A VARIATIONAL FORMULATION FOR INTERPOLATION OF …Vol. 1 (2013) pp. 1-16. A VARIATIONAL FORMULATION...

Proceedings of the Project Review, Geo-Mathematical Imaging Group (Purdue University, West Lafayette IN),Vol. 1 (2013) pp. 1-16.

A VARIATIONAL FORMULATION FOR INTERPOLATION OF SEISMICTRACES WITH DERIVATIVE INFORMATION

FREDRIK ANDERSSON AND JENS WITTSTEN

Abstract. We construct a variational formulation for the problem of interpolating seismic data in the case ofmissing traces. We assume that we have derivative information available at the traces. The variational problemis essentially the minimization of the integral over the smallest eigenvector of the structure tensor associated withthe interpolated data. This has the physical meaning of penalizing the local presence of more than one direction inthe interpolation. We show that the solution to the variational problem also satisfies an elliptic partial di↵erentialequation. Moreover, this can be obtained by considering the steady state solution of a non-standard anisotropic di↵u-sion problem. We show existence and uniqueness for this type of anisotropic di↵usion. In particular, the uniquenessproperty is important as it guarantees that the unique solution to the variational problem that we constructed canbe obtained by the numerical schemes we propose.

1. Introduction. Let f f x, y be a two-dimensional functon1. We assume that we knowvalues of f along vertical lines (traces) x xl. In addition we assume that we have knowledge ofthe values of the derivative of f along the traces x xl. Since we know f along the traces, the only

addition information are the values of fx x, yx xl

. Assuming that f is continuous, this means that

we know the values of f approximately in a vicinity of the traces. The problem then arises how toextrapolate this information. One simple way of extrapolation is to extend f using the derivativeinformation and then use linear interpolation between di↵erent traces. This is illustrated in themiddle upper panel of Figure 1. Another simple approach is to take use the extrapolation using thederivative information, and construct the interpolated image by picking the extension closest to thetraces where there is information. This approach is depicted in the upper right panel of Figure 1.

Both the interpolation results obtained in the upper middle and upper right panels of Figure 1are physically unrealistic. We would rather expect a smooth transition between the left trace andright trace. An apparent problem is the introduction of crossing events or discontinuities.

We are interested in phrasing the interpolation problem in terms of a variational problem. Inthis formulation, we would like to penalize the “artifacts” produced by the two naive approachesmentioned above. A good tool for measuring artifacts of this kind is by means of structure tensors.The structure tensor of an image f is defined by

(1.1) Tf x, yK fx

2 K fxfyK fxfy K fy

2 ,

for some regularization function K. In this work we assume that K is a Gaussian.Note that without the convolution with K, the tensor

fx2 fxfy

fxfy fy2 ,

is of rank one for each fixed point x, y . The eigenvectors are f2x f2

y and 0, respectively, and theeigenvectors are parallel to rf and perpendicular to rf , respectively. Now, if f locally describesa plane wave, then the convolution will describe a local averaging over areas where the eigenvectorof the tensor above remain the same (parallel respectively perpendicular to rf). The (positive)eigenvectors of Tf will thus have the property that one will be large while the other one will be closeto zero.

However, at regions where f depart from locally resembling a plane wave, the smallest eigenvalueof Tf will no longer be close to zero. Regions where the smallest eigenvalue of Tf not small thus

Mathematics, Faculty of Science, Lund University, P.O. Box 118, SE-221 00 Lund, SwedenKyoto University, Japan, [email protected] seimic applications the variable y would usually be time and the letter t would be used instead of y. Since

we will use time for other purposes we use y instead of t

1

2 F. ANDERSSON AND J. WITTSTEN

Fig. 1. The left upper panel shows a simple synthetic data set. The middle and right upper panels showstwo simple way of interpolation the interior of the image using derivative information f

x

. In the lower panels thesmallest eigenvalues of the structure tensor associated with the upper images are displayed. Note the large di↵erencein magnitude for the three cases.

indicate that f does not locally look like a plane wave. The bottom panel of Figure 1 show thesmallest eigenvalues (as function of x and y), for the corresponding three images in the upper panels.Note the variation in magnitude for the three di↵erent cases. The eigenvalues have been normalizedin relation to the maximum values of the largest eigenvalues of the structure tensor2.

This motivates the construction of the following variational problem

ming

s2 Tg x, y dx dy such that g xl, y f xl, y ,

where s2 Tg x, y denotes the smallest eigenvalue of the structure tensor at the point x, y .

In the next section we formalize this choice of variational problem, and show that solutions toit can be found by means of solutions to certain partial di↵erential equations.

2. Problem description. In Weickert [12, Chapter 2], a result by Catte et al [3] concerningisotropic di↵usion filters and its generalization to the anisotropic case is discussed. The result ofCatte et al [3] gives existence and uniqueness of solutions to the problem

u t div g rG� u ru 0 on 0, T ⌦,

u 0 u0,

2To avoid aliasing e↵ects originating from the periodic convolutions, the eigenvalue maps are depicted for x

14

instead of x

12 .

