South America. To con-struct a useful 3D model of the slab interface, we mustinterpolate these scattered depths to obtain slab depthas a uniformly sampled function of latitude and longi-tude.

A large variety of interpolation methods are possi-ble. Several previous studies have attempted to modelthe slab geometry in subduction zones, particularly indeeper regions below the seismogenic zone. For exam-ple, Syracuse and Abers (2006), produced hand-drawncontours to match general Wadati-Beniof Zone struc-ture beneath volcanic arcs; others have produced moregeneralized multi-regional (Bevis and Isacks, 1984) andglobal (Gudmundsson and Sambridge, 1998) models.

Hayes et al. (2009) and Hayes et al. (2012) producedSlab1.0 which is a 3D subduction geometry compilationof approximately 85% of the subduction zones world-wide. They used estimates of slab strike derived fromearthquake source mechanisms, depths of the earth-quakes that those mechanisms represent, and a collec-tion of other complimentary datasets to constrain thegeometry of the subduction interface. Their method fitsthe data with a 3D non-planar surface interpolating be-tween a series of 2D sections which sample the slabgeometry every 10 km along the strike of the subduc-tion zone. Each 2D section is constructed using Hermitesplines and consists of two parts: a polynomial fit of or-der 2 or 3 to shallow data (depths≤80 km) splined witha polynomial fit of order 3 or 4 to intermediate and deepdata.

The slab geometry model produced by Hayes et al.(2012) is distinguished from its predecessors becauseit (1) focuses mainly on the shallowest part of sub-ducting slabs while simultaneously attempting to honorthe deeper structure of the slab, (2) filters the earth-quake data sets used to include only well-located eventswith thrust mechanisms associated with the subduc-tion process, and (3) includes additional data sets, suchas bathymetry and subduction interface interpretationsobtained from active-source seismic profiles.

Although Hayes et al. (2012) use strike data to de-termine the azimuth of the 2D cross sections they fitwith splines, they do not incorporate strike data directlyin their polynomial fitting.

In this paper, we use the same filtered subduction-related earthquake and active-source seismic data setsas Hayes et al. (2012), but use a different method to pro-duce a 3D model for the subduction interface in SouthAmerica. Our tensor-guided method is more direct inthat we interpolate all of the scattered depth samples atonce to obtain slab depth as a function of longitude andlatitude. We use the strike directions and the earth’scurved surface geometry to construct a metric tensorfield that guides the interpolation of slab depths.

A metric tensor field provides a measure of dis-tance that need not be Euclidean. For the interpolationshown in Figure 1b, the tensor field depicted by thewhite ellipses describes geodesic distance measured onthe curved surface of the earth. The ellipses are elon-gated because they account for the curvature of theearth’s surface when projected onto an equi-rectangularlongitude-latitude coordinate system.

The tensors used to guide the data fitting, shown aswhite ellipses with variable elongations and orientationsin Figure 1c, account for not only the curvature of theearth’s surface but also the estimates of slab strike direc-tions taken from moment tensor and focal mechanismdata. The ellipses are more eccentric in areas where theslab dips more steeply. The orientations of the ellipsesare determined using the slab strike directions.

The metric tensor fields shown in Figure 1b and 1cguide interpolation (and fitting) by decreasing distancesin directions in which the ellipses are elongated, the di-rections in which slab depths are more highly correlated.These metric tensor fields therefore provide models ofspatial correlation that may yield more accurate mod-els of subducting slab interfaces.

Most interpolation methods require, either explic-itly or implicitly, a model of spatial correlation for thedata to be interpolated. However, some methods forinterpolation do not permit spatially varying modelsof spatial correlation like those depicted in Figure 1.For example, kriging methods in geostatistics do noteasily allow for nonstationary anisotropic correlationmodels (Boisvert et al., 2009). In particular, covariancefunctions commonly used in kriging are not guaran-teed to remain valid (positive definite) when used with

Tensor-guided fitting of subducting slab depths 263

non-Euclidean distance measures (Curriero, 2006). Theblended neighbor interpolation method used in this pa-per to produce the slab models shown in Figure 1 wasdeveloped by Hale (2009) specifically for the purpose ofpermitting the use of spatially varying models of spatialcorrelation.

Another distinguishing aspect of our data fittingmethod is incorporation of uncertainties associated withdata. When estimates of such uncertainty are available,the data fitting process must be capable of properly(in a statistical sense) accounting for them. In otherwords, where significant uncertainty exists, we wish forour model to honor the broad bounds provided by theestimates of uncertainty rather than exactly matchingeach data point.

We show that the blended neighbor interpolationmethod (with a slight modification) is capable of ac-counting for the data uncertainties. This is achieved byaltering the smoothness of the blended neighbor interpo-lation solution so that the prediction errors statisticallymatch the specified uncertainties.

In this paper we first describe the process used tocompute metric tensor fields (e.g., the tensors displayedin Figure 1) and then provide the details of our statisti-cal approach for incorporating data uncertainties in ourfitting method. Finally, we apply our fitting method tothe scattered slab depths shown in Figure 1a to producea new surface for the subducting Nazca slab beneathSouth America and compare it with the same surfacefrom Slab1.0 produced by Hayes et al. (2012).

