Examples of reverse-time migration of single-channel ... - CiteSeerX

GEOPHYSICS,

VOL. S7. NO. 4 (APRIL

1992); P. 577-586, 8 FIGS.

Examples of reverse-time migration of single-channel, ground-penetrating radar profiles

Elizabeth Fisher*, George A. McMechan*, A. Peter Annant, and Steve W. CoswayS ground, and we enter the domain of electromagnetic induction methods. The seismic analogy is no longer appropriate and the depth imaging methods of transient EM (Macnae and Lamontagne, 1987; Eaton and Hohmann, 1989) are the GPR equivalent. The kinematic similarities between radar and seismic wave propagation may be exploited in data processing. Many techniques that have been extensively developed for processing of seismic data can, and have been, directly applied to radar data when recorded with the same survey configurations. The main change required is a resealing, which may be simply implemented by changing input parameters, rather than the data. It has long been recognized that a raw radar or seismic reflection time section, consisting of traces with a constant source-receiver antenna separation (or “constant-offset”), presents a distorted, unfocused image of the subsurface structure (Stern, 1929; Barringer, 1965; Harrison, 1970). The purpose of migration is to take a reflection profile (a function of survey position and time) and to produce a focused image of reflectivity correctly positioned in space. Previous attempts using migration for radar data generally operated on time picks from a single reflector (Harrison, 1970; Jezek et al., 1985; Fisher et al., 1989). Recent examples include imaging of satellite-based synthetic aperture radar data (Rocca et al., 1989) and of ground-penetrating radar profiles (Hogan, 1988). There are a number of algorithms available for migration of radar reflection profile data, but few have actually been used. Hogan (1988) uses Kirchhoff diffraction migration for ground-penetrating radar data; this is an integral formulation that has the advantage of being directly applicable to data for all source-receiver antenna separations, but is difficult to implement efficiently when the propagation velocity varies in space. Other algorithms that may be borrowed from reflection seismology include all those based on scalar wave propagation. A good summary of these techniques may be found in Gardner (1985) and Yilmaz (1987).

ABSTRACT

A single-channel, ground-penetrating radar (GPR) profile portrays a distorted, unfocused image of subsurface structure due to apparent position shifts associated with dipping reflectors and to diffractions from corners and edges. A focused image may be produced from such data by using any of the migration algorithms previously developed for seismic data; we use reverse-time migration based on the scalar wave equation. Field work was performed over a simple stratigraphic soil sequence and a complicated fluvial environment. In the migrated images, reflector continuity is enhanced and the level of detail available for highresolution interpretation is significantly increased.

INTRODUCTION

Analysis of the theory of electromagnetic and elastic body wave propagation reveals a number of similarities (Szaraniec, 1976, 1979; Ursin, 1983; Lee et al., 1987; Zhdanov, 1988). By comparing the differential equations, identifications of operators and variables that play corresponding roles can be made. Both radar and acoustic pulses propagate with finite velocities that depend on the material properties and each are reflected and diffracted by local changesin the medium. The dynamic behaviors are different (with regard to amplitude attenuation and dispersion), but the kinematic behaviors (such as pulse propagation times) are the same. This is a consequence of the fact that displacement currents dominate conductive currents at frequencies where GPR is effective. Under these conditions, an electromagnetic pulse propagates with virtually no dispersion and has a velocity controlled by the dielectric properties of the material alone. At low frequencies or in high conductivity environments where conduction currents dominate, GPR is no longer an appropriate method; electromagnetic fields diffuse into the

Manuscript received by the Editor February 28, 1991;revised manuscriptreceived October 9, 1991. *University of Texasat Dallas, Centerfor Lithospheric Studies, P. 0. Box 830688. Richardson, TX 75083-0688. $ Sensors and Software, Inc., 5566 Tomken Rd., Mississauga, Ontario, Canada L4W IP4. 0 1992 Societyof Exploration Geophysicists. All rights reserved. 577 Downloaded 10 Mar 2010 to 130.79.19.73. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/

578

Fisher et al.