INTERPOLATION OF SEISMIC TRACES WITH DERIVATIVE INFORMATION 3

where g is a function exemplified by g t 1 t2 1. Here ⌦ is the open set 0, 1 0, 1 in R2,and G� is a Gaussian filter given by

G� x1

2⇡�exp

x 2

4�.

One would also need to impose some boundary conditions on the solution u t, x on the boundary� of ⌦, for example

u ⌘ 0 on 0, T �,

where ⌘ is the outward normal direction. (Note that the type of boundary conditions a↵ects thedefinition of the convolution G� u, since one needs to extend the domain of definition of u inan appropriate way.) The generalization of this problem to the anisotropic case involves replacingg with a smooth, symmetric positive definite di↵usion tensor acting on the structure tensor K⇢

ru� ru� ; the convolution with the Gaussian K⇢ is understood to be componentwise, and u�

denotes the regularization G� u.We are interested in a closely related problem, derived from the following interpolation problem.

We shall throughout this note assume that all functions are real valued unless stated otherwise. Fora 2 2 symmetric matrix M , let M U⌃UT be the singular value decomposition of M wheres1 s2 are the elements of the diagonal matrix ⌃ and U is unitary. Let g : R R be a function tobe specified later, and for real parameters a, b and c, consider the map L : R3 R given by

(2.1) L a, b, c g s1 s2,a bb c

Us1 00 s2

UT .

For f C1 R2 we let fx and fy denote the partial derivatives, and similarly we denote by La, Lb

and Lc the partial derivatives of L. Let K be a Gaussian function, and define the structure tensorTf by (1.1). We will identify Tf x, y with a vector in R3, also denoted by Tf x, y , and permit usto write L Tf to represent the map from R2 to R given by

L Tf x, y L Tf x, y L K fx2 x, y ,K fxfy x, y ,K fy

2 x, y .

We also permit us to evaluate the partial derivatives of L at Tf x, y in this way.Let ⌦ R2 be the open set ⌦ 0, 1 0, 1 , and let � C ⌦ satisfy µ � x, y 1 for

all x, y ⌦, where µ 0 is a positive constant. For a given function f0 C ⌦ consider now thefunctional

(2.2) f⌦

1 � x, y f x, y f0 x, y 2dxdy⌦� x, y L Tf x, y dxdy.

We remark that we throughout this note always will assume that if given a function u defined on ⌦,the convolution K u of u and a Gaussian K is defined as K u, where u is a linear and continuousextension of u to R2. Since we will be considering Neumann boundary data below, the extensionwill be given by

u x, y u x, y , 1 x 0, 0 y 1,(2.3)

u x, y u x , y , 0 x 1, 1 y 0,

and so on. We shall not distinguish between u and its extension u. A necessary condition for fto be a minimizer of (2.2) is found using standard variational calculus (compare with [13]): for allfunctions ' (in some suitable function space defined in terms of appropriate regularity and boundary


conditions) we have

0d

dt f t'

t 0(2.4)

2⌦

1 � f f0 'dxdy

⌦� 2 La Tf K fx'x

Lb Tf K fx'y fy'x 2 Lc Tf K fy'y dxdy.

Let I denote the integral in the right-hand side. Since K is radial, we have K u x, y v x, y dxdyu x, y K v x, y dxdy for all u and v, which together with a partial integration argument gives

I⌦

2 K � La Tf fx'x K � Lb Tf fx'y fy'x

2 K � Lc Tf fy'y dxdy

⌦ x2 K � La Tf fx K � Lb Tf fy

yK � Lb Tf fx 2 K � Lc Tf fy 'dxdy.

Here, the integrand in the last integral can be written as the product of ' and the factor

2 divK � La Tf

12K � Lb Tf

12K � Lb Tf K � Lc Tf

rf .

Dividing by a factor 2, and working within the distributional framework to restrict our attentionto test functions ' C0 ⌦ , we can therefore express the necessary condition (2.4) for f to be aminimizer of the functional (2.2) in the sense indicated by saying that f must be a solution to theequation

(2.5) 1 � f divK � La Tf

12K � Lb Tf

12K � Lb Tf K � Lc Tf

rf 1 � f0

in D ⌦ .For simplicity, let us now consider the case � 1, although the arguments used treating this

case are valid also for any � of the type introduced above. For technical reasons, we will also replacethe structure tensor Tf with a regularized version utilizing the Gaussian filter G� introduced above,which we shall also denote by Tf , but this should not cause any confusion. Let f� denote theregularization G� f , and consider the structure tensor

TfK G� fx

2 K G� fx G� fyK G� fx G� fy K G� fy

2 .