2 THE METRIC TENSOR FIELD

The methodology described in this paper relies on theblended neighbor interpolation method developed byHale (2009). As shown in appendix A, this method con-sists of two steps, where each step requires a metrictensor field D(x) that defines a measure of distance, orequivalently, a model of spatial correlation.

Different tensor fields may be used in different situ-ations. Hale (2009, 2010) shows examples where metrictensor fields are derived from uniformly sampled images.In other situations, where an image is not available, itmay be possible to derive the metric tensor fields fromother types of secondary data.

In interpolating subducting slab depths, primarydata to be interpolated are estimated slab depths; sec-ondary data, from which we derive the tensor field, areestimated slab strikes. The slab depths and strikes aremeasurements acquired on the curved surface of theearth as a function of longitude and latitude. There-fore, any metric tensor field that we use to guide theinterpolation of depths must account for this curvature.

Hale (2011) studies tensor guided interpolation ofscattered data on arbitrary non-planar surfaces and pro-vides a general recipe for construction of a tensor field

Strike

Dip

A’

B’C’

B

CA

D

D’

N

γ

δ

Figure 2. Depths are most highly correlated in the strike

direction of the slab. Point D is located somewhere between

points A and B in the slab strike direction. In this configu-

ration, although D’ is closer to C’, the depth at D’ is more

similar to the depth at A’ and B’ than it is to the depth at

C’. γ denotes the strike angle defined as the azimuth of the

strike and measured relative to geographic north N . Dip is

perpendicular to the strike direction. δ denotes the dip angle

measured relative to horizontal.

needed for guiding such an interpolation. Here, we fol-low the same methodology. However, we do not needthe same level of generality as provided by Hale (2011).Therefore, in what follows, we will reproduce some ofthe relevant results presented in Hale (2011) and showthe steps of implementing them for addressing the spe-cific problem at hand.

We define the earth’s surface parametrically by amapping from 2D longitude-latitude coordinates u to3D Cartesian coordinates x(u) : U ∈ R2 → X ∈ R3,and interpolate slab depths in the 2D space U . In thisspace each location u is specified by longitude φ andlatitude θ.

The tensor field required to guide a blended neigh-bor interpolation of depths is constructed in two steps.First, we ignore the curvature of the earth and definea tensor field in an infinitesimal plane tangent to theearth’s surface, where we can assume a flat geometry.Next, we modify this strike tensor field to account forthe curvature of the earth.

2.1 Strike tensor field

We base our method for constructing the strike tensorfield on the fact that the slab depths should be mosthighly correlated in the slab strike directions; the spatialcorrelation of the slab depths is lowest in the slab dipdirection (perpendicular to the strike direction). Thisfact follows intuitively (Figure 2) from the definitionsof the strike and dip directions for a dipping plane. Putanother way, the dipping structure of the slab results in


an anisotropic model of spatial correlation for the slabdepths. This anisotropy is proportional to the dip angleor slope of the slab; the anisotropy is higher for steeperslopes.

The metric tensor D in the eikonal equation A3 isa 2 × 2 symmetric and positive definite matrix (Hale,2009) with two orthonormal eigenvectors s and d andcorresponding to positive and real eigenvalues λs andλd. D can be graphically represented by an ellipse (e.g.,see white ellipses in Figure 1) elongated in the direc-tion of the eigenvector corresponding to the maximumeigenvalue and with axes proportional to the squareroots of eigenvalues (Hale, 2011). In general, for a tensorfield that represents an anisotropic model of correlation,these two eigenvalues are not equal. Here we assume0 < λd ≤ λs.

In the parametric space U ∈ R2, the desiredstrike tensor field D(u) must be designed so that non-Euclidean distances to samples in the strike directionare shorter; such samples therefore get more weightin the interpolation. This design can be achieved bypointing eigenvector s in the slab strike direction andeigenvector d in the slab dip direction (perpendicularto strike), i.e.,

s(u) =

�cos γ(u)sin γ(u)

�and d(u) =

�− sin γ(u)cos γ(u)

�, (1)

where γ(u) represents the estimated strike angle of theslab (the angle between the estimated strike directionand geographic north) at location u on the earth’s sur-face.

Now we use eigen-decomposition to construct eachtensor D as

D = λsssT + λddd

T , (2)

where s and d are the eigenvectors defined in equation1, and λs and λd are their respective eigenvalues. In con-structing the strike tensor field for guiding the blendedneighbor interpolation of slab depths, what is impor-tant is the ratio of the eigenvalues and not their actualsizes; i.e., the aspect ratio of the tensor ellipses is im-portant, not the actual size of their axes. Therefore, wecan normalize the eigenvalues and use λs(u) = 1 andsome value of 0 < λd ≤ 1 in equation 2. We let

λd(u) =1

1 + η tan2 δ(u), (3)

where δ(u) is the slab dip angle at location u and η isa non-negative real parameter.

Note that with λs = 1 the aspect ratio (eccen-tricity) of the tensor field ellipses is determined by λd.Therefore, using larger values for η amounts to increas-ing the eccentricity of these ellipses and hence the degreeof anisotropy in our model for spatial correlation of slabdepths at locations where tan(δ), the slope of the slab,is nonzero.

