CWP-592

Defining regions in seismic images by flattening

Derek Parks1, William Harlan2, and Dave Hale11Center for Wave Phenomena, Colorado School of Mines, Golden CO 80401, USA2Landmark Graphics, 1805 Shea Center Drive, Highlands Ranch CO 80129, USA

ABSTRACTFlattening a seismic image removes the effects of geologic processes and trans-forms the image into layers as they were deposited in geologic time. In a flattenedimage, we can easily define regions that correspond to geologic layers.Estimates of local dip are required to compute the shifts in depth (or time)needed to flatten every event in a seismic image. Our flattening process usesstructure tensors to estimate the local dip of every sample in a seismic imageand a regularized least squares inversion to solve for shifts that are consistentwith these local dips.

Key words: flattening, structure tensors, inversion

1 INTRODUCTION

Our goal is to create an algorithm to allow for the easydefinition of regions within a 3D seismic image. Oftenthese regions are geologic layers, and to define such lay-ers in seismic images, we might track top and bottomisochron surfaces. Figure 1b displays an example of acomplete isochron surface with no holes. For surfaceslike these, traditional horizon tracking methods may re-quire more user interaction than is necessary.

The algorithm presented by Lomask et al. (2006)computes such surfaces by flattening a seismic image.This flattening algorithm reverses the effects of geologicprocesses and transforms an image back into layers asthey were laid down in geologic time by solving for ashift field (Figure 2b). This shift field provides a map-ping between the original image and the flattened image.

After the image has been flattened (Figure 2d),tracking isochron surfaces of constant geologic time isonly a matter of selecting a plane of constant time inthe flattened image and reversing the flattening process.Thus, by calculating the shifts needed to flatten the vol-ume, all of the isochron surfaces in the image are trackedin 3D at once.

Our process for flattening a seismic image onlyvaries in the details from Lomask et al. (2006). Bothalgorithms calculate the shift field from dips of seismicreflections. Therefore, the first step of the flattening pro-cess is the estimation of a local dip at every sample of a

seismic image. Our method uses structure tensors (vanVliet and Verbeek, 1995) to compute the unit normalvector of the best fitting plane in a local window aroundevery sample in a seismic image. After the dip is esti-mated, both algorithms solve for the flattened image byapplying a vertical shift to every sample in the inputimage. Our method computes the shift field by using aregularized least squares inversion. Posing the shift fieldcalculation as an inversion problem allows us to take ad-vantage of preexisting inversion frameworks, such as theone described by Harlan (2004).

2 THE STRUCTURE TENSOR

Our implementation of the flattening process uses thestructure tensor (Fehmers and Hocker, 2003) or thegradient-squared tensor (van Vliet and Verbeek, 1995)to efficiently calculate the normal vector of the best fit-ting plane in a local window around every sample of aseismic image.

The gradient of an image can be used to obtain in-formation about the orientation of features in an image.Unfortunately, we cannot use the simple gradient to es-timate the orientation of features in an image becausegradients with similar orientation but opposite signs willcancel each other. Computing the outer product of gra-dients resolves this problem and causes gradients withopposite directions to reinforce each other. After the

68 D. Parks, W. Harlan, & D. Hale

(a) (b)

(c)

Figure 1. (a) A screenshot of a 3D seismic image. (b) Thesame seismic image with an isochron surface tracked in 3D.(c) A close up of the isochron surface shown in (b).

outer product of gradient has been computed, one canobtain the orientation of the structure in an image atdifferent scales by averaging over a local window.

Smoothing of the gradient outer products may beimplemented using recursive Gaussian filters (RGF)(Hale, 2006). To perform this local averaging, we con-volve the image with a Gaussian function. The recursivenature of the RGF allows for a more efficient computionthan a simple convolution. RGFs are also separable fil-ters. This means that applying a 1D version of the filterin all three coordinate directions is equivalent to apply-ing a 3D version of the filter. Therefore, we only need toimplement the RGF in 1D, and apply it independentlyin each coordinate direction.