We shall thus consider the di↵erential equation

divK La Tf

12K Lb Tf

12K Lb Tf K Lc Tf

rf 0

in D ⌦ . One way to solve this problem is to solve the nonlinear di↵usion problem

ut t, x, y divK La Tu t K Lb Tu t

K Lb Tu t K Lc Tu tru t 0,(2.6)

u 0 u0,


where u t : ⌦ R plays the role of the function f above, and then to consider the steady statesolution

f x, y limt

u t, x, y .

Comparison with the problem studied by Weickert in [12, Chapter 2] shows that we have replacedthe di↵usion tensor D : R2 2 R2 2 acting on a structure tensor of the form K⇢ rf� rf� ,that is, a map

(2.7) rf� D K⇢ rf� rf� ,

by the matrix valued map

(2.8) rf�K La Tf K Lb Tf

K Lb Tf K Lc Tf.

Here we also wish to mention the study by Hahn and Lee [6], which concerns a problem similar toours but with a di↵erent setup.

To find a suitable function g in the definition (2.1) of L, we now digress to discuss the map(2.8). Without loss of generality, we may assume that

L a, b, c g s1 s2,a bb c

Us1 00 s2

UT

with s1 s2 and

Usin ✓ cos ✓

cos ✓ sin ✓

for ✓ ✓ a, b, c . It is straightforward to check that

L a, b, c g a sin2 ✓ 2b cos ✓ sin ✓ c cos2 ✓ a cos2 ✓ 2b cos ✓ sin ✓ c sin2 ✓,

where ✓ ✓ a, b, c is a solution to the equation

(2.9) c a cos ✓ sin ✓ b cos2 ✓ sin2 ✓ 0.

Introduce the auxiliary functions and ' given by

s1 a, b, c a sin2 ✓ 2b cos ✓ sin ✓ c cos2 ✓, ✓ ✓ a, b, c ,

s2 ' a, b, c a cos2 ✓ 2b cos ✓ sin ✓ c sin2 ✓, ✓ ✓ a, b, c .

By virtue of (2.9), a straightforward computation shows that the partial derivatives of and ' aregiven by

a sin2 ✓, b 2 cos ✓ sin ✓, c cos2 ✓,'a cos2 ✓, 'b 2 cos ✓ sin ✓, 'c sin2 ✓.

The partial derivatives La, Lb and Lc of L are therefore found to be

La a, b, c cos2 ✓ g a, b, c sin2 ✓,

Lb a, b, c 2 cos ✓ sin ✓ 1 g a, b, c ,

Lc a, b, c sin2 ✓ g a, b, c cos2 ✓,


where ✓ ✓ a, b, c . Hence, we have

divK La Tf

12K Lb Tf

12K Lb Tf K Lc Tf

rf div Sfrf

with the matrix Sf given by

Sf Kcos2 ! cos! sin!

cos! sin! sin2 !g Tf sin2 ! g Tf cos! sin!

g Tf cos! sin! g Tf cos2 !

where

(2.10) ! ✓ Tf .

Note that if Tf ⇠, ⌘ and ' Tf ⇠, ⌘ are the largest and the smallest eigenvalues of Tf ⇠,⌘ ,respectively, then(2.11)

Sf x,y Ksin ✓ Tf cos ✓ Tf

cos ✓ Tf sin ✓ Tf

g Tf 00 1

sin ✓ Tf cos ✓ Tf

cos ✓ Tf sin ✓ Tfx, y .

Proposition 1. Let F be the Frobenius norm on the space S of symmetric 2 2 matrices.

Let V be the set of maps R2 S . Let ⌥ be the map from a subset of V into V given by ⌥ : Tf Sf

for f L R2, so that ⌥Tf x, y Sf x,y . If g : R R is Lipschitz continuous, then there is a

constant C depending only on K and g such that

⌥Tu x, y ⌥Tv x, y F Su x,y Sv x,y F C sup⇠,⌘ ⌦

Tu ⇠,⌘ Tv ⇠,⌘ F

for all u and v in L R2.

Proof. First note that the map on S given by

⌥ :a bb c

Ug s1 00 1

UT ,a bb c

Us1 00 1

UT ,

is Lipschitz continuous by [9, Theorem 5.2]. Therefore, there is a constant CLip depending only ong such that

⌥A ⌥B F CLip A B F

for all A and B in S . With

Sf x,ysin ✓ Tf cos ✓ Tf


g Tf 00 1


cos ✓ Tf sin ✓ Tfx, y ,

so that Sf x,y K Sf x, y , this means that the map on S given by Tf x,y Sf x,y satisfies

Su x,y Sv x,y F CLip Tu x,y Tv x,y F , u, v L R2 .

Now

Su x,y Sv x,y F K Su Sv x, y F K Su Sv F x, y ,

which can be seen to hold by approximating the convolution by a Riemann sum, convergent in C .Estimation of the right-hand side gives

Su x,y Sv x,y F K L1 R2 sup⇠,⌘ R2

Su ⇠,⌘ Sv ⇠,⌘ F

K L1 R2 CLip sup⇠,⌘ R2

Tu ⇠,⌘ Tv ⇠,⌘ F ,


which completes the proof.We shall now estimate Tf L ⌦ . Note that

Tf K G� fx2 sin2 ! 2 K G� fx G� fy cos! sin! K G� fy

2 cos2 !.