As shown in section 4, a value for the parameter

η can be determined using a 1D line search and cross-validation. At this point, however, we assume that thevalue for η is known and proceed to the next step whichis modifying the strike tensor D in equation 2 to de-fine a metric tensor field that accounts for both strikedirections and the curvature of the earth.

2.2 Accounting for curvature of the earth

The geodesic distance between two points on a non-planar surface is non-Euclidean. This distance can becalculated by solving an eikonal equation written in theparametric space in which the curved surface is defined(Weber et al., 2008).

In interpolation of subducting slab depths, dis-tances are non-Euclidean because of anisotropy in themodel for spatial correlation of depths (inferred fromestimated strike directions and the dip angles) and alsobecause of the curvature of the earth’s surface. There-fore, to correctly specify the non-Euclidean measure ofdistance, the desired metric tensor field must accountfor a combined effect of slab strikes and curvature ofthe earth.

We approximate the shape of the earth by a sphere.The gradient operator expressed in spherical coordi-nates is

∇ =∂∂r

r+1

r cos θ∂∂φ

φ+1r

∂∂θ

θ. (4)

If we denote the radius of the earth by a constant R,then on the earth’s surface r=R and equation 4 becomes

∇ =1

R cos θ∂∂φ

φ+1R

∂∂θ

θ. (5)

Note that in this equation, R cos θ and R are the respec-tive metric scale factors for longitude φ and latitude θin the spherical coordinate system. Equation 5 can bewritten in matrix form as

∇ =

�1

R cos θ 00 1

R

� � ∂∂φ∂∂θ

�(6)

or equivalently, following the notation of Hale (2011),as

∇ = F−T∇u, (7)

in which F and ∇u are defined by

F =

�R cos θ 0

0 R

�(8)

and

∇u =

� ∂∂φ∂∂θ

�. (9)

∇u denotes the gradient operator in the parametriclongitude-latitude space U ∈ R2 defined in section 2.Replacing the gradient operator in eikonal equation A3with the right hand side of equation 7, we can write the


(a) (b)

Figure 3. Scattered estimates of subducting slab depths (a) and estimates of uncertainties (b) specified as the standard

deviation of the errors associated with depth data. The solid black line represents the west coast of South America. The data

points depicted here have been gridded according to the procedure described in the text and consist of 2675 earthquake source

locations, 344 measurements from active-source seismic surveys (data points along linear tracks perpendicular to the coastline),

and 2057 measurements from bathymetry surveys (data forming a stripe, parallel to the coastline).

eikonal equation in the parametric space U as

∇utT F−1 D F−T ∇ut = 1. (10)

In this equation, D is the anisotropic spatial model forcorrelation (the strike tensor field) defined in equation 2.

In eikonal equation A3, matrix D, sandwiched be-tween the two gradients, is the metric tensor. Similarly,in eikonal equation 10, matrix Du, defined as

Du = F−1 D F−T , (11)

is sandwiched between two gradients and therefore canbe regarded as the metric tensor in parametric space U .Using Du we can write equation 10 entirely in paramet-ric space U as

∇ut(u) · Du(u) · ∇ut(u) = 1. (12)

This equation is mathematically similar to equation A3in every aspect except that it is expressed in thelongitude-latitude parametric space U . We can use ten-sor field Du defined in equation 11 to guide the blendedneighbor interpolation of slab depths on the non-planarspherical surface of the earth.

3 INTERPOLATING SLAB DEPTHS

3.1 Data and initial gridding

Primary data used in this study are scattered estimatesof depth of the subducting slab in South America (Fig-ure 3a). Each depth estimate is considered to be a ran-dom variable with an associated uncertainty (Figure


(a) (b) (c)

Figure 4. Scattered strike data (a) are interpolated on the curved surface of the earth to construct a uniformly sampled strike

field (b). The strike field shown in (c) is the result of the 40th iteration of interpolating the scattered strikes guided along the

slab dip direction. White ellipses in (b) represent the tensor field G that was used to guide the interpolation on the curved

surface of the earth. These ellipses are elongated as they account for the curvature of the earth’s surface when projected onto

an equi-rectangular longitude-latitude coordinate system. The white ellipses in (c) are elongated in the slab dip direction and

are used to guide the interpolation of strikes in the dip direction.

3b). Following Hayes and Wald (2009), we assume a nor-mal vertical probability density function (pdf) for deptherrors. For earthquake data, the half-width (standarddeviation σ) of the normal pdf is chosen based on thelocation uncertainty reported in the earthquake catalogfrom which the data were extracted. For active seismicand bathymetry data, the pdf half-width σ is based onan assumed uncertainty related to time-to-depth con-version or estimates of sediment thickness (Hayes et al.,2012).

Secondary data consist of scattered estimates of thestrike direction of the slab (Figure 4a). The strike dataare taken (Hayes and Wald, 2009) from best-fitting dou-ble couples of CMT solutions when available (using thegCMT catalog, www.globalcmt.org).

As a first step, we use a uniform rectangular meshwith grid cells of size 0.1◦ × 0.1◦ to grid the depthand strike data separately. The value assigned to each

grid cell containing bathymetry, active-source seismic,or strike data is the average of all sample values con-tained in the cell. No values are assigned to the gridcells that contain no strike or depth data.