When computing the local average of the gradientouter products, the local window is defined in terms ofthe half-width σ of the Gaussian function that we areconvolving with the image. Since, we apply smoothingindependently in each coordinate direction, we can ad-

(a)

(b)

(c)

(d)

Figure 2. (a) A vertical slice of the seismic image f(x, y0, z)shown in Figure 1. (b) A vertical slice of the calculated shiftfield s(x, y0, z) needed to flatten the input image f(x, y0, z).Blue indicates the section will be shifted up, and conversely,red sections will be shifted down to make the output imageflat. (c) The seismic image and the shift field superimposed.(d) The flattened image g(x, y0, z).

Flattening 69

Figure 3. A 2D slice of planarity values (equation 2) calcu-lated from the image in Figure 1a. This is the same slice ofthe 3D image shown in Figure 2a. White indicates a planarityvalue close to one which indicates a highly planar event atthat location. Conversely, black indicates a planarity valueclose to zero and a non-planar event at that location.

just the half-width of the Gaussian function in each di-rection independently. A half-width of four samples, ineach coordinate direction, was used in the computationof images shown in this paper.

We now have all of the components of the structuretensor G which is defined as

G(x, y, z) =

0@ < gxgx > < gxgy > < gxgz >< gygx > < gygy > < gygz >< gzgx > < gzgy > < gzgz >

1A ,

(1)where < · > is a Gaussian smoothing operator witha half-width of σ. The x, y, and z components of thegradient are represented by gx, gy, and gz, respectively.

The eigenvector corresponding to the largest eigen-value of G is the normal to the best fitting plane inthe local window. Note that G is symmetric positive-definite. Therefore, we can efficiently solve for its eigen-values and eigenvectors. This is important because thisdecomposition is performed for every sample in an im-age.

We are able to use the two largest eigenvalues of Gto estimate the planarity p(x, y, z) of events:

p(x, y, z) =λ1(x, y, z)− λ2(x, y, z)

λ1(x, y, z)∈ [0, 1], (2)

where λ1 is the largest eigenvalue and λ2 is the secondlargest eigenvalue. The planarity is close to one whenthere is a well defined planar event in the local windowand close to zero when the event is noisy or non-planar.We use the planarity value p(x, y, z) to estimate thequality of our normals when solving for the shift fields(x, y, z).

3 2D SHIFT FIELD CALCULATION

To easily introduce our process of solving for the shiftfield, we first consider the problem in 2D and later ex-tend the process to 3D. We also assume that the verticalaxes of our images is depth. However, the flattening pro-cess works equally well for either depth or time.

Once the local dips have been estimated, the nextstep in the flattening process is to find the distance ev-ery sample in the seismic image must be shifted to flat-ten the image (Figure 4a). These shifts will only movea sample up or down in depth. They cannot move asample into a different trace. All samples in a trace arenot statically shifted together. A trace is allowed to bestretched in some parts and squeezed in others to makethe final image flat.

The required shift at every sample comprises ourshift field s(x, z). A positive value in the shift field in-dicates the corresponding sample needs to be pusheddown to greater depth to create a flat image. Conversely,a negative value of the shift field indicates the sampleneeds to be moved up. The shift field s(x, z) is thenused to map an input image f(x, z) to a flattened im-age g(x, z).

Figure 4c illustrates how we solve for this shift atevery sample. The partial derivative of the shift fieldwith respect to x is equal to the x component of thenormal vector divided by the z component of the normalvector. This ratio also equals the tangent of the dipangle θ:

∂s(x, z)

∂x=

nx(x, z)

nz(x, z)= tan θ. (3)

Therefore, the shift field s(x, z) is calculated by inte-grating all the dip tangents nx(x, z)/nz(x, z) along anevent:

s(x, z) =

Z x

0

nx(η, z)

nz(η, z)dη. (4)

Thus, the shift at some location (x0, z0) is a function ofall of the dip tangents nx(x, z)/nz(x, z) along the event.