By trivially estimating the sine and cosine in the expression above, and applying Holder’s inequality,we have

Tf L ⌦ K L1 R2 xG� f 2L R2(2.12)

2 xG� f L R2 yG� f L R2 yG� f 2L R2 .

A change of variables shows that

xG� f x, yk,` Z2 ⌦

xG� x ⇠ k, y ⌘ ` f ⇠ k, ⌘ ` d⇠d⌘.

By virtue of the definition (2.3) of the extension of f used to define the convolution with G�, it isstraightforward to check that the Cauchy-Schwartz inequality then gives the estimate

xG� f x, yk,` Z2

xG� k x , ` y L2 ⌦ f L2 ⌦ .

Given x, y R2, there are integers M and N such that x, x 1 M 1,M 1 and y, y 1N 1, N 1 . To simplify notation, let ⌧ x,y be the translation operator ⌧ x,y u ⇠, ⌘ u ⇠ x, ⌘ y ,and introduce the function F ⇠, ⌘ xG� ⇠, ⌘ so that

⌧ x k,y ` F L2 ⌦ xG� k x , ` y L2 ⌦ .

By a change of variables, this norm satisfies

⌧ x k,y ` F2L2 ⌦

M 1

M 1⇠ k 2e

⇠ k 2

2� d⇠N 1

N 1e

⌘ ` 2

2� d⌘

2 1 k M 2ek M 1 2

2� 2e` N 2

2�

for all k M 1 and ` N 1. Trivially we also have

M 1

M 1⇠ M 2e

⇠ M 2

2� d⇠ 2 andN 1

N 1e

⌘ N 2

2� d⌘ 2.

A straightforward computation then gives that

k,` Z2

⌧ x k,y ` F L2 ⌦ 2 2 2n 1

1 n en 1 2

4� 2 2 2n 1

en 1 2

4� ,

which implies that

(2.13) xG� f L R2 C� f L2 ⌦

where the constant only depends on G�. We have the same estimates for the terms involving yderivatives in (2.12), from which we therefore obtain

Tf L R2 4C2� K L1 R2 f 2

L2 ⌦ .

By comparing the definitions of and ', it is also clear that

' Tf L R2 4C2� K L1 R2 f 2

L2 ⌦ .


Recall that Tf is the largest eigenvalue of the matrix Tf . Since the entries on the diagonalin Tf are nonnegative, it follows that Tf 0 for all f C1 R2 . Hence, if g is a positive anddecreasing on the positive half-axis with limt g t 0 then for all f in a bounded subset ofL2 ⌦ it follows that g Tf is bounded away from zero from below, uniformly with respect tof . If in addition g 0 1 we have

(2.14) 0 ⌫0 g Tf 1

for all f in a bounded subset of L2 ⌦ . This allows us to conclude that the matrix Sf is positiveand uniformly bounded away from zero from below with respect to f B, where B L2 ⌦ isbounded.

Lemma 2. The smallest eigenvalue of the matrix Sf is positive and uniformly bounded away

from zero from below with respect to f B, where B is a bounded subset of L2 ⌦ .

Proof. Let A and B be two (symmetric) positive semidefinite matrices. Then

(2.15) minz 1

zT A B z minz 1

zTAz minz 1

zTBz

with equality only if the eigenvectors corresponding to the smallest eigenvalue of A and B, respec-tively, are parallel. Note that the smallest eigenvalue of

cos2 ! cos! sin!cos! sin! sin2 !

g Tfsin2 ! cos! sin!cos! sin! cos2 !

is equal to g Tf , which follows from (2.14) and since both the two matrices above are of rank1. From applying (2.15) to the Riemann integral definition defining the convolution with K we thenconclude that the smallest eigenvalue of Sf is larger than ⌫0 K 1.

We remark that Weickert [12] introduces a condition on the di↵usion tensor D in (2.7) which hecalls “uniformly positively definiteness”, similar to that given by Lemma 2; compare with conditionC on p. 58 in [12].

The arguments used in the proof of Lemma 2 also show that

(2.16) rv TSfru CB ru rv

for all f B, where B L2 ⌦ is bounded. With the notation introduced above, we have thefollowing result.

Theorem 3. Let u0 L2 ⌦ . Then there is a unique function u u t, x in C 0, T ;L2 ⌦L2 0, T ;H1 ⌦ such that u is a solution to

(2.17)

u

tdiv Suru 0 in 0, T ⌦,

Suru, n R2 0 on 0, T �

u 0 u0,

where � is the boundary of ⌦, n is the outward normal direction on � and , R2is the usual scalar

product in R2. Moreover, this solution is in C 0, T ⌦ .

