Travel time inversion with a priori information on illuminated ... - Wiley

JOURNAL

OF GEOPHYSICAL

RESEARCH,

VOL.

100, NO. B2, PAGES 2151-2160, FEBRUARY

10, 1995

Travel time inversion with a priori information on illuminated parts of the reflectors L. R. Jannaud and F. Delprat-Jannaud Pre-stack Structural Interpretation Research Consortium, Institut Franqais du P6trole Rueil-Malmaison, France

Abstract. Since the solution of reflection tomography is underdetermined, the use of a priori information is essentialto compute a model that is satisfactory from a geological point of view. We propose to compute a model that, classically, matches in a least squares sense the observed travel times and that also satisfies some additional useful constraints on the portions of reflectors illuminated by the rays. The implementation of this method requires the computation of the Jacobian of the impact point of a ray on a reflector with respect to the model. This computation is performed by the adjoint state technique, which efficiently produces exact derivatives. The utility of this approach is illustrated on a field data set acquired on fault structures. The method produces a model that matches the kinematic reflectors do not cross the fault.

information

under

the constraint

Introduction

The goal of travel time inversion is the determination of the subsurface structure (reflector geometries and wave propagation velocities) from arrival times picked on a multioffset seismic data set. Introduced by Bois et al. [1971] for transmission seismology and by Bishop et al. [1985] for reflection seismology,tomography has proved its efficiency. However, it is widely recognized that the solution of reflection tomography is, by construction, strongly underdetermined [Bube and Resnick, 1984; Bube et al., 1989; Bickel, 1990; Scales et al., 1990; Stork, 1992a, b]. To overcome this difficulty, a priori information [Delprat-Jannaud and Lailly, 1993] on the regularity of the model (reflector and velocity curvatures) has to be introduced. However, even with such information, the solution generally remains underdetermined, and the computed model is not geologically satisfactory.

In this paper, we introduce geological information on the portions of reflectors that are determined by reflection tomography: we control the location of the impact points of the rays on the reflectors. We can thus constrain the rays to focus on a given portion of the reflector or, on the contrary, constrain them to illuminate a wide region of the reflector. We can also constrain the distance between two reflectors, impose that some reflectors do not cross each other. After presenting the classical formulation of reflection tomography and its limitation on a raw seismic data set, we present a method to control the location of the impact points of the rays on the reflectors. Since our optimization code deals with linear constraints, the implementation of this a priori information requires the computation of the Jacobian of the impact point of a ray with respect to the model for a fixed source and a fixed receiver. This computation relies on the adjoint state technique, a technique which is rarely used in tomography but which is very efficient. It is detailed in this

that certain

paper. We illustrate the method on two examples: a synthetic one for which the data have been computed by two-point ray tracing and a field data set acquired on fault structures, in which we require that some sedimentary reflectors

not to cross the fault.

Limitation of Travel Time Inversion in the Presence of Fault Structures

The data of travel time inversion are composed of a set of

observed traveltimestøbs(i,Xs, xR) associated with some picked reflection i (1 -< i -< I), some shot S located at Xs, and some receiver R located at x R. To simplify the notations, we assume shots and receivers to be located at the surface.

The

extension

of the results

to buried

shots

and

receiversis straightforward. Tøbsdenotesthe set of observed travel

times.

We consider a two-dimensional rectangular domain of the subsurface. The horizontal coordinate ranges between Xmin and Xmax.The vertical coordinate (or depth) rangesbetween Zminand Zmax-Within this rectangular domain, an Earth model m is defined by a velocity distribution and the reflector geometries. The number of these reflectors is the number of picked reflections I. The geometry of the ith

reflector (1 -< i -< I) is describedby a function Zi(x) which gives the depth of the reflector at any lateral position x between Xmin and Xmax. In order to exploit an efficient ray tracer, the velocity field is described by a smooth slowness

squared distribution [Virieuxet al., 1988],u2(x, z), for any X • [Xmin, Xmax]and any z • [Zmin, Zmax].

Two-pointray tracingcalculates traveltimestcal(i,xs, x•; m) for rays emitted by shot S in model m, which propagate downward to reflector i, are reflected, and then propagate upward to be recorded by receiver R. T(m) denotes the set of calculated travel times corresponding to the considered acquisition pattern. The aim of travel

Copyright 1995 by the American Geophysical Union.

time inversion

is to find the models that

produce ray-traced travel times that best match the observed travel times [Bishopet al., 1985].The least squaresformulation of this problem consistsin minimizing the objective function

Paper number 94JB02873. 0148-0227/95/94JB-02873505.00 2151

2152

JANNAUD

AND DELPRAT-JANNAUD:

Shot location (km) 15

REFLECTION

TOMOGRAPHY

Here we are interested in structural models in which the

20

sedimentaryreflectorsdo not crossthe fault. The goalof this paper is to proposean extensionof travel time inversion that controlthe locationsof the impactpoints of the rays on the

reflectors.We will useit on the previouslypresenteddataset to take fault structuresinto accountby constrainingselected reflectors not to cross the fault.

Extension of Travel to Fault Structures

Time Inversion

We have seenin the previoussectionthat to simplifythe

modeldescription,reflectorsweredefinedfromXmin to Xma x. They are consequentlyallowed to cross each other, which is not satisfactory in the case of fault structures. Besides,

traveltimeinversiondoesnotdeterminereflectorsfromXmi n to Xmax:reflectorsare only determinedat the impactpoints of the rays, i.e., at the points where the rays meet the reflectors.The other parts of the reflectorsare not informed Figure 1. Travel times picked in the near-offset section by the data and reflect a priori model constraints.We will (163 m). Travel timeshave alsobeenpickedfor largeroffsets exploitthe geologicalinformationat the impactpointsof the rays to constrain the selected reflectors to be above (or (up to 2500 m). below, accordingto the consideredexample)the fault (Fig-

C(m)= I T(m)- TøbSll 2

(1)

ure 3).

To simplify the notation, we will consider a model with

which measures the misfit between calculated and observed

two reflectors:the first one describedby Z•(x), and the travel times. This inverseproblemis nonlinearand is solved secondone (the fault) describedby Z2(x), which we will iteratively by a Gauss-Newtonmethod [Presset al., 1992] constrain to lie below the first reflector (Figure 3). The whichconsistsin minimizingthe quadraticobjectivefunction: extension of travel time inversion to fault structures consists

Ck(&m) = IaSm - [Tøbs-T(mk)][I 2. (2) fa istheJacobian of theforward modeling operator about

in minimizing (1) subject to the set of constraints:

Zl[Xilmp(m)) -