For earthquake depth data, we use a weighted av-eraging scheme described in Appendix B, i.e., the depthvalue and the uncertainty assigned to a grid cell thatcontains more than one earthquake sample are com-puted according to equations B2, B4, and B5. This isbecause the uncertainties associated with earthquakedepths are variable and uncorrelated. By using thisweighted averaging scheme, we ensure that more cer-tain depth estimates are given more weight than lesscertain estimates.

Figure 3a shows the location and values (depictedas colors) of the 5076 depth samples after gridding, andFigure 3b shows the uncertainties associated with thesesamples on a longitude-latitude map. The location and


values of the 458 gridded strike samples are depictedin Figure 4a. The colors represent the estimates of thestrike angle γ of the slab measured relative to geographicnorth.

3.2 Interpolation of the strikes

To perform a blended neighbor interpolation of slabdepths, we must construct a metric tensor field usingthe process described in section 2. This tensor field mustbe defined at each point on the interpolation grid. Asshown by equations 1 and 2, to define the metric ten-sor field D at each grid point, we need the value of thestrike angle γ(u) at that point. Therefore, we must firstinterpolate the scattered strikes.

As shown in Figure 4a, strike data are only avail-able in the region near the coastline. This makes it moredifficult to produce accurate estimates of the strike an-gles for deeper parts of the slab away from the coast.

However, the strikes of the slab are most highlycorrelated in the dip direction that is perpendicular tothe strike direction. This follows from the definition ofstrike and dip for a dipping plane (Figure 2). Therefore,to obtain more accurate estimates of the strike angles,we guide the interpolation of the scattered strikes in thedip direction using an iterative scheme which consists ofthree steps:

(i) Interpolate the strikes using tensor-guided in-terpolation with a metric tensor field that accounts foronly the curved geometry of the Earth’s surface. In thiscase, the non-Euclidean distances are simply geodesic.Therefore, we can use D = I in equation 11 to constructa geometry tensor field G defined as

G = F−1F−T . (13)

Figure 4b shows the result of the interpolation of scat-tered slab strikes as a uniformly sampled strike fieldγg(u). Tensor field G used to guide this interpolation isrepresented by the white ellipses.

(ii) Compute a dip direction field γd(u) using thestrike field γg(u) as

γd(u) = γg(u) + 90◦ (14)

because the dip direction is perpendicular to the strikedirection.

(iii) Construct a tensor field (white ellipses in Fig-ure 4c) based on γd(u) and use it to guide a new inter-polation of scattered strikes, in the dip direction. Theresult is a new strike field we call γ(u).

This is the end of the first iteration. A new iter-ation starts by going back to step (ii) but this time,instead of γg(u) in equation 14, we use γ(u) obtainedin the previous iteration. This process converges (after40 iterations) to the strike field shown in Figure 4c.

3.3 Interpolation of the depths

With the strike field γ(u) obtained in section 3.2, wedefine tensor field D according to equation 2, where weset λs = 1 and compute λd using equation 3 with η = 92.We chose this value for η using the procedure describedin section 4.

Next we compute Du, according to equation 11, bymultiplying D on the left by F−1 and on the right byFT . This is equivalent to combining the strike tensorfield D with geometry tensor field G defined in equation13 to get the combined tensor field Du.

At this point we have all the components needed toperform a blended neighbor interpolation of slab depthson the surface of the earth. Figure 5 shows the resultsof two blended neighbor interpolations of slab depthsguided by two different tensor fields. In Figure 5a, thetensor field G used to guide the interpolation (shownas white ellipses) accounts for only the curvature of theearth’s surface. In Figure 5b, the tensor field Du usedto guide the interpolation (shown as white ellipses) ac-counts for both that curvature and the strike directions.

4 CROSS-VALIDATION

The solution to any interpolation problem is non-uniqueand depends on the method used for interpolation;the interpolants shown in Figure 5 are just two so-lutions among infinitely many. These solutions wereboth obtained using the blended neighbor interpolationmethod. However the tensor fields used to guide the in-terpolations were different.

A possible criterion for assessing the results of aninterpolation method is to analyze the accuracy of itspredictions. We use a 10-fold cross-validation technique(Kohavi, 1995) to find an optimal value for the param-eter η in equation 3.

In 10-fold cross-validation, input samples are ran-domly partitioned into ten mutually exclusive subsets(the folds) of approximately equal size (Kohavi, 1995).Subsequently ten iterations of interpolation and vali-dation are performed such that within each iteration adifferent subset of samples is held out for validation andthe union of the remaining 9 subsets (the training set)is used for interpolation.

The validation process involves computing the nu-merical difference between interpolated values and test-set sample values normalized by the estimates of un-certainty (standard deviation of error) associated withthe samples. Based on this normalized difference, a di-mensionless error is assigned to each sample in the testset. Therefore, after 10 iterations, a cross-validation er-ror is assigned to all data samples. The accuracy of theinterpolation method can then be assessed by analyz-ing these cross-validation errors. For instance, we cancompute the root mean square (rms) of the normalized


(a) (b)

Figure 5. A strike-ignorant interpolation (a) of slab depths on the earth’s surface, in which the metric tensor field used to

guide the interpolation of depths accounts for only the difference between Euclidean and geodesic distance and a strike-guided

interpolation (b) in which the metric tensor field also accounts for estimated strike directions. Ellipses represent the metric

tensor fields.

errors and use it as a measure of accuracy of an inter-polation method.

