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GEOPHYSICS, VOL. 74, NO. 2 �MARCH-APRIL 2009�; P. A7–A11, 6 FIGS.10.1190/1.3054136

ime-lapse image registration using the local similarity attribute

ergey Fomel1 and Long Jin1

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ABSTRACT

We present a method to register time-lapse seismic imagesbased on the local-similarity attribute. We define registrationas an automatic point-by-point alignment of time-lapse im-ages. Stretching and squeezing a monitor image and comput-ing its local similarity to the base image allows us to detect anoptimal registration even in the presence of significant veloc-ity changes in the overburden. A by-product of this process isan estimate of the ratio of the interval seismic velocities in thereservoir interval. We illustrate the proposed method anddemonstrate its effectiveness using both synthetic experi-ments and real data from a time-lapse experiment in the Durifield, Indonesia.

INTRODUCTION

Time-lapse seismic monitoring is an important technology for en-ancing hydrocarbon recovery �Lumley, 2001�. At the heart of theethod is comparing repeated seismic images in an attempt to iden-

ify changes that indicate fluid movements in the reservoir.In general, time-lapse image differences contain two distinct ef-

ects: �1� shifts of image positions in time caused by changes in seis-ic velocities and �2� amplitude differences caused by changes in

eismic reflectivity. The data-processing challenge is to isolatehanges in the reservoir itself from changes in the surrounding areas.ross-equalization is a popular technique for this task �Rickett andumley, 2001; Stucchi et al., 2005�. A number of different cross-qualization techniques have been applied successfully to estimatend remove time shifts between time-lapse images �Bertrand et al.,005; Aarre, 2006�. An analogous task exists in medical imaging,here it is known as the image-registration problem �Modersitzki,004�.

In this paper, we use the local-similarity attribute �Fomel, 2007a�or automatic quantitative estimation and extraction of variable time

Manuscript received by the Editor 23 July 2008; revised manuscript receiv1The University of Texas at Austin, John A. and Katherine G. Jackson Sc

inlong@ig.utexas.edu.2009 Society of Exploration Geophysicists.All rights reserved.

A7

hifts between time-lapse seismic images. A similar technique haseen applied to multicomponent image registration �Fomel et al.,005�. As a direct quantitative measure of image similarity, local at-ributes are suited perfectly for measuring nonstationary time-lapseorrelations. The extracted time shifts also provide a direct estimatef the seismic-velocity changes in the reservoir. We demonstrate anpplication of the proposed method with synthetic and real data ex-mples.

THEORY

The correlation coefficient between two data sequences at and bt isefined as

c �

�t

atbt

��t

at2�

t

bt2

�1�

nd ranges between 1 �perfect correlation� and �1 �perfect correla-ion of signals with different polarity�. The definition of the localimilarity attribute �Fomel, 2007a� starts with the observation thathe squared correlation coefficient can be represented as the productf two quantities c2 � pq, where

p �

�t

atbt

�t

bt2

�2�

s the solution of the least-squares minimization problem,

minp

�t

�at � pbt�2, �3�

nd where

q �

�t

atbt

�t

at2

�4�

ptember 2008; published online 4 February 2009.Geosciences, Austin, Texas, U.S.A. E-mail: sergey.fomel@beg.utexas.edu;

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A8 Fomel and Jin

s the solution of the least-squares minimization problem

minq

�t

�bt � qat�2. �5�

nalogously, the local similarity � t is a variable signal defined as theroduct of two variable signals pt and qt that are the solutions of theegularized least-squares problems

minpt

��t�at � ptbt�2 � R�pt�� �6�

nd

minqt

��t�bt � qtat�2 � R�qt�� , �7�

here R is a regularization operator designed to enforce a desired be-avior such as smoothness.

Shaping regularization �Fomel, 2007b� provides a particularlyonvenient method of enforcing smoothness in iterative optimiza-ion schemes. If shaping regularization is applied iteratively withaussian smoothing as a shaping operator, its first iteration is equiv-

lent to the fast local crosscorrelation method of Hale �2006�. Fur-her iterations introduce relative amplitude normalization and com-ensate for amplitude effects on the local-image similarity. Choos-ng the amount of regularization �smoothness of the shaping opera-or� affects the results. In practice, we start with strong smoothingnd decrease it when the results stop changing and before they be-ome unstable.

The application of local similarity to the time-lapse image-regis-ration problem consists of squeezing and stretching the monitor im-ge with respect to the base image while computing the local similar-ty attribute. Next, we pick the strongest similarity trend from the at-ribute panel and apply the corresponding shift to the image.