The purpose of this paper is to present some examples of spatial imaging of radar reflectivity using reverse-time migration. This approach has been widely applied in the seismic context (McMechan, 1989) and has the advantage of easily incorporating arbitrary velocity variations and steeply dipping structure. We will consider only constant-offset data. The geological settings selected are nearly ideal for GPR data acquisition for the purpose of demonstrating the application of reverse-time migration. THEORY

Data acquisition and preprocessing

Acquisition of a single-channel, constant-offset radar reflection profile involves fixing the separation between the sourceand receiver antennas and recording one trace at each of a number of equally spaced positions along the survey line. The trace is usually associated with the location midway between the antennas. Reverse-time migration assumes that the source and receiver antennas are coincident (i.e., “zero-offset” in seismic jargon, or “monostatic” in GPR jargon). Bistatic data collected using a finite antenna offset (s) may be corrected by a time- and offset-dependentshift to smaller times to approximate the corresponding monostatic data before processing (Figure 1) using (cf. Yilmaz, 1987, pp. 157-166) t*(o) = t2(x) + x?/v,?,,,

increases with time The lower panel shows the same data after interpolation in time and midpoint and correction for the finite source-receiver separation. The latter explains the shift to earlier times which effectively removes the first pulse from the trace; this pulse correspondsto propagation in the air. Velocity estimation

A crucial component to all migration algorithms is the estimation of the velocity distribution from the earth’s surface down to the greatest depth to which imaging is to be performed. Velocities of radar wave propagation generally decreasewith depth, in contrast to seismic velocities, which generally increase. In the present context of two-dimensional imaging, lateral, as well as vertical, velocity variations may be present. There are at least four ways in which velocities may be estimated.

LX4

I

0 O

m

I

s

p

osition

R-

(1)

where t(0) is the estimated time at zero offset, I(X) is the observed time at offset X, and V rms is the rms velocity from the earth’s surface down to the reflector. This is identical to the normal move-out (NMO) correction of seismic processing. It results in a stretching of the data, and therefore, each trace must be resampled at a constant time increment before it can be input to reverse-time migration. When steeply dipping events are present, more comprehensive approachesfor correction to zero-offset, such as dip moveout, are required (cf. Yilmaz and Claerbout, 1980; Deregowski and Rocca, 1981; Bolondi et al., 1982; Hale, 1984). Preprocessing of the data also includes removal of the characteristic low-frequency background from the data so it has approximately zero mean, and application of a timedependent exponential amplitude gain that compensates empirically for amplitude attenuation with depth. The latter clearly distorts amplitudes but allows small reflections associated with deep structure to become visible. This supports the main objective of migration, which is to determine the shape and location of the reflectors and diffractors that are present at depth. In addition, a smooth taper is applied to both the spatial and temporal edges of the data before migration to reduce artifacts related to the finite data aperture (cf. Chang and McMechan, 1989). Figure 2 illustrates preprocessing of a typical singlechannel radar profile. Each trace in the upper panel has had its exponential baseline subtracted and is plotted at its respective midpoint position with an amplitude gain that

&I

reflector

depth

0:

X .

offset

time FIG. I. Offset correction. Data mapping from finite to zero source-receiver antenna separation. The time t(x) for a reflection observed from source (S) to receiver (R) antennae separated by distance x is larger (by At) than t(O), which would be observed if the source and receiver were coincident at the midpoint position m.

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Migration of Single-channel GPR Data Radar propagation velocities (V) in materials with low electrical loss (and so are amenable to radar sounding) may be determined (e.g., Davis and Annan, 1989) from V = cK - “2,

(2)

where c is 3 x lo* m/s (the propagation velocity of electromagnetic waves in free space), and K is the real part of the (dimensionless) complex dielectric permittivity relative to free space. [The imaginary part of the dielectric permittivity is associated with the wave attenuation (cf. Davis and Annan, 1989), as is the imaginary part of a complex velocity in seismic propagation (cf. Kind, 1976; Martinez and McMechan, 1990).] Thus, propagation velocities may be estimated from laboratory measurements of the dielectric permittivity from drill cores (cf. Redman et al., 1990). A second approach to velocity estimation is tomographic imaging based on transmission times. This has been done most often, in both radar and seismic contexts, with data