Now we explain how to use cross-validation to findan optimal value for η in equation 3. The idea is toconstruct (according to the process described in sec-tion 2) different tensor fields using different values forη, and compute a separate tensor-guided interpolantusing each of these tensor fields. We then compute across-validation rms normalized error for each of thesetensor-guided interpolants. The optimum value for η is

the one corresponding to the interpolant with minimumcross-validation rms normalized error.

One difficulty with this approach is that equation 3requires knowledge of dip angles of the slab. To computethe dip angles, we smooth the strike ignorant interpolantof Figure 5 in order to obtain an approximate initial slabmodel qi(u). While inaccurate, this initial model is goodenough for the purpose of approximating dip angles as

δ(u) ≈ tan−1 �∇qi(u)�. (15)


η

η

Figure 6. Cross-validation rms normalized error computed

for the strike-guided interpolant as a function of η. The nor-

malization involves dividing the cross-validation error at each

data point by the estimate of the uncertainty (standard de-

viation) associated with that point. The rms error curve has

a minimum of 2.82 at η = 92 denoted by the red cross. The

area within the red ellipse is magnified in the subfigure to

show the location of the minimum. The value of the rms nor-

malized error at η = 0 corresponding to the strike-ignorant

interpolation is 6.34.

Substituting this approximation into equation 3 yields

λd(u) =1

1 + η�∇qi(u)�2, (16)

which is more straightforward to implement than equa-tion 3.

Figure 6 summarizes the results of the cross-validation process used to determine the best choice forparameter η. The blue curve shows the cross-validationrms normalized error as a function of η. The most ac-curate strike-guided interpolant (the one with the min-imum cross-validation rms error) is given by η = 92.

For a quantitative comparison of the strike-ignorantand strike-guided interpolants shown in Figure 5, wecontrast their respective cross-validation rms normal-ized errors inferred from the curve in Figure 6. The rmsnormalized cross-validation error for strike-guided in-terpolation with a tensor field constructed using η = 92is approximately 2.82. The same error for the strike-ignorant interpolation guided by a tensor field con-structed using η = 0 is more than three times larger,i.e., 6.34.

5 ACCOUNTING FOR DATAUNCERTAINTIES

The interpolant shown in Figure 5b matches all scat-tered input depth samples. This is expected as the in-terpolation error must be (by definition) zero at the lo-cation of input samples. However, this interpolant doesnot represent a geophysically reasonable model for thesubduction interface because unlike the interpolant sur-

face of Figure 5b, a model for the slab geometry must besmooth and without abundant inflection points (Hayesand Wald, 2009).

Figure 8a is a vertical cross section showing a profile(green curve) of the interpolated slab depths. Note theabundant fluctuations and excessive curvatures in thegreen curve. If real, such fluctuations would imply un-warranted bending strains and stresses within the struc-ture of the slab, especially in the shallow parts wherethe slab is still relatively cold and brittle. Therefore,the fluctuations observed in the interpolated depths aremore likely the result of error in depth estimates and donot represent the true geometry of the slab.

To account for data uncertainties and to obtaina smoother model for the slab, we modify the tensor-guided interpolation method to obtain a data fittingmethod. For this modification, we utilize the given esti-mates of data uncertainties and incorporate them in astatistically plausible manner into the modeling proce-dure.

5.1 From interpolation to data fitting

The tensor-guided procedure described so far is basedon the blended neighbor interpolation method (summa-rized in appendix A). The first step of the blended neigh-bor method can be interpreted as scattering the valuesfk corresponding to the nearest known sample pointsxk to construct a nearest neighbor interpolant p(x).

The second step of the blended neighbor methodcan be interpreted as smoothing p(x) to obtain ablended neighbor interpolant q(x). The extent of thissmoothing at each location x is proportional to the non-Euclidean distance t(x) from x to the nearest knownsample (Hale, 2009). This means that at the location ofknown samples, where t(x) = 0, no smoothing is appliedso the solution to blending equation A4 is q(x) = p(x);Hence, the interpolation condition q(xk) = fk is satis-fied and the interpolant q(x) matches all scattered data.

Satisfying the interpolation condition is an implicitrequirement of any interpolation method. However, insituations where there is uncertainty associated withdata, this condition may not be desirable.

In blended neighbor interpolation, the constraint ofsatisfying the interpolation condition can be relaxed byadding a function w(x) to the term t2(x) in equationA4 so that it becomes

q(x)− 12∇ · (t2(x) + w(x))D(x) · ∇q(x) = p(x).

(17)

This modification implies that the amount of smooth-ing applied to p(x) at every location x (including thelocations of known samples xk) can be controlled bychoosing a proper function w(x). If w(xk) �= 0 then thesolution q(x) of the equation above no longer matchesall the scattered samples. In this case, q(x) does not in-terpolate but, rather, fits the scattered data. Therefore,


instead of interpolation, we shall refer to the modifiedscheme described above as tensor-guided fitting.

Different fitting functions can be obtained using dif-ferent smoothing functions w(x) in equation 19. In prac-tice, we may require a fitting solution that models somereal phenomenon. Therefore it is important to choose aw(x) that results in the desired fitting solution.

