[PDF] Resolution limits for crosswell migration and traveltime tomography

View - Download Resolution limits for crosswell migration and traveltime tomography

Previous PDF

Next PDF

Geophys. J. Int. (1996) 127,427-440

Resolution limits for crosswell migration and traveltime tomography

Gerard T. Schuster

Geology and Geophysics Department, College of Mines and Earth Sciences, University of Utah, 717 W. B. Browning Building, Salt Lake City,

UT 84112-1183, USA

Accepted 1996 July 9. Received 1996 July 9; in original form 1995 April

10 SUMMARY

Equations are derived for the point-scatterer response of the crosswell migration and traveltime tomography operators. These formulas are used to estimate the limits of spatial resolution in reflection migration images and traveltime tomograms. In particular, for a crosswell geometry with borehole length

L, well offset 2x0, source wavelength I,

and a centred point scatterer I show the following. (1) The vertical resolution APg of the reflection migration image is equal to

21x0/L under the far-field approximation. Under the Fresnel approximation,

AYg x 2Ix,/L - 212/L, which says that images become better resolved with an increase in aperture and a decrease in wavelength and well offset. (2) The horizontal resolution APg of the migration image is equal to 161xi/L2. The lateral resolution of the migrated image is worse than the vertical resolution by a factor of 8x,/L (where xo/L > 1 under the far-field approximation). (3) The vertical resolution AFmo of the traveltime velocity tomogram is proportional to

6. This estimate agrees with that of a previous study. However, the tomographic

image of the slowness perturbation behaves as a non-local cosine function along the depth axis, whereas the migration image behaves as a localized squared sinc function in the depth coordinate. This is consistent with the empirical observation that interface

boundaries are more sharply resolved by migration than by traveltime tomography. (4) The horizontal resolution of the slowness image in a traveltime tomogram is

equal to ( 4x0/L),/w, a factor more than 4x0/L worse than the vertical resolution. (5) For N, sources and N, geophones, the dynamic range of the migrated image is proportional to N,Ng. The dynamic range of the slowness tomogram is proportional to JW. Many of the estimates for the resolution limits have simple geometrical interpretations. For example, the minimum vertical (horizontal) resolution in a migrated reflectivity section corresponds to the minimum vertical (horizontal) stretch that a migrated wavelet undergoes in going from the time domain to the depth domain. In addition, the minimum vertical (horizontal) resolution in a traveltime tomogram corresponds to the minimum vertical (horizontal) width of the wavepath at the scatterer location Key words: crosswell, Fraunhofer approximation, migration, resolution, tomography.

INTRODUCTION

Migration and traveltime inversion of crosshole seismic data have proved to be important tools for imaging interwell geology (Lines 1988; Stolt & Benson 1986). Reservoir engineers can use good-quality reflectivity and slowness images to under- stand the detailed geology of reservoirs, leading to the optimal recovery of hydrocarbons.

To optimize the use of reflectivity and slowness imaging we need to understand the resolution limitations of both

crosswell migration and traveltime tomography. Towards this goal I have derived, under the far-field approximation, the response of both the crosswell migration and traveltime tomography operators to localized velocity perturbations. These perturbation responses reveal the resolution of the migrated reflectivity and tomographic slowness images as a function of the aperture length, the perturbation offset and the source frequency. This knowledge can lead to the better design

Q 1996 RAS 427 Downloaded from https://academic.oup.com/gji/article/127/2/427/591028 by guest on 23 October 2023

428 G. T. Schuster

of crosswell surveys, more efficient discretization of images, and a clearer understanding of the opportunities and limi- tations in crosswell imaging.

GENERALIZED IMAGING FORMULA

For the crosswell geometry in Fig. 1, the following generalized imaging formula can be used to represent either high-frequency Kirchhoff migration or transmission traveltime tomography:

L2 L2

m(s a)= I-,, j-,, mg, rs, r, 0) x exp {iwCz(r,, r) + z(rs, r) - z(r,, r,)ll 4rS, r,, w) dz, dz,. (1) Here, d(rs, r,, w) represents the data measured at the geophone location r, with a source at r,; T(x,x') represents the direct- wave traveltime from x to x'; and m(r, w) represents the image of the model at the interwell location r, for source frequency w. B(r,, r,, r, w) contains geometrical spreading terms, and the integrations are over the source z, and geophone z, depth coordinates, where each of the two wells has a length of L= lLll + ILzl. Note that the integral expression is the same as the Fresnel-Kirchhoff integral, except that the obliquity factor has beeen neglected (Elmore & Heald 1969). The assumption implicit in the Fresnel-Kirchhoff integral is that the aperture is large compared with the wavelength, i.e.

1 < L.

For a point scatterer, eq. (1) reduces to different imaging equations for special values of

B and d.