579

from cross-well surveys (Olhoeft, 1988; McMechan et al., 1987). This is another example of the same processingbeing applied to both data types as a result of their kinematic similarities. The standard approach to in-situ velocity estimation, for both seismic (cf. Yilmaz, 1987, pp. 166-183) and radar (cf. Clougb, 1976; Annan and Davis, 1976) data is based on equation (I). Given the traveltimes of observed reflections from a series of depths, at a number of offset (x) positions for a common midpoint (m in Figure l), equation (1) may be solved for the rms velocity (V,,) down to each reflector. This, in turn, may be recursively solved for the propagation velocity as a function of depth (Dix, 1955). Another more recent approach to velocity estimation involves iterative migration. If the correct migration velocity distribution is used, the migrated image will be well focused everywhere. When migration velocities are too low, diffractions will not be collapsed completely and diffraction tails

.oo

b . =: FIG. 2. Data preprocessing.(a) Data after removal of an exponential baseline drift and application of time-dependent scaling; (b) the same data after interpolation and correction for the finite-source-receiver separation. Resampling was to 0.30 ns intervals in time and to 0.03 m intervals in horizontal position. For clarity, only every second trace is plotted in (a); every ninth trace is plotted in (b).

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580 will

Fisher et al. extend from

depths. Where

the

will have propagated extend

from

depths.

Observation

locations

velocities

toward

diffractor

allows

trarily

greater

are too high, diffractions

past their focus and diffraction

the

pp. 276282)

diffractor

migration

locations

toward

complicated

appropriately

tails will

For the examples implementation

shallower

(cf. Yilmaz,

1987,

is used

to extrapolate

adjustments

to be

equation

is (cf. Mitchell,

approaches

such

as combining

migration

are also possible (cf. Bording

examples

below,

straints (when

below,

of these phenomena velocity

tomography

et al., 1987). For the

we use a combination

available),

and

and iterative

of drill

can

be represented

the

data

as an

of point diffractors.

a second-order,

of the two-dimensional

appropriate

made. Other

structures

chosen spatial distribution

finite-difference

scalar wave equation

backward

in time

This

1969)

a2u

a?U

?+ ax

azZ=

I a2u ~v2(x, Z) ar2 ’

(3)

hole conwhere

migration.

U is the scalar wavefield

horizontal

space coordinate,

being propagated,

z is the vertical

x is the

space coordi-

Properties of the zero-offset geometry and model parameterization In the zero-offset is recorded

geometry,

the reflected

at each coincident

necessarily

incident

perpendicularly

source-to-reflector

Thus the traveltimes

to twice

the reflector-to-reciever

complete

zero offset reflection

by simultaneously

initiating

times. section

upward to recorders everywhere

(Loewenthal

In general, performed

is one-half

using a numerical associated

tude of the virtual Reflections the reflector ical,

time-stepping

constructed

velocity

is distributed

through-

solution

in time

is parameterized

through

with

each point

the

dif-

at that point.

constructive

via Huygen’s slices where

duced at all recorder

be

as having a

interference

points of high reflectivity

solution,

in time

may

on a computational

associated

from adjacent trajectory

sources at this energy

propagation

to the local reflectivity

are generated

of diffractions

ct=ti

a

with each grid point. The magni-

source

is proportional

this fact,

1976).

extrapolation

grid. In this case, the medium

fractor

the

where finite reflectively

point diffractor

Using

ct=t.

surface through a velocity

of

the wavefield

as the

may be synthesized

and propagating

et al.,

m

are equal

a sequence of virtual

at the earth’s

that

is

paths are super-

of such reflections

time f = 0, along each reflector distribution

location,

at each reflector

and reflector-to-receiver

imposed.

out a model,

radar energy that

source-receiver

principle. “zero-offset” one time

along

In a numersection

sample

is

is pro-

points at each time step in the solution.