For example, in modeling the subducting slab ge-ometry, the uncertainties are not the same for differ-ent types of data and different locations (Figure 3b).The estimates of data uncertainties (variance of the er-rors) are relatively small for shallow sections of the slab(bathymetry and active-source seismic data) and largerfor deeper sections of the slab (earthquake data).

Therefore, the smoothing function w(x) must bedesigned so that the amount of smoothing applied to thelocation of each data sample is proportional to specifieddata uncertainties. One such smoothing function can bedefined as

w(x) = sσ2(x), (18)

where s is a positive scalar parameter and σ(x) is asmooth model of standard deviation that approximatesthe actual standard deviation of the error associatedwith data. This model of standard deviation is shownin Figure 7.

Using w(x) defined above in equation 17, we obtain

q(x)− 12∇ · (t2(x) + sσ2(x))D(x) · ∇q(x) = p(x).

(19)

The smoothing parameter s in equation 19 controlsthe smoothness of the fitting solution q(x). Note thattensor-guided interpolation is a special case (with s = 0)of the more general tensor-guided fitting (with s > 0).

The question left to be answered is, therefore, howto choose the smoothing parameter s.

5.2 Choosing the smoothing parameter

In real problems, there are often constraints that makeone fitting solution preferable to other possible solu-tions. For instance, in constraining the geometry of asubduction slab, a geophysically reasonable model forthe slab is expected to be smooth. Nevertheless, such amodel must not be so smooth that sampled depths arecompletely disregarded.

Therefore, to obtain an optimum fitting function,we must find a balance between the smoothness of q(x)and the degree to which it honors the information givenby data. Before explaining how to find this balance, wedefine some new terms.

We define the standardized data error (error in eachdepth sample value) as

ek =µk − fk

σk(20)

Figure 7. An approximate smooth model σ(x) of the stan-

dard deviation of the error associated with data. This model

is used to design a spatially varying smoothing function.

where k is the sample index, µk is the expected valueof slab depth (true depth) at location xk of the sample,fk is the sample value, and σk is the uncertainty or thestandard deviation of the error associated with the sam-ple. Note that in equation 20, µk and ek are unknownquantities.

Recall that each depth sample is assumed to be arandom variable with a normal distribution N (µk,σk).Therefore, the standardized data error ek is expected tobe a random variable with standard normal distribution.From this, we infer that the collection of all standard-ized data errors E, computed according to equation 20,also constitutes a population that is a standard normaldistribution, i.e.,

E ∼ N (0, 1). (21)

Similarly, we define the standardized fitting error


(residual) at location xk of each sample as

rk =q(xk)− fk

σk(22)

where k is the sample index, q(xk) is the value of thefitting function at location xk of the sample, fk is thesample value, and σk is the uncertainty or standard de-viation of the error associated with the sample.

Our goal is to find a fitting function q(x) that cor-rectly estimates the true slab depths. If this goal is at-tained for some optimum fitting function qop(x), thenwe have qop(xk) = µk and hence, by equations 20 and22, rk = ek. This implies that, for the optimum solu-tion, the collection of all standardized fitting errors Rwill have a standard normal distribution, i.e.,

R|qop(x) ∼ N (0, 1). (23)

This can be used as a criterion to choose the opti-mum smoothing parameter s in our data fitting method.In other words, we can analyze fitting errors associatedwith fitting solutions computed using different smooth-ing parameters, and then choose the one for whichR ∼ N (0, 1) as the optimal smoothing parameter s.

Note that our assumption about the normality ofthe distribution of the fitting errors might not be ac-curate. Therefore, when assessing the distribution ofthe fitting errors, it is important to use a robust statis-tic which is not severely affected by the potential out-liers. One such statistic is the interquartile range (IQR).For N (0, 1), the IQR is the range of values from -0.674 to 0.674 containing 50% of the population. Thus,for an optimum fitting solution, half of the standard-ized fitting errors are expected to lie within the range[−0.674, 0.674].

To find the right smoothing parameter, we startfrom the interpolation solution (i.e., from s = 0) andthen gradually increase the smoothness parameter un-til we reach a point where half of the fitting errors fallwithin and half fall outside of the IQR for a standard-ized normal distribution. We choose the smoothing pa-rameter s for which this condition is satisfied to be theoptimal fitting parameter and use it to compute the op-timum fitting solution.

6 RESULTS AND DISCUSSION

The final result of applying our tensor-guided fittingprocedure to slab data is shown in Figure 8c. Comparedto the interpolating model (shown in Figure 8b), thisfitting model is smoother and is therefore likely to bemore geophysically reasonable. The smoothing param-eter s = 0.56, employed in the tensor-guided fitting toproduce the model in Figure 8c was chosen using themethod described in section 5.2.

The fitting solution obtained by Hayes et al. (2012)in the Slab1.0 subduction slab compilation is shown

in Figure 8d. A cross-section profile of Slab1.0 is con-trasted with the tensor-guided fitting and interpolatingprofiles in Figure 8a. In this cross section, the Slab1.0,the tensor-guided fitting, and the tensor-guided inter-polating profiles are the blue, red, and green curves,respectively. Compared to the tensor-guided interpola-tion, the tensor-guided fitting profile is clearly smoother.