( 1) m(r, w) in eq. (1) represents the migrated reflectivity image if

B= ~/~l~,--~l1~,-~l~

and d = IreJection data[. The associated imaging algorithm is a type of Kirchhoff migration,

CROSSWELL GEOMETRY

:O,Lz> s

Source Weli Geophone Well

Figure 1. Point scatterer located at r,=(x,, z,), where the wells are offset by a distance X. (2) Eq. (1) represents the image of the slowness field obtained from transmission traveltime tomography if

B = 2sOw3 Ir, - rg\/( Ir, - r /I r, - rl )

and d = Az, where AT is the traveltime residual (see eq. 8 in

Schuster

& Quintus-Bosz 1993). The goal of the next two sections is to analyse the response of the migration and traveltime tomography equations to a localized perturbation in the velocity distribution. This will lead to an estimate of the resolution limits for these imaging algorithms

RESOLUTION OF MIGRATED

REFLECTIVITY IMAGES

Ideally, the migration image of a point scatterer in a homo- geneous medium should be a point in space but in practice it is a smeared version of a point scatterer, where the spatial width of the smear is determined by the source frequency, scatterer location and the source-receiver configuration.

The magnitude

of the monochromatic response of a point source at r, (see Fig. 1) and a point scatterer at r, is represented by 1 Inserting the above equation into eq. (l), setting B = l/lrg -rIlr, -rJ, and replacing the traveltimes by the appropriate scaled distances we obtain a Kirchhoff-like imaging formula: m(r, w) - L2 L2 exp[ik(~rg-r~+~rs-r~-~r,-ro~-~rg-ro~)] - lL1 LL1

Ir, - r /I rs - r II rs - ro II rg - ro I

x dz, dz, . (3) Eq. (3) is the starting point for analysing the resolution of a migrated image under the far-field approximation.

Vertical resolution of a migration image

The vertical resolution of a migrated crosswell image will now be analysed for the point scatterer in Fig.

1. If the wavefront is

'nearly' planar by the time it reaches the interrogation point r then the klr, - rl exponent in eq. (3) can be approximated by z: - 2z,z klr,-rl=kr 1+1, (4) z kz: r 2r J = kr - kz, - + - + higher-order terms x kr - kz,?, (5) r where r = d-. This approximation is appropriate if Irl> L2/A i.e. the distance from the source point to the interrog- ation point /r - rs/ is larger than the aperture length L times the aperture length in units of wavelengths (Hanish 1981,
p. 41). The characterization of distance as r - z,z/r is known as the far-field, planar or Fraunhofer approximation (Steinberg & Subbaram 1991). The z,' term corrects for curvature effects

0 1996 RAS, GJI 127,421-440 Downloaded from https://academic.oup.com/gji/article/127/2/427/591028 by guest on 23 October 2023

Crosswell resolution limits 429

in the wavefront not accounted for by the planar approximation.

Applying the far-field approximation to eq.

(3) results in L2 Lz exp[-i(k,z, + k,z,)] dzgdz,, (6)

J-Ll J-Ll

m(r, o) x A where k, = k(z/r' - zo/rb), k, = k(z/r - zo/ro), (7) and A = exp[ik(r + r' - ro - rb)]/rrOr'rb. By taking the geo- metrical spreading factors outside the integral we have assumed the 'paraxial' approximation; i.e. the imaging distance from the well is large compared with the borehole length so that IL1 + Lz12<Integration of eq. (6) yields

4A exp[ -i(L2 - L,)(k, + k,)/2] sin(k,L/2) sin(k,L/2)

m(r, w) = k, k, (81 This equation will now be used to define the vertical resolving limits of migration.

Definition

1. The resolution limit A of an imaging algorithm is

defined to be the minimum distance in which two point scatterers of equal strength can be separated and yet still be distinguished from one another in the image section. From eq. (8) and

Rayleigh's criterion (Elmore

& Heald 1969), the minimum distance is artiJcially defined to be when the central peak of one sinc function falls on the first minimum of the other pattern (see Fig. 2a). However, we will adopt the conservative Rayleigh criterion illustrated in Fig. 2 (b), i.e., the minimum distancequotesdbs_dbs47.pdfusesText_47

[PDF] limit of resolution microscope

[PDF] Limitation de l'intensité: le coupe-circuit

[PDF] limitation de vitesse allemagne

[PDF] limitation de vitesse autoroute france

[PDF] limitation de vitesse en agglomération

[PDF] limitation de vitesse hors agglomération

[PDF] limitation de vitesse la plus élevé

[PDF] limitation de vitesse usa

[PDF] Limitation des risques de contamination et d'infection

[PDF] limitationde l'intensité:le coupe circuit

[PDF] Limite

[PDF] limite 0/infini

[PDF] limite calcul

[PDF] limite conventionnelle d'élasticité

[PDF] limite cosinus