Reverse-time migration Reverse-time reflection

migration

section

of a monostatic,

may be implemented

versing the steps described more,

1983; Baysal

the recorded

recorder

position,

order (Figure

above (McMechan,

et al.,

inserting

1983). Migration

data as boundary

in constant-time

3). One constant-time

time step in the numerical

solution

the mesh, from the recorder of the recorded original

trapolation

conditions

slices

is continued

back

and diffracted

in reverse-time at each drive

with the time reverse

backwards to time

where

diffracted).

The wavefield

wavefield.

An

example

3. As described

may be considered

energy

by

at each

to synchronously

locations,

propagating

the spatial positions

Figure

1983; Whit-

is performed

slice is inserted

when all points are imaged

all the reflected

radar by re-

traces. This produces a reconstruction

wavefield

condition)

digital,

numerically

in time =

of the

This ex-

0 (the

imaging

simultaneously has traveled

they were originally

(i.e., back to

reflected

or

present at time = 0 is the migrated for a point above,

diffractor

the diffraction

as the elementary

is shown

in

from a point

response

since arbi-

FIG. 3. Reverse time migration of data from a point diffractor. (a) The synthetic zero-offset section for a point diffractor. (b), (c), and (d) the midpoint-depth (m-z) plane (a vertical slice through the earth) at three widely spaced time steps during reverse-time extrapolation; ti (far from the image time), tj (intermediate), and t = 0 (the image time). In (d), all the energy in the time section (a) has migrated back to produce an image of the point; the two small artifacts that extend faintly below the focus are due to the data truncation at the spatial edges of the profile. The dotted lines show the relationship between the data traces and the boundary values in the m-z plane at I = ti (see text). For clarity, only every fourth trace that was used is plotted in (a). After McMechan (1983).

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J

581

Migration of Single-channel GPR Data nate, V(x, z) is the local propagation velocity, and t is time This wave equation makes no assumptionsabout the specific physical problem being solved. It is equally applicable to all propagating waves that are, or can be reasonably approximated by, a scalar response (U) in a spatially varying velocity distribution [ V( x, z)]. Solution of equation (3) is via an explicit, second-order central finite-difference scheme. At a representative time step (at time ti) three U wavefields are involved: U(x, z, ti), I/(X, Z, ti_l), and I/(X, Z, ti-2). The response U(X~, Zj, ri) at the internal grid point ( xk , zj) (located at the intersection of the jth horizontal grid line with the kth vertical grid line) at time li is

Data were recorded in the PC on a hard disk and transferred to 3.5 in floppy disks for storage and further processing. The radar data are acquired by positioning the antennas at the measurement station and then stacking multiple repetitions of the digitally acquired transient waveform. The station interval is selected to minimize or eliminate spatial aliasing and is determined by the operating frequency of the radar system (Davis and Annan, 1989). The center frequency and the antenna separation are defined by the target, the host geologic environment, and the depth of exploration. The details are survey-specific and are given for each of the case history examples that follow. EXAMPLES

U(Xk. Zj, ti) = 2(1 - 2A2)U(Xk, -

u(xk,

+ u(xk-I + u(xk,

zj,

ti-2)+A2[u(Xk+Iv

7 zjv ti-I) zj-t

Zj, fi-1)

9

+ u(xkv

zjv ti-1)

In this section, the reverse-time migration algorithm described above is illustrated through the processing of two data sets from eastern Ontario, Canada: (1) a simple soil

zj+l 1 ti-I)

ri-I)I,

(4)

where A = V(xI;, zj)Arlh, At is the time step between and h is the grid successive U wavefields (At = ti-li_,), increment in both x and z directions. The condition for local stability is At < hV-‘2-l’* (Mitchell, 1%9). The estimated propagation velocity V(x, z) is input to the wavefield extrapolation part of the migration. V( x, z) needs only to be a slowly varying average function; details are not required. In fact, it is a disadvantage to have sharp changes, as these would produce secondary reflections during the migration extrapolation (Loewenthal et al., 1987), and thus, artifacts in the final image. The velocity transitions in the velocity distributions used (cf. Figure 4a) should be at least one wavelength in thickness. Some of the advantages of reverse-time migration are (1) it can be used with survey lines having arbitrary topographic changes(McMechan and Chen, 1990), (2) it can be used with multiple- as well as single-offset data (Chang and McMechan, 1986), (3) it can be directly applied to threedimensional data (Chang and McMechan, 1989), and (4) it can be modified for use in very small computers (Harris, 1990). Discussions and alternative implementations are presented by Baysal et al. (1983), Loewenthal and Mufti (1983), and Levin (1984). INSTRUMENTATION AND SURVEY PROCEDURE