In Figures 4a and 4b, we sketch a hypothetical seis-mic event before and after flattening, respectively. Toflatten the event, the sample at (x0, z0) in the inputimage f will be moved to (x0, z1) in the output imageg:

g(x0, z1) = f(x0, z0). (5)

In our current algorithm, the output image g iscomputed by interpolating samples from the input im-age f . Consequently, when computing the output at(x0, z1), we must find the corresponding sample at(x0, z0) in the input image. The x coordinate of thislocation is known but the z coordinate is not known.Fortunately, we can use the shift field to find the z co-ordinate at which to interpolate the input image. Fig-ure 4a shows that the unknown z coordinate z0 is equalto the output location z1 minus the shift at the event

70 D. Parks, W. Harlan, & D. Hale

s(x0, z0):

z0 = z1 − s(x0, z0). (6)

Combining equations 5 and 6 yields

g(x0, z1) = f(x0, z1 − s(x0, z0)). (7)

Unfortunately, the right-hand-side of this relation stillhas an unknown z0. The location (x0, z0) of the eventin the input image is a function of the shift s(x0, z0)at the location of the event. To overcome this circulardependency, we assume that the shift at the locationof the event in the input is approximately equal to theshift at the output location:

s(x0, z0) ≈ s(x0, z1). (8)

With this approximation we can calculate one sam-ple g(x0, z1) via

g(x0, z1) ≈ f(x0, z1 − s(x0, z1)). (9)

We can generalize this to all samples in the output im-age:

g(x, z) ≈ f(x, z − s(x, z)). (10)

From equation 4, we know that this approximationis best if the integral of dip tangents at (x0, z0) is equalto the integral of dip tangents at (x0, z1):Z x

0

nx(η, z0)

nz(η, z0)dη ≈

Z x

0

nx(η, z1)

nz(η, z1)dη. (11)

This is equivalent to assuming that all of the events areclose to parallel or that the depths z0 and z1 are almostequal. Thus, as |z0 − z1| approaches zero, the error inour approximation approaches zero.

Since, in practice, the error is never zero, the out-put image g(x, z) may not be completely flattened, evenafter the flattening process has been applied. We com-pensate for this error by repeatedly applying the flat-tening process (dip estimation, shift field calculation,and flattening) until |z0 − z1| becomes negligible. Fig-ure 5 shows a graph of the average dip angle (θ fromequation 3) per flattening iteration for the 3D image inFigure 1.

4 3D SHIFT FIELD CALCULATION

In 3D, we use the same process and assumptions as in2D. Our mapping from an input image f(x, y, z) to aflattened image g(x, y, z) becomes

g(x, y, z) ≈ f(x, y, z − s(x, y, z)). (12)

The major difference, in 3D, is that the normal vec-tor n(x,y, z) of the best fitting plane at every samplenow includes a y component. Thus, we have two diptangents for every sample in the input image. Whereas,in 2D, there was only one dip tangent for every sample

Figure 4. (a) A cartoon of a 2D seismic image f(x, z) withonly one event. The arrow shows the amount of shift that isneeded to flatten the event at one point. (b) The resultingoutput image g(x, z) which shows the original event afterflattening the input image f(x, z). (c) A close up of the squarein (a) showing the local normal vector n(x, z) of the reflectorand the amount of shift needed as x changes.

(equation 3). Now, there is one dip tangent for the xcomponent of the normal

nx(x, y, z)

nz(x, y, z)(13)

and another for the y component of the normal

ny(x, y, z)

nz(x, y, z). (14)

Accordingly, there are now two relevant partial deriva-tives that relate to the shift field s(x, y, z)

∂s(x, y, z)

∂x=

nx(x, y, z)

nz(x, y, z)(15)

and

∂s(x, y, z)

∂y=

ny(x, y, z)

nz(x, y, z). (16)

Flattening 71

Figure 5. A graph of the average dip angle θ (equation3) versus flattening iteration for the 3D image in Figure 1.Note that, in practice, the dip angle is never reduced to zerobecause of noise in the image.

With two partial derivatives, we can no longer simplyintegrate the dip tangents to solve for the shift fields(x, y, z). Instead, we use a least squares approach tohonor both x and y dip tangents in the computation ofthe shift field. This approach has the advantage thatwe never need to explicitly define the operator thatis applied to the dip tangents to solve for the shiftfield s(x, y, z). We only need to formulate the shift fields(x, y, z) in terms of its partial derivatives, as in equa-tions 15 and 16.

This least squares approach has another advantagein that it takes the form of an inversion problem. Ina typical inversion problem we attempt to find somemodel m that when transformed by the function t(·)approximates some data d:

d = t(m). (17)

Formulating the shift field calculation as an inversionproblem allows us to use the geophysical inversionframework of Harlan (2004).

