Z-99 High-resolution transforms and amplitude preservation - PGS

1
Z-99 High-resolution transforms and amplitude
preservation
Michel Schonewille 1 and Paul Zwartjes 2
1
PGS Geophysical, PGS Court, Halfway Green, Walton-on-Thames, Surrey KT12 1RS, UK;
2
Delft University of Technology, Delft, the Netherlands
Abstract
High-resolution transforms have several advantages over standard least squares transforms. However, the highresolution
transform can show extra artifacts, which may compromise amplitude preservation. In addition,
amplitudes may be affected if the induced sparseness is too strong. In this paper the amplitude preservation of
high-resolution transforms is studied using synthetic data. Firstly, the high-resolution parabolic Radon transform
is shown to improve both the preservation of primary AVO and demultiple efficiency related to standard least
squares methods. Secondly it is shown that a high resolution Fourier/Radon regularization provides an improved
interpolation of data with large gaps, and less edge effects. Finally, it is shown that that the artifacts can be
suppressed by using a frequency independent sparseness prior. This prior can be derived in a cost efficient
manner from a previous (adjacent) common midpoint gather.
Introduction
The high resolution transform proposed by Sacchi and Ulrych (1995) uses a sparseness prior for the curvature
direction, which gives it several advantages compared with standard least squares transforms. For the parabolic
Radon transform (Hampson, 1986), it improves the separation of primaries and multiples that have a small
move-out difference (Hunt, 1996). High resolution Fourier regularization shows a better reconstruction in large
gaps than standard least squares Fourier regularization (Zwartjes and Duijndam, 2000). However, the highresolution
transform can cause smearing of energy in the transform domain in the direction in which no
sparseness is imposed, i.e. the time direction (see e.g. Sacchi and Ulrych, 1995 Fig. 4a; Hargreaves and Cooper
2001, Fig. 2). This smearing raises the question of amplitude preservation. In addition, amplitude preservation
may not be optimal if the induced sparseness would be too strong. For example, if a perfectly parabolic event
with AVO variation is focused onto a single curvature parameter, then the AVO variation is lost. In this paper,
the amplitude preservation (especially important for AVO analysis and time-lapse seismic) of high-resolution
transforms is studied using a number of synthetic examples.
High resolution Radon transform and AVO
Fig. 1a shows a simple synthetic data set that was used to test the high-resolution parabolic Radon transform.
The events are perfectly parabolic, but the curvatures are not exactly equal to the curvature parameters that are
used for the Radon transform (the primaries also have small curvature). Moreover, most primaries and multiples
have varying amplitudes with offset. In Fig. 1b, the same data with primaries only are shown. Fig. 1c shows the
result after demultiple using the least squares transform. For a comparison with the high resolution transform, we
will look at the difference between the gather after demultiple (Fig. 1c) and the modeled multiple free gather (Fig
1b). In Fig. 1d this difference (1c-1b) is shown for the least squares transform, and in Fig. 1g for the highresolution
transform (the displays show clipped data, note the gray-scale bars next to the plots). The highresolution
transform shows a better multiple elimination (there is less energy in the plot), but also some smearing
(or ringing) of energy in the time direction (e.g. between 0 and 0.8 second in Fig. 1g). The Radon spectra are
shown in Fig. 1e and 1h respectively. The amplitude preservation is illustrated in Fig. 1f for a constant amplitude
primary. The solid line is the ideal modeled AVO variation, the dashed line the amplitude after least squares
EAGE 64th Conference & Exhibition — Florence, Italy, 27 - 30 May 2002