The data presented in this paper were obtained with a PC-based, digital radar system’ with a system performance factor of 155 dB; output voltages of 400 V or 1000 V; a repetition rate of 30 kHz; a programmable recording time duration, sampling interval, and stacking; and interchangeable dipole antennas. The antennas have a radiation pattern with maximum amplitude for propagation at the critical refraction angle (cf. Simmons et al., 1972). This angle decreases with depth due to downward refraction as the dielectric permittivity [equation (2)] decreases with depth.

‘pulseEKK0 IV, manufacturedby Sensors & Software, Inc., Mississauga,Canada.

FIG. 4. Migration of the soil stratigraphy data in Figure 2b. (a) The velocity distribution used for migration; the shaded zone denotes a linear transition between 0.0648 and 0.507 m/ns. (b) The result of reverse-time migration. For comparison, (c) is the result of Kirchhoff migration using a constant velocity of 0.0648 m/ns. For clarity, only every ninth trace is plotted in (b) and (c).

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582

Fisher et al.

stratigraphic sequence, and (2) a complicated fluvial environment. Example 1: Soil stratigraphy

The data (Figure 2a) were obtained using 100 MHz antennas with a constant offset of 1 m; observations are 0.25 m apart. At each recording location, 128 traces were stacked to improve the signal-to-noise ratio. The output voltage was 400 V and the time sample increment was 0.8 ns. As part of the preprocessing,the data were resampled in both time and spaceto give 15-20 samples per wavelength at the dominant frequency (see caption to Figure 2). This is necessary to keep numerical grid dispersion down to an acceptable level during the finite-difference computations. Borehole log data were available near the survey line; the prominent reflection near 12 ns was found to correlate with an interface between a thinly bedded sand underlain by a very fine grained (clay-sized) soil. The velocity (Figure 4a) of the sand above the reflector was constrained by its known depth in the borehole and both velocities were independently confirmed (D. Redman, personal communication) by direct measurements of dielectric constants of the drill cores. It is desirable, but not necessary to have a priori velocity information since reliable estimates are directly obtainable from the data, as described above. Figure 4b shows the migrated image produced by reversetime migration of the preprocesseddata in Figure 2b using the velocity distribution in Figure 4a. For comparison, Figure 4c contains the image produced independently using Kirchhoff migration with a constant velocity equal to that of the upper layer in Figure 4a (see introduction above); both images are very similar. The relative amplitudes of the shallow and deep reflectors differ in the two plots because reverse-time migration implicitly corrects for geometrical divergence, whereas Kirchhoff migration does not. In this example, only minor changes in the shape of the reflection are introduced by migration because the reflector is fairly flat, except near the right side. The time axis, of course, has been changed to depth as part of the migration. Example 2: Fluvial environment For this example, the data were obtained using 100 MHz antennas with a constant offset of I m and observations were 1 m apart. At each survey location, 64 traces were stacked to improve the signal-to-noise ratio. The output voltage was 400 V and the time sample increment was 0.8 ns. As in the example above, the data were resampled in both time and space to give 15-20 samples per wavelength (see caption to Figure 5). Since there were no boreholes near this line, a migration velocity function was estimated directly from the data. The assumed form of this function is exponential, which is consistent with velocity estimates from multichannel data acquired on other lines in the area (Fisher, 1991). Specific coefficients were determined by iteratively adjusting the velocities and reverse-time migration to progressively reduce artifacts of over- and under-migration. The resulting velocity function is v(z) = (0.03 + o.07~~-“~‘75y37z),