However, our fitting profile (red) is not as smoothas the is Slab1.0 profile (blue). This is because Hayeset al. (2012) use polynomial splines of degrees 2,3, or 4to fit the data in 2D sections. Therefore, Slab1.0 solutionis forced to be smoother than the tensor-guided fittingresult in the cross section shown here. Nevertheless, ourmodel honors the data more precisely than the Slab1.0model.

Also, note that the 3D slab surface produced by ourfitting method (Figure 8c) is smoother than the Slab1.0model (Figure 8d) along the direction parallel to thecoastline. One reason for this is that unlike the methodused by Hayes et al. (2012) which requires interpolat-ing between interpolated 2D profiles along the coastline,tensor-guided fitting is performed in one step. Anotherreason is that the tensor-guided fitting solution is guidedalong the strike directions of the slab.

The difference between the depths predicted by thetensor guided method and the depths predicted by theSlab1.0 model is in most areas 0 − 40 km. The tensor-guided fitting and interpolating solutions change con-cavity with their slopes approaching horizontal at theeastern edge of the model (see Figure 8a around dis-tance 900-1070 km and depth 400-600 km). This is dueto the use of a zero-slope boundary condition in solvingthe system of partial differential equations 19.

Figure 9 shows the histogram of standardized fit-ting errors rk associated with the optimum fitting so-lution computed using smoothing parameter s = 0.56.Note that this histogram does not exactly represent anormal distribution. The long tails of the distributionobserved here are not expected for a normal distribu-tion. Therefore, this study might benefit from assuminga different type of distribution (e.g., exponential) for thedata and fitting errors.

Also, note that the histogram shown in Figure 9 isnot centered at 0, which indicates a bias towards posi-tive fitting errors. As explained below, the observed biasis a consequence of the geometry of the problem, andthe smoothing applied in our fitting procedure.

The slab geometry is almost horizontal in areas nearthe coastline. Therefore, the tensors used in the tensor-guided fitting procedure in these areas have low eccen-tricity (e.g., see the nearly circular ellipses to the leftof the coastline in Figure 5). One consequence of usingthese tensors in the smoothing step is that the fitting so-lution q(x) in the bathymetry area receives contributionfrom the deeper sections of the slab. Thus, q(xk) is morelikely to overestimate the slab depth at locations xk

of the bathymetry samples. This means that the stan-


AB

AB

AB

A B

Tensor-guided interpolation

Tensor-guided fitting

Slab1.0

(b) (d)(c)

a)

Figure 8. A cross section (a) showing the profiles of three different slab models. The slab model obtained by tensor-guided

interpolation (b) is compared with the model obtained by tensor-guided fitting (c) and the same model from Slab1.0 (d)

produced by Hayes et al. (2012). Line segment AB shows the geographical location of the vertical cross section shown in (a).

The gray crosses in (a) are the orthogonal projection of all data points (the points shown in white in (b), (c), and (d)) that lie

within a rectangular window of width 100 km centered on the vertical plane of section AB. The gray dots in (b), (c), and (d)

denote the location of scattered data points.


rk

s = 0.56

Figure 9. Histogram of the standardized fitting errors rk for

the optimal smoothing function computed using smoothing

parameter s = 0.56.

dardized fitting errors defined in equation 22 are morelikely to be positive than negative for the bathymetrydata. Recall that about 40% of the depth samples inthis study are bathymetry data densely clustered nearthe coastline (Figure 3). Therefore, the positive fittingerrors statistically outnumber the negative fitting errorsand a positive bias in the distribution of errors is cre-ated.

7 CONCLUSION

Additional information provided by secondary data canbe used to guide the interpolation or fitting of a primaryset of spatially scattered data. This is done through con-struction of a metric tensor field that defines a modelfor spatial correlation. The tensor-guided fitting methodpresented here is capable of utilizing tensors that areboth anisotropic and spatially variable.

We constructed a tensor field based on the earth’scurved surface geometry and strike estimates of the sub-ducting slab in South America and used it to guide theinterpolation of the scattered estimates of slab depths.

Proper handling of data uncertainties is an impor-tant aspect of data fitting methods. If estimates of un-certainties associated with data are available, they canbe easily incorporated into the tensor-guided fitting pro-cedure to produce a statistically consistent fit to thescattered data. We used the slab data to demonstratethe capability of our data fitting method in handlingdata uncertainties.

In other areas of geoscience, such as explorationgeophysics, atmospheric, oceanic, and environmentalstudies, there are problems and applications similar tothe one considered in this paper, where the spatial cor-relation model for one scattered dataset can be inferredfrom another and where estimates of uncertainty asso-

ciated with data are available. Therefore, the process ofconstructing a metric tensor field and accounting for theuncertainties described here can serve as an example forapplying tensor-guided fitting in such problems.

8 ACKNOWLEDGMENTS

First author wishes to thank Simon Luo for the help-ful discussions on this paper and Diane Witters for herinstructions on revision and editing. This research wassupported by the sponsors of the Center for Wave Phe-nomena at the Colorado School of Mines.

APPENDIX A: BLENDED NEIGHBORINTERPOLATION

Blended neighbor interpolation is specifically designedto facilitate tensor-guided interpolation of scattereddata (Hale, 2010).