Radon demultiple, and the dotted line the amplitude after high-resolution Radon demultiple. In Fig. 1i a similar
plot is shown for a primary with AVO variation. Both results indicate that better amplitude preservation is
obtained using high-resolution Radon demultiple.
2
High resolution Radon transform with constant weighting
The smearing of energy in the time direction as shown in Fig. 1g can effectively be suppressed by using a
frequency independent sparseness prior (assuming no dispersion). This sparseness prior is the average over a
frequency range of the sparseness priors derived using the frequency dependent transform (Sacchi and Ulrych,
1995). The frequency range should be chosen where the data have a good signal to noise ratio and are unaliased.
Preferably it is chosen at frequencies that show good resolution in the curvature parameters, i.e. not the lower
frequencies. In Figs. 1j to 1l the absolute Radon spectra are shown for the different methods using a logarithmic
scale to show the artifacts. From left to right the standard least squares transform (1j), the high resolution
transform (1k) and the high resolution transform with frequency independent prior (1l) are shown. The smearing
of energy in the time direction present in Fig. 1k is not visible in Fig. 1l. In practice, the frequency independent
prior can be derived in a cost efficient manner from a previous (adjacent) common midpoint gather.
Fourier/Radon regularization of time-lapse data
In Figure 2 an example is shown of the application of Fourier/Radon regularization to two synthetic data sets
(“base” and “monitor”) with the same subsurface model but different sampling (simulating a time-lapse
repeatability test). The acquisition parameters are: 8 streamers, with 75m separation and 240 channels, group
interval 12.5m and shot interval of 18.75m (flip-flop). The data are regularized using a two-step approach (see
Duijndam et al., 1999). First the data are regularized along streamers such that the inline midpoints are exactly at
cross-lines. Then the data is sorted to cross-lines, and the irregularly sampled cross-line midpoint and offset
coordinates are regularized using a combined Fourier/parabolic Radon transform (Fourier for the midpoint
coordinate). The sampling in the equivalent cross-line, for the base- and monitor survey is shown in Fig. 2b and
2c. The base and monitor data have been regularized, and the normalized RMS difference is shown in Fig. 2d for
the least squares transform, and in Fig. 2e for the high-resolution transform. The normalized RMS has been
computed in a time window of 1.5s-2.5s, which included a reflection from a layer which is dipping 13 degrees in
the cross-line direction. The three main conclusions that can be drawn from these results are that the highresolution
transform provides much improved reconstruction in the gap, reduced edge effects, and a comparable
(if not slightly better) repeatability at the well sampled parts. Note that good time-lapse repeatability does not
directly mean that amplitude preservation is good, since base and monitor data may have been affected in the
same way. However, it has been verified (but not shown here) that the amplitude preservation is also good in the
case of AVO variations and amplitude variations in the midpoint direction.
Discussion and conclusions
On these synthetic data sets, the high resolution transforms show significantly reduced edge effects and much
better reconstruction in large gaps than standard least squares transforms. The smearing in the time direction
observed with the high resolution transforms can be effectively suppressed by using the proposed method with a
frequency independent sparseness prior. The good amplitude preservation means that the high-resolution
transforms are well suited for demultiple before AVO analysis, and for regularization of time-lapse data. Finally,
it is remarked that the difference in performance between the high-resolution transforms and the standard least
squares transforms may be smaller for field data, due to noise and a higher number of curvatures present.
References
Duijndam, A., Koek, A., Ongkiehong, L., Romijn, R. and Schonewille, M., 1999, A general reconstruction scheme for
dominant azimuth 3-D seismic data, 69th Ann. Internat. Mtg: Soc. of Expl. Geophys., 1441-1444. Hampson, D., 1986,
Inverse velocity stacking for multiple elimination, J. Can. Soc. Expl. Geophys., 22(1):44-55.
Hargreaves, N. and Cooper, N., 2001, High-resolution radon demultiple, 71st Ann. Internat. Mtg: Soc. of Expl. Geophys.,
1325-1328.
Hunt, L., Cary, P. and Upham, W., 1996, The impact of an improved Radon transform on multiple attenuation, 66th Ann.
Internat. Mtg: Soc. of Expl. Geophys., 1535-1538.
Sacchi, M.D., and Ulrych, T.J., 1995, High-resolution velocity gathers and offset space reconstruction, Geophysics, 60, 4,
1169-1117.
Zwartjes, P. and Duijndam, A., 2000, Optimizing reconstruction for sparse spatial sampling, 70th Ann. Internat. Mtg: Soc. of
Expl. Geophys., 2162-2165.

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Figure 1: Comparing conventional and high resolution parabolic Radon demultiple: (a) input data gather;
(b) primaries only; (c) data after multiple elimination with conventional Radon demultiple; (d) difference
between modelled primaries and data after conventional Radon demultiple; (e) Radon domain for
conventional transform; (f) amplitude of first primary: solid line - modelled AVO variation; dashed line -
conventional Radon; dotted line - high resolution Radon; (g) similar plot as Fig. 1d, but now for the high
resolution transform; (h) Radon domain for the high resolution transform; (i) amplitude of second primary:
solid line - modelled AVO variation; dashed line - conventional Radon; dotted line - high resolution Radon;
(j) logarithmic plot of absolute Radon spectrum for the conventional transform; (k) logarithmic plot of
absolute Radon spectrum for the high resolution transform; (l) logarithmic plot of absolute Radon spectrum
for the high resolution transform with frequency independent prior.
EAGE 64th Conference & Exhibition — Florence, Italy, 27 - 30 May 2002

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Figure 2: Comparing conventional and high resolution Fourier/Radon regularization for a synthetic time-lapse
data set: (a) Subsurface model; (b) sampling in cross-line midpoint, offset for the base data (no feathering); (c)
sampling for monitor data set; (d) RMS difference between the regularized base and monitor data between 1.5s
and 2.5s, using the conventional least squares Fourier/parabolic Radon regularization; (e) RMS difference for the
high resolution Fourier/Radon regularization.
Acknowledgements
We would like to thank PGS geophysical for the permission to publish this paper.