Here, Hk ⌦ for nonnegative integers k N is the Sobolev space of distributions with weakderivatives of order k belonging to L2 ⌦ . This is a Hilbert space with the norm

v Hk ⌦

↵ k

↵v 2L2 ⌦

1 2

where we use the standard multi-index notation ↵ ↵1x

↵2y for ↵ N N, with ↵ ↵1 ↵2.

Moreover, Lp 0, T ;Hk ⌦ is the space consisting of all distributions u such that u t Hk ⌦ for


almost every t 0, T , equipped with the norm

u Lp 0,T ;Hk ⌦

T

0u t p

Hk ⌦ dt

1 p

, p 1, k N.

When p , this is to understood as the essential supremum norm ess sup0 t T u t Hk ⌦ .Note also that the type of boundary conditions imposed in Theorem 3 a↵ects the definition of theconvolution G� u, since one needs to extend the domain of definition of u as done in (2.3).

Proof. The idea is to adapt the proof of [3, Theorem 2.1] to our situation. We first determinethe existence of a weak solution to (2.17). To this end, let H1 ⌦ denote the dual of H1 ⌦ , andintroduce the subspace W 0, T of L2 0, T ;H1 ⌦ defined by

W 0, T w L2 0, T ;H1 ⌦ :dw

dtL2 0, T ; H1 ⌦ .

Let w W 0, T L 0, T ;L2 ⌦ satisfy

(2.18) w L 0,T ;L2 ⌦ u0 L2 ⌦

and consider the auxiliary problem Ew given by

Ew

u t t, v L2 ⌦⌦

rv TSw t ru t dxdy 0 for all v H1 ⌦ a.e. in 0, T ,

u 0 u0,

where , L2 ⌦ denotes the usual inner product on L2 ⌦ . Let A t;u, v ⌦ rv TSw t rudxdyfor u and v in H1 ⌦ . By Lemma 2 it follows by virtue of (2.18) that

A t; v, v ⌫⌦

rv 2dxdy ⌫ v H1 ⌦ ⌫ v L2 ⌦ for almost every t 0, T ,

where ⌫ is a positive constant independent of w (but depending on u0). Similarly, property(2.16) together with an application of the Cauchy-Schwartz inequality implies that A t;u, vC u H1 ⌦ v H1 ⌦ for almost every t 0, T , where the constant is independent of w. We may thenapply a result due to J. L. Lions, see Brezis [2, Theoreme X.9], and conclude that the problem Ew

has a unique solution Uw L2 0, T ;H1 ⌦ C 0, T ;L2 ⌦ with dUw dt L2 0, T ; H1 ⌦ .In particular, we have Uw W 0, T .

Following [3, pp. 188-189], we now deduce that U : w Uw preserves a nonempty, convex andweakly compact subset W0 of W 0, T , defined by

W0 w W 0, T : w 0 u0, w L 0,T ;L2 ⌦ u0 L2 ⌦ ,

w L2 0,T ;H1 ⌦ C1, dw dt L2 0,T ; H1 ⌦ C2 .

By considerations of compact inclusions of Sobolev spaces if follows that U is weakly continuous onW0, see the theorem of Rellich and Kondrachov in Brezis [2, Theoreme IX.16]. Since W 0, T iscompactly embedded in L2 0, T ;L2 ⌦ , the Schauder fixed point theorem can then be applied toconclude the existence of an element u Uu in W0, see for example Friedman [5, p. 189]. Thus wehave found a weak solution u to the problem (2.17).

Regularity of solutions. By a bootstrap argument follows that u t H1 ⌦ for all t 0, fromwhich we conclude that u t H2 ⌦ for all t 0. Iterating the argument and using the generaltheory of parabolic equations [2, 7], we conclude that u C 0, T ⌦ , and that u is a strongsolution to (2.17). In particular, the boundary condition in (2.17) must hold; compare for examplewith [2, Etape D, p. 180].


Uniqueness of solutions. Let u and v be two solutions to (2.17). They will then satisfy

du t dt div Su t ru t 0,

dv t dt div Sv t rv t 0.

The di↵erence of the two equations above can be written

(2.19) d u t v t dt div Su t ru t rv t div Su t Sv t rv t .

Multiplying (2.19) with u t v t and integrating over ⌦ gives after partial integrations

⌦

1

2

d

dtu t v t 2dxdy

⌦ru t rv t T Su t Sv t rv t dxdy(2.20)

⌦ru t rv t TSu t ru t rv t dxdy.

Recall that if u is a solution then t u t L2 ⌦ is continuous, and that u L 0,T ;L2 ⌦

u0 L2 ⌦ . Since v is also a solution, Lemma 2 implies that we can find a constant ⌫ such that

⌦ru t rv t TSu t ru t rv t dxdy ⌫ ru t rv t 2

L2 ⌦

for all t 0, T . Hence,

(2.21)1

2

d

dtu t v t 2

L2 ⌦ ⌫ ru t rv t 2L2 ⌦

⌦ru t rv t T Su t Sv t rv t dxdy,

where, by an application of the Cauchy-Schwartz inequality, the right-hand side is bounded by

sup⌦

Su t Sv t spec ru t rv t L2 ⌦ rv t L2 ⌦ .