We simplify notation by packing the samples of theshift field s(x, y, z) into a vector s such that

s[i + Nx(j + Nyk)] = s(xi, yi, zi), (18)

where Nx, Ny, and Nz are the number of samples inthe x, y, and z directions, respectively, and xi, yi, andzi are the x, y, and z locations of our sampled valuesfor every sample in the shift field. The index of the shiftfield s runs from 1 to N = Nx × Ny × Nz for everysample in the original shift field s(x, y, z), packed oneafter another:

s ≡

26664s(1)s(2)

...s(N)

37775 . (19)

We perform a similar packing to create a vector dthat contains both x and y dip tangents:

d ≡

266666666666664

nx(1)/nz(1)...

nx(N)/nz(N)

ny(1)/nz(1)...

ny(N)/nz(N)

377777777777775. (20)

Note that the vector d contains 2N elements.Next, we create a vector that contains both hori-

zontal partial derivatives of the shift field packed in thesame way as d. This vector also contains 2N values sothat it can be related to d. To create this vector, wedefine a horizontal gradient operator:

∇xy ≡

2664∂/∂x

∂/∂y

3775 . (21)

When the horizontal gradient ∇xy is applied to the shiftfield vector s, we obtain a vector of length 2N that con-tains both sets of horizontal partial derivatives:

∇xys =

266666666666664

s,x(1)...

s,x(N)

s,y(1)...

s,y(N)

377777777777775. (22)

Using our newly defined vectors, we can combine equa-tions 15 and 16 into one expression:

∇xys = d. (23)

This is an overdetermined system. There are N un-known shifts s and 2N equations. Equation 23 takesthe form of a typical inversion problem where we wishto find the model that when transformed is equal to adataset (equation 17). In our case, the shift field vectors is the model, the horizontal gradient ∇xy is the trans-form, and the dip tangents vector d is the dataset thatwe wish to fit. Using this information we create an errorfunction that the inversion framework will minimize tofind the shift field s:

E = [d−∇xys]T [d−∇xys] . (24)

Note that for steeply dipping reflectors, nx/nz andny/nz become very large, even infinite. To avoid this,

72 D. Parks, W. Harlan, & D. Hale

we redefine d to be

d ≡

266666666666664

nx(1)...

nx(N)

ny(1)...

ny(N)

377777777777775. (25)

Then, we define a new vector nz that contains the zcomponents of the estimated normal vectors packed ina similar way to d:

nz ≡

266666666666664

nz(1)...

nz(N)

nz(1)...

nz(N)

377777777777775. (26)

Note that the vector nz contains 2N samples. Insteadof dividing the x and y components of the estimatednormal vectors by the z component, we multiply thehorizontal gradient of the shift field by nz. Therefore,equation 23 becomes

nz ×∇xys = d. (27)

and our new error function is

E = [d− nz∇xys]T [d− nz∇xys] . (28)

Next, we weight the data misfit error differently.We would like to allow for more misfit in areas where weknow that dataset is of poor quality, and conversely, wewould like to force the error function to have less misfitin areas where we have good quality data. We use theplanarity value computed from the structure tensor asan indicator of data quality. The planarity, defined byequation 2, varies from zero to one. It will be close toone where there is a strong planar event and close tozero elsewhere. We force the error function to put moreemphasis on the parts of the image that have a strongerplanarity value by packing the planarity value for everysample into a diagonal matrix P:

P =

266666666666664

p(1)

. . .

p(N)

p(1)

. . .

p(N)

377777777777775. (29)

P is a 2N × 2N matrix. We incorporate the planarityvalue into the error function by scaling each sample bythe corresponding planarity value:

E = [d− nz∇xys]T P [d− nz∇xys] . (30)

An iterative search for solutions to equation 30 willintroduce high-frequency variations in early iterations.We would prefer that the framework solve for an oversimplified shift field, if it has not fully converged. Toaccomplish this, we add a regularization term to ourerror function. The regularization term will discouragecomplexity in the shift shift field by increasing the errorE when the shift field contains large variations. We ap-ply a derivative-like roughening operator R to the shiftfield in the regularization term. This operator will en-hance the high frequencies of the shift field, but unlike atrue derivative, this operator will not remove a constantvalue from the shift field. This has the added benefit offorcing the magnitude of the shift field s to be small.Forcing the shifts to be small reduces the error in theshift field calculation caused by our earlier approxima-tion (equation 8) that shifts are small. Therefore, oursolution to the shift field solves for the smallest possibleamount of shift that can account for the dip tangents.This means that we do not need to shift all traces inreference to a particular trace. All samples in the im-age will be shifted by the minimum amount needed toflatten the image.