In addition to its use for image registration, the estimated localime shift is a useful attribute by itself. Time-shift analysis has beenpplied widely to infer reservoir compaction �Hatchell and Bourne,005; Tura et al., 2005; Janssen et al., 2006; Rickett et al., 2007�. Be-ause the time shift has a cumulative effect, it is helpful to computehe derivative of the time shift, which can relate the time-shifthange to the corresponding reservoir layer. Rickett et al. �2007� de-ne the derivative of time shift as time strain and find it to be an intu-

tive attribute for studying reservoir compaction.What is the exact physical meaning of the warping function w�t�

hat matches the monitor image I1�t� with the base image I0�t� by ap-lying the transformation I1�w�t��? One can define the base travel-ime as an integral in depth:

t � 2�0

H0

dz

v0�z�, �8�

here v0�z� is the base velocity and H0 is the base depth. A similarvent in the monitor image appears at time

w�t� � 2�0

H1

dz

v1�z�� �

0

t��t

v̂0�� �v̂1�� �

d� , �9�

here H1 is the monitor depth, v̂0�t� and v̂1�t� are seismic velocitiess functions of time rather than depth, and �t is the part of the timehift caused by the reflector movement

�t � 2�H0

H1

dz

v1�z�. �10�

In a situation where the change of �t with t can be neglected, aimple differentiation of the function w�t� detected by local similari-y analysis provides an estimate of the local ratio of the velocities:

dw

dt�

v̂0�t�v̂1�t�

. �11�

f the registration is correct, the estimated velocity ratio outside ofhe reservoir should be close to one. The local velocity ratio can beonnected to other physical attributes that are related to changes inaturation, pore pressure, or compaction.

EXAMPLES

D synthetic data

Figure 1a shows a simple five-layer velocity model, where wentroduce a velocity increase in one of the layers to simulate a time-apse effect.After generating synthetic image traces, we can observeFigure 2a� that the time-lapse difference contains changes not onlyt the reservoir itself but also at interfaces below the reservoir. Addi-ionally, the image amplitude and the wavelet shape at the reservoirottom are incorrect. These artifact differences are caused by timehifts resulting from the velocity change.

After detecting the warping function w�t� from the local-similari-y scan �Figure 3� and applying it to the time-lapse image, the differ-nce correctly identifies changes in reflectivity only at the top andhe bottom of the producing reservoir �Figure 2b�. To implement theocal similarity scan, we use the relative stretch measure s�t�

w�t�/t. When the two images are aligned perfectly, s�t� � 1. De-iations of s�t� from one indicate possible misalignment. Finally, wepply equation 11 to estimate interval-velocity changes in the reser-oir and observe a reasonably good match with the exact syntheticodel �Figure 1b�.

D synthetic data

Figure 4a shows a more complicated 2D synthetic example. Inhis experiment, we assume that the changes occur in the reservoir asell as in the shallow subsurface �Figure 4b�. The synthetic dataere generated by convolution modeling. After computing local

imilarity between the two synthetic time-lapse images �Figure 5�,e apply the extracted stretch factor to register the images. Figure 4c

nd d compares time-lapse difference images before and after

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Time-lapse image registration A9

egistration. Similarity-based registration effectively removes arti-act differences above and below the synthetic reservoir. As men-ioned, the local similarity cube is an important attribute by itself andncludes information on uncertainty bounds for the local stretch fac-or, which reflects the uncertainty of the reservoir parameter estima-ion.

D field data

Finally, Figure 6 shows an application of the proposed method toime-lapse images from steam-flood monitoring in the Duri field, re-roduced from Lumley �1995a, 1995b�. Before registration, real dif-erences in the monitor surveys after 2 months and 19 months arebscured by coherent artifacts, caused by velocity changes in thehallow overburden and in the reservoir interval �Figure 6a�. Simi-arly to the results of the synthetic experiments, local similarity reg-stration successfully removes artifact differences above and belowhe reservoir level �Figure 6b�. After separating the time-shift effect

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igure 1. �a� A 1D synthetic-velocity model before �solid line� andfter �dashed line� reservoir production. �b� True �solid line� and es-imated �dashed line� interval-velocity ratio.

rom amplitude changes, one can image the steam-front propagationore accurately using time-lapse seismic data. We expect our meth-

d to work even better on higher-quality marine data.

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igure 2. The 1D synthetic seismic images and the time-lapse differ-nce �a� initially and �b� after image registration.

igure 3. �a� Local similarity scan for detecting the warping functionn the 1D synthetic model. Red indicates strong similarity. The blackurve shows an automatically detected trend.

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igure 4. �a� Synthetic model. �b� Time-lapse change containing differences in the reservoir interval and the shallow overburden. �c� Initial time-

apse difference image. �d� Time-lapse difference image after registration.

igure 5. Local similarity scan for the 2D synthetic model.

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CONCLUSIONS

We propose a method of time-lapse image registration based on anpplication of the local similarity attribute. The attribute provides amooth, continuous measure of similarity between two images. Per-urbing the monitor image by stretching and squeezing it in timehile picking its best match to the base image enables an effective

egistration algorithm. The by-product of this process is an estimatef the time-lapse seismic velocity ratios in the reservoir interval.

Using synthetic and real data examples, we have demonstrated thebility of our method to achieve an accurate time-domain image reg-stration and to remove artifact time-lapse differences caused by ve-

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igure 6. Application to Duri field data. Base image and time-lapseifferences: �a� before registration, reproduced from Lumley1995a�; �b� after registration.

ocity changes. Unlike some alternative cross-equalization methods,

ur method is not influenced by amplitude differences and can ac-ount for velocity changes in the shallow overburden.

ACKNOWLEDGMENTS

We thank David Lumley for permission to use results from hish.D. thesis. We also thank Aaron Janssen, Brackin Smith, and Aliura for inspiring discussions and two anonymous reviewers forelpful suggestions. This publication is authorized by the director,ureau of Economic Geology, The University of Texas atAustin.

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