Here, A spec is the spectral radius norm of the matrix A, which satisfies A spec A F , the secondnorm being Frobenius norm. We now claim that

(2.22) supx,y ⌦

Su t,x,y Sv t,x,y F C u t v t L2 ⌦ ,

where the constant depends only on K, G�, g and the initial value u0. Admitting this for themoment, the previous discussion and an application of Young’s inequality then gives

1

2

d

dtu t v t 2

L2 ⌦ ⌫ ru t rv t 2L2 ⌦

⌫

2ru t rv t 2

L2 ⌦

C2

2⌫u t v t 2

L2 ⌦ rv t 2L2 ⌦ .

After subtracting ⌫ ru t rv t 2L2 ⌦ from both sides, another estimation gives

d

dtu t v t 2

L2 ⌦

C2

⌫u t v t 2

L2 ⌦ rv t 2L2 ⌦ .

Together with the initial condition u 0 v 0 0, an application of Growall’s inequality to thisestimate now shows that u t v t 2

L2 ⌦ is constant on 0, T . Since u 0 v 0 0, uniquenessfollows.

It remains to prove (2.22). By Proposition 1 we have that

(2.23) Su t,x,y Sv t,x,y F C sup⇠,⌘ ⌦

Tu t,⇠,⌘ Tv t,⇠,⌘ F ,


where C only depends on K and g . To simplify notation, let us for the moment suppress theparameter t. Note that Tu Tv is the symmetric matrix

(2.24) Tu Tva bb c

,

where a straightforward computation shows that a, b and c can be written as

a K xG� u v xG� u v ,

b K yG� u xG� u v xG� v yG� u v ,

c K yG� u v yG� u v .

Let as before a, b, c and ' a, b, c denote the largest and smallest eigenvalues of the symmetricmatrix given by (2.24). In view of the expressions for a, b and c, estimates similar to (2.12) and(2.13) show that

Tu x,y Tv x,y F C K,G� u L2 ⌦ v L2 ⌦ u v L2 ⌦ .

Reintroducing the parameter t, recall that since u and v are solutions to (2.17), we have that thenorms u t L2 ⌦ and v t L2 ⌦ are bounded by u0 L2 ⌦ for all t 0, T . By virtue of (2.23)this gives (2.22), which completes the proof.

3. Numerical aspects. For solving the problem (2.5) we consider the steady state solutionto

(3.1) ut t, x, y � x div S u t, x, y ru t, x, y 1 � x u t, x, y f0 x, y .

Written out explicitly, this is an parabolic equation containing non-zero contribution from the mixedderivatives, i.e.,

(3.2)u

ta

2u

x22b

2u

x yc

2u

y2d

u

xe

u

y1 � x u f0 ,

where the coe�cients a, b, c, d and e depend on the solution u. Due to the averaging in thecomputation of Su they vary slowly compared to u which heuristically makes feasible to instead solvethe linearized equations. The fixed point argument in the existence part of Theorem ?? formalizesthis claim. For the solution of (3.1), we therefore compute a, b, c, d and e given the solution u attime t. As t increases we then recompute the coe�cient functions. Due to the averaging e↵ect, thecoe�cient function does not in practice need to be recomputed at each time step, but multiple timesteps can be taken before it is needed to update them.

For the solution of parabolic di↵erential equations, we can choose between using an explicitor implicit method. Explicit methods have the advantage that they are easy to implement, andthat each time step can easily be computed in a fast manner. However, they require that the timesteps taken are small in order to be stable. If large timesteps are desirable, it is advantageous toinstead use an implicit method. We provide details for how to deal with both explicit and implicitimplementations.

First, let us consider the computation of the coe�cient functions a, b, c, d and e for a fixed timet. The mollified version of the derivatives G�ux t, x, y and G�uy t, x, y can be computed rapidlybe means of FFT. Also by using the FFT, we can rapidly compute the functions K G�ux t, x, y 2,K G�ux t, x, y G�uy t, x, y and K G�uy t, x, y 2. For the computation of S, let

a K G�ux t, x, y 2, b K G�ux t, x, y G�uy t, x, y , c K G�uy t, x, y 2.

The angle ! in (2.10) is then given by

! arctanc a c2 2ac a2 4b2

2b


at each point x, y , and the largest eigenvector s1 of the matrix

a b

b c

it holds that

s1a c a2 c2 2ac 4b2

2.

Using the formulas above, we can compute the ✓ x, y and s1 x, y explicitly, without having loopingthrough each point x, y and make eigenvalue decompositions of 2 2 matrices.

We now have explicit formulas for the computation of elements

a bb c



g Tf 00 1


cos ✓ Tf sin ✓ Tf,

and we may thus obtain

a bb c

Su t ,

by using (2.11) and FFT to compute the element-wise convolutions with K.Now, let us consider an explicit discretization. The simplest version is to use an scheme which

is first order approximate in time and second order approximate in the spatial discretization. Wethus approximate

u

tt, x, y

u t �t, x, y u t, x, y

�t

and use standard stencils for the spatial discretizations, i.e.