We account for differences in the magnitudes of thedata misfit term and the regularization term by intro-ducing a scaling ε to the regularization term. This scal-ing ε may be used to adjust for differences in the unitsof the two terms. The scaling ε also allows us to adjustthe importance of keeping the shifts smooth and smallversus the importance of accurately fitting the dip tan-gents. We use the heuristic from Harlan (2004) that theshifts “should be allowed to attain magnitudes 10 to 100times their most reasonable values before the two termsare equal”. Adding the regularization term to our errorfunction gives

E = [d− nz∇xys]T P [d− nz∇xys] + ε [Rs]T [Rs] . (31)

Instead of solving for the shift field s directly, wesolve for a rough version of the shift field s. We solvefor this rough version of the shift field by performing achange of variables (Harlan, 1995):

s = Ss, (32)

where S is a Gaussian smoothing filter (Hale, 2006) in allthree dimensions. We have used a Gaussian smoothingfilter with half-widths of four, four, and ten samplesin the x, y, and z directions, respectively, to computethe images in this paper. Our error function, after thechange of variables, becomes

E = [d− nz∇xySs]T P [d− nz∇xySs] + ε [RSs]T [RSs] .(33)

Flattening 73

Next, we explicitly defined the roughening operatorR as the inverse of the smoothing operator S. Therefore,

RS = I. (34)

Now, we see the roughening operator R has all of theproperties that we described earlier. It will enhancehigh-frequency variations in the shift field because weknow that the Gaussian smoothing filter S will atten-uate high-frequency variations. It will also not removeconstant shifts from the shift field s because applyinga smoothing filter to a constant shift field will have noeffect.

Using equation 34, to simplify the error function,we obtain

E = [d− nz∇xySs]T P [d− nz∇xySs] + ε sT s. (35)

This error function was minimized in all of the flatteningexamples shown in this paper.

5 FAULTS

Our current algorithm has limited success flatteningseismic images with faults. In Figure 6a, we show a3D synthetic seismic image with a large vertical faultrunning down the middle of the image. The calculatedshift field (Figure 6d) smoothly transitions from nega-tive shift to positive shift across the fault. These shiftsdo not fully flatten the image. To accurately flatten theimage, the shifts should be very sharp at the fault plane.This leads to the inaccurate isochron surface in Fig-ure 6c. This isochron surface should be vertical at thefault, but it linearly transitions from one side of thefault to the other.

This artifact may be a result of our regularizationterm which forces the shift field to be smooth. Thissmoothing can obscure discontinuities in structures atfaults. We would instead prefer to have a spatially vary-ing smoother that does not smooth across faults (wherethe planarity is low), while smoothing elsewhere. Wehope to add this improvement to future versions of ouralgorithm.

6 CONCLUSIONS

We have presented an algorithm that allows for the com-putation and display of isochron surfaces. Our flatteningprocess is an extension of that by Lomask et al. (2006).This process uses structure tensors to efficiently calcu-late estimates of the local dip in the seismic image andemploys a least squares inversion to compute the verti-cal shifts needed to flatten the image. Isochron surfaces,calculated from a flattened image, can then be used todefine regions within a seismic image.

(a) (b)

(c)

(d)

Figure 6. (a) A 3D synthetic seismic image with a fault run-ning vertically through the middle of the volume. (b) A 3Disochron surface for one of the layers in the synthetic seismicvolume shown in (a). (c) A close-up of the isochron surfaceshown in (b) that highlights the undesirable smoothing ofthe isochron surface across the fault. (d) A 2D slice throughthe center of the shift field that is used to flatten the imagein (a).

ACKNOWLEDGMENTS

We thank Landmark Graphics and all of the CWP spon-sors for their funding of this research. The Teapot Domeseismic data used in this research were provided by theRocky Mountain Oilfield Testing Center and the U.S.Department of Energy. This research would not have

74 D. Parks, W. Harlan, & D. Hale

been possible without the patient guidance of Craig Art-ley and Andreas Rueger.

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