If we assume the scattered data to be interpolatedare a set

F = {f1, f2, ..., fK} (A1)

of K known sample values fk ∈ R corresponding to aset

X = {x1,x2, ...,xK} (A2)

of K known sample points (coordinates) xk ∈ Rn, thenthe goal of interpolation is to use the know samples toconstruct a function q(x) : Rn → R such that q(xk) =fk.

In the blended neighbor interpolation method(Hale, 2009), the interpolant q(x) is constructed in twosteps:(i) Solve the eikonal equation

∇t(x) · D(x) · ∇t(x) = 1 , x /∈ χ ;

t(xk) = 0 , xk ∈ χ (A3)

for t(x): non-Euclidean distance from x to the nearestknown sample xk, andp(x): the value fk of the sample at point xk nearest tothe point x.(ii) Solve the blending equation

q(x)− 12∇ · t2(x)D(x) · ∇q(x) = p(x) (A4)

for the blended neighbor interpolant q(x).In the equations above, D is a metric tensor field

which defines the measure of distance in space by pro-viding the anisotropic and spatially varying coefficientsof the eikonal equation. In n dimensions, the metric ten-sor field D is a symmetric and positive definite n × nmatrix (Hale, 2009).


APPENDIX B: COMBININGMEASUREMENTS HAVING RANDOMUNCORRELATED ERRORS AND KNOWNVARIANCES

Here, we discuss a way to combine independent mea-surements of a quantity using a weighted averagingscheme.

Consider N independent measurements(x1, x2, ..., xN ) of the same quantity where eachmeasurement has an unknown expected value anderror, i.e.,

xi = µ+ ei, i = 1, 2, ..., N (B1)

where µ is the true value of the quantity we wish toestimate, and ei is the error in each measurement.

We assume that the errors are not correlated, theirexpected values are zero, and their variances σ2

i areknown.

We let the combined measurement x to be aweighted average (linear combination) of the individualmeasurements xi,

x =N�

i=1

wixi, (B2)

requiring that the weights wi must satisfy the unbiased-ness condition

N�

i=1

wi = 1. (B3)

Using these assumptions, the variance of the combinedmeasurement can be expressed as

σ2 =N�

i=1

w2i σ

2i . (B4)

By minimizing this variance with respect to wi

(subject to the unbiasedness constraint of B3) theweights wi can be determined as

wi =

1σ2i

N�

j=1

1σ2j

, i = 1, 2, ..., N. (B5)

Using these weights, we can compute the best (mini-mum variance) estimate for the combined measurementx and its variance σ2 using equations B2 and B4, respec-tively. Note that to arrive at these results, we assumedno particular statistical distribution for the measure-ment errors.

REFERENCES

Bevis, M., and B. L. Isacks, 1984, Hypocentral trendsurface analysis: Probing the geometry of benioffzones: J. of Geophys. Res., 89, 6153–6170.

Boisvert, J. B., J. G. Manchuk, and C. V. Deutsch,2009, Kriging in the presence of locally varyinganisotropy using non-euclidean distances: Mathemat-ical geosciences, 41, 585–601.

Curriero, F. C., 2006, On the use of non-euclidean dis-tance measures in geostatistics: Mathematical geol-ogy, 38, 907–926.

Gudmundsson, O., and M. Sambridge, 1998, A region-alized upper mantle (rum) seismic model: J. of Geo-phys. Res., 103, 7121,7136.

Hale, D., 2009, Image-guided blended neighbor inter-polation of scattered data: SEG, Expanded Abstracts,28, 1127–1131.

——–, 2010, Image-guided 3D interpolation of bore-hole data: SEG, Expanded Abstracts, 29, 1266–1270.

——–, 2011, Tensor-guided interpolation on non-planar surfaces: Technical Report CWP-696, Centerfor Wave Phenomena, Colorado School of Mines.

Hayes, G. P., and D. J. Wald, 2009, Developing frame-work to constrain the geometry of the seismic ruptureplane on subduction interfaces a priori - a probabilis-tic approach: Geophys J. Int, 176, 951–964.

Hayes, G. P., D. J. Wald, and R. L. Johnson, 2012,Slab1.0: A three-dimensional model of global sub-duction zone geometries: Journal of Geophysical Re-search, 117, 1–15.

Hayes, G. P., D. J. Wald, and K. Keranen, 2009, Ad-vancing techniques to constrain the geometry of theseismic rupture plane on subduction interfaces a pri-ori: Higher-order functional fits: Geochemistry Geo-physics Geosystems, 10, 1–19.

Kohavi, R., 1995, A study of cross-validation and boot-strap for accuracy estimation and model selection:Proceedings of International Joint Conference on Ar-tificial Intelligence, 1995.

Syracuse, E. M., and G. A. Abers, 2006, Global com-pilation of variations in slab depth beneath arc vol-canoes and implications: Geochemistry GeophysicsGeosystems, 7, 1–18.

Wang, K., and J. He, 2008, Effects of frictional be-havior and geometry of subduction fault on coseismicseafloor deformation: Bulletin of the Seismological So-ciety of America, 98, 571–579.

Weber, O., Y. S. Devir, A. M. Bronstein, M. M. Bron-stein, and R. Kimmel, 2008, Parallel algorithms forapproximation of distance maps on parametric sur-faces: ACM Transactions on Graphics, 27, 104:1–104:16.

Documents

Tensor-guided ﬁtting of subducting slab depths