2

x2

1

�2x

0 0 01 2 10 0 0

,2

x y

1

4�x�y

1 0 10 0 01 0 1

,2

y21

�2y

0 1 00 2 00 1 0

.

for a second order scheme, and

1

12�2x

0 0 0 0 00 0 0 0 01 16 30 16 10 0 0 0 00 0 0 0 0

,1

144�x�y

1 8 0 8 18 64 0 64 800 0 0 0 08 64 0 64 81 8 0 8 1

,

1

12�2y

0 0 1 0 00 0 16 0 00 0 30 0 00 0 16 0 00 0 1 0 0

.

for a fourth order spatial scheme. The explicit discretization then reads

u t �t, x, y u t, x, y �t a2u

x2t, x, y 2b

2u

x yt, x, y c

2u

y2t, x, y

du

xt, x, y e

u

yt, x, y 1 � x u t, x, y f0 x, y .


Implicit methods are numerical schemes where u t j�t, x, y for j N appear on the righthand side of the equation (3.2). Hence, in order to obtain u t �t, x, y when u t j�t, x, y , j N0,it is necessary to solve a system of linear equations. Using generic methods for inverting the matricesthat the stencils give rise to will prohibitively slow. However, for (short) finite di↵erence stencils inone dimension, solving the linear system of equations can be done in linear time, since the matrixdescribing the linear system of equations will be diagonal dominant. For instance, for the case wherethe second order spatial stencil 1 21 for the approximation of

2

x2 , the matrix will the tridiagonal,and hence one can make use of for instance Thomas algorithm [11, §2.4].

However, when solving problem in two dimensions or higher, this approach will not work.Column stacking (or any other ordering of elements) of the unknowns on a two-dimensional grid,will give rise to diagonal entries far away from the main diagonal. For instance, if the u t, x, y forfixed t is represented on an N N lattice, then there will be contributions (at least) from the N :tho↵ center diagonal, and standard approaches like LU -factorizations will fairly dense. A remedy tothis problem when discretizing the Laplace operator 4

2

x2

2

y2 is to make use of alternating

direction implicit (ADI) methods, [4, 10]. The idea is to split the operator into two parts, onethat is only acting in the x variable and one that is only acting in the y-variable. In this way,one can transfer the two-dimensional problem into solving two problems containing essentially onlyspatial derivatives in one variable (either x or y), and solve those problem by the one-dimensionalapproach discussed above (for instance using Thomas algorithm). These methods are well knownand commonly used.

The problem that we are dealing with can, however, not be directly treating using ADI-methods.This is because of the presence of the mixed derivate term

b2

x y.

There are suggestions for how to generalize the ideas behind ADI to also include the presenceof mixed derivatives in the literature, although this case is much less known that the case withno mixed terms. One early such reference is [8], where a two-level first order accurate (in time)unconditionally stable scheme is presented. In this paper, we will follow the path suggested in [1].

Define the forward and backward time di↵erence operators by

Q u t, x, y u t �t, x, y u t, x, y ,

and

Q u t, x, y u t, x, y u t �t, x, y ,

respectively. A factored scheme then reads [1, pp. 19–20 ]

1 !�ta2

x21 !�tc

2

y2Q ↵Q u

�t1 ⇠2

a2

x22b

2

x yc

2

y21 ⇠ ✓

1

2Q u 2!�tb

2

x yQ u

↵!�t a2

x2c

2

y2Q u

⇠

1 ⇠↵ Q u,

for constants !, ↵, ✓ and ⇠ satisfying

✓2 1 ⇠

2 1 ↵ 1 2⇠

1 ⇠

, ⇠1

2, 1 ↵ 1, !

✓

1 ⇠.

For details about the choices of parameters e↵ect the properties of the numerical scheme, cf. [1].The factored scheme can be implemented as follows


1. Compute the auxiliary function u x, y by solving

1 !�ta2

x2u

�t1 ⇠2

a2

x22b

2

x yc

2

y21 ⇠ ✓

1

2Q u 2!�tb

2

x yQ u

↵!�t a2

x2c

2

y2Q u

⇠

1 ⇠↵ Q u,

2. Compute the auxiliary function u x, y by solving

1 !�tc2

y2u u

3. Update for the solution u at time t �t.

u t �t, x, y u 1 ↵ u t, x, y ↵u t �t, x, y .

Note that the first step can be implement fast since it only involves derivatives in the x-direction,and can thus be solved by using a tridiagonal (second order) or pentadiagonal (fourth order) solver.The same thing holds for the second step, but now the derivatives are only in the y-direction.