(5)

where 71 is in mins and z is the depth in m. This produces well-focused features throughout most of the migrated image (Figure 5b). It is not easy to appreciate many of the details and differences between the input and migrated data in Figure 5 as only every thirtieth trace is plotted, and the amplitude scaling in plotting was chosen to show only the main structural features. To allow more detailed comparisons and evaluations, we have extracted three portions of the plots in Figure 5, and redisplayed them in expanded form in Figures 6, 7, and 8. Figure 6 contains expanded views of a portion near the upper left corners of the input (time) and migrated output (depth) sections extracted from the data plotted in Figure 5 (refer to the plot axes of Figures 6 and 5 for the exact locations). The relatively flat reflector A is, as expected, only slightly shifted by migration. The continuity of reflections at B, C, D, and G is significantly improved, and the energy that produced interference near C and D has moved up to reveal what are interpreted as two lapping sedimentary sequences, E and F. Figure 7 contains expanded views of a portion near the lower left corners of the input (time) and migrated output (depth) sections. Refer to the plot axes of Figures 7 and 5 for the exact locations. Migration has enhanced reflector continuity throughout (especially near A and B), distortions are unraveled (C and D), and it has become clear that the crossing energy near D [in Figure 7 (a)] is actually a continuous reflector dipping to the left upon which lies a pinched out reflector dipping to the right. Figure 8 contains expanded views of a portion near the center right area of the input (time) and of the migrated output (depth) sections. Refer to the plot axes of Figures 8 and 5 for the exact locations. As expected, the anticlinal structure A has been narrowed in its horizontal extent by migration. The continuity of reflectors near B and C has been significantly improved and, as in Figure 6, energy that produced interference at these locations in the time section moves up (to areas D and E, respectively) to show two lapping sedimentary sequences that were not visible in the unmigrated data. The rest of the section has been similarly improved; migration has clearly been beneficial in correctly positioning reflectors in space and thereby increasing the potential for correct interpretation of fine details of the sedimentary sequence.This increasein accuracy is a result of the processing and could not be achievedby decreasingthe data sampling intervals or increasingthe sourcebandwidth alone. Increasing the center frequency would increase the resolution of the shallowestfeatures at the expenseof penetration depth. DISCUSSIONAND SYNOPSIS The case history data above were acquired in geological settings suitable for GPR sounding. They were selected as being nearly ideal for the purpose of demonstrating the applicability of reverse-time migration to GPR data. Use of the scalar wave equation requires only that a wave be propagated and that a spatially dependent propagation velocity can be defined. It has no assumption about the physical interpretation or origin of the scalar wave. Elastic

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583

Migration of Single-channel GPR Data (multicomponent) seismic migration algorithms (cf. Sun and McMechan, 1986; Teng and Dai, 1989) are not directly applicable to processing of radar data as they are tied too closely to the dynamics of the seismic context (they explicitly incorporate the coupling of compressional and shear waves during propagation and involve a vector rather than a scalar wave equation). Electromagnetic waves are also vector waves, but at that level, the radar/seismic correspondences are less easy to exploit because the dynamic (amplitude) behaviors are not similar. In the presentation of the algorithm above, we emphasized the similarities between the kinematic properties of radar and seismic data as justification for applying migration to GPR data. Migration, however, is expensive, and the potential benefits are data-dependent so caution is recommended. In the example in Figures 2 and 4, no significant improvement is obtained; a simple time-to-depth conversion using a

constant velocity would have been sufficient. On the other hand, where reflectors dip steeply (and hence reflector positions are unreliable in the raw data), or where the structure changes suihciently sharply so that visible diffractions are produced, or where velocity changesspatially, migration can produce significant improvements (Figures 6, 7, and 8). The main criterion in deciding to migrate is whether the resulting increase in accuracy and resolution is required to obtain the level of detail desiredfor subsequentinterpretation. Migration of radar data is applicable only when the data satisfy the same assumptions that are inherent in migration of seismic reflection data. These are that the kinematic aspects of wave propagation (reflection, diffraction, refraction, and energy transport along raypaths) satisfy the laws of geometrical optics and that propagation is linear and nondispersive. Fortunately, these conditions are met by radar data when electrical conductivity is low (