CWP-786January 2014

Automatic multicomponent image registration

Advisor: Prof. Dave Hale Committee Members: Andreas Rüger, PhD Prof. Terence Young

- Master of Science Thesis - Geophysics

Center for Wave PhenomenaColorado School of MinesGolden, Colorado 80401

303.384.2178 http://cwp.mines.edu

Stefan Compton

Defended on January 10, 2014

AUTOMATIC MULTICOMPONENT IMAGE REGISTRATION

by

Stefan Compton

ABSTRACT

Multicomponent seismic images are composed of different combinations of downgoing

and upgoing wavefields. Each wave mode has different propagation velocity and polariza-

tion direction and thus carries unique, direction-dependent information about the subsurface.

Differences in propagation velocity cause events in converted- or shear-wave images to appear

at