We briefly show some numerical results. The left upper panel of Figure 2 shows a syntheticdata set consisting of sums of parabolic Gaussian wave packets of the form

gj x, y e �j y yj � x xj2 2 2⇡i⌘j y yj � x xj

2 2

, � 0,

where �j describes the Gaussian decay,⌘j wavelet frequency, �j the parabolic curvature, and wherexj and yj denotes center locations. Hence

f x, yj

gj x, y .

We assume that we know f along 11 vertical lines (traces) x xl. The upper right panel of Figure2 shows f at the know traces, and where the unknown traces have been blanked out. The tracesare also depicted in the lower right panel of Figure 2. In the lower left panel of the figure, thereconstruction by solving (2.5) by means of the anisotropic di↵usion problem (3.1) is shown.

4. Conclusions. We have designed a method for the interpolation of seismic data when thedata is either very sparsely undersampled, or where there are large gaps in the data. We assume thatin addition to knowledge of data along traces, we also know the x-derivative of f at the locations ofthe traces. This means that we have local knowledge of the function in a neighborhood around thetraces. We then formulate the interpolation problem as a variational problem, where we minimimizethe second derivative of the structure tensor. This is done in order to minimize the local presenceof several directions in the reconstructed data. The solution of the variational formulation can beexpressed in terms of an elliptic partial di↵erential equation. One way to compute the solution tosuch an equation, is by considering the steady state solution of a parabolic equation constructedfrom the elliptic part. This has the advantage that it can be solved by a time stepping approach.

We show existence and uniqueness of the nonlinear parabolic (anisotropic di↵usion) problemthat we have derived. This is important as it reveals that we can solve the variational problemby solving the nonlinear di↵usion problem. Because of the nonlinearity of the anisotropic di↵usionproblem, it is not obvious that solutions are unique. The kind of anisotropic di↵usion that we havederived is slightly di↵erent from the one that is usually considered. The proofs are based on ideasfrom [3], but applied to the more complicated setting that follows from the designed variationalproblem.


Fig. 2. Example on a synthetic data set consisting of parabolic Gaussian wave packets. The top left panelshows the original data. The information available is depicted in the two right panels, while the result from doinginterpolation by solving the variational problem is shown the lower right panel.

5. Acknowledgements. This research has been supported by the JSPS postdoctoral fellow-ship program, the Craaford Foundation, the Swedish Foundation for International Cooperation inResearch and Higher Education, the Swedish Research Council and the members of the Geomathe-matical Imaging Group: BGP, ExxonMobil, PGS, Statoil and Total.

REFERENCES

[1] Richard M Beam and RF Warming. Alternating direction implicit methods for parabolic equations with a mixedderivative. SIAM Journal on Scientific and Statistical Computing, 1(1):131–159, 1980.

[2] Haım Brezis. Analyse fonctionelle, volume 5. Masson, 1983.[3] Francine Catte, Pierre-Louis Lions, Jean-Michel Morel, and Tomeu Coll. Image selective smoothing and edge

detection by nonlinear di↵usion. SIAM Journal on Numerical Analysis, 29(1):182–193, 1992.[4] Jim Douglas, Jr. On the numerical integration of ˆ2uxˆ2+ˆ2uyˆ2=ut by implicit methods. Journal of the

Society for Industrial & Applied Mathematics, 3(1):42–65, 1955.[5] Avner Friedman. Partial di↵erential equations of parabolic type, volume 196. Prentice-Hall Englewood Cli↵s,

NJ, 1964.


[6] Hahn J and Lee C-Oi. A nonlinear structure tensor with di↵usivity matrix based on image. Technical ReportApplied Mathematics Research Report, 08 01, KAIST, January 2003.

[7] Olga A Ladyzenskaja and Vsevolod A Solonnikov. Nn ural ceva, linear and quasilinear equations of parabolictype, translated from the russian by s. smith. translations of mathematical monographs, vol. 23. AmericanMathematical Society, Providence, RI, 63:64, 1967.

[8] S McKee and AR Mitchell. Alternating direction methods for parabolic equations in two space dimensions witha mixed derivative. The Computer Journal, 13(1):81–86, 1970.

[9] H Mohebi and A Salemi. Analysis of symmetric matrix valued functions. Numerical functional analysis andoptimization, 28(5-6):691–715, 2007.

[10] Donald W Peaceman and Henry H Rachford, Jr. The numerical solution of parabolic and elliptic di↵erentialequations. Journal of the Society for Industrial & Applied Mathematics, 3(1):28–41, 1955.

[11] William H Press, Brian P Flannery, Saul A Teukolsky, and William T Vetterling. Numerical Recipes in FOR-TRAN 77: Volume 1, Volume 1 of Fortran Numerical Recipes: The Art of Scientific Computing, volume 1.Cambridge university press, 1992.

[12] Joachim Weickert. Anisotropic di↵usion in image processing, volume 1. Teubner Stuttgart, 1998.[13] Joachim Weickert and Martin Welk. Tensor field interpolation with pdes. In Visualization and processing of

tensor fields, pages 315–325. Springer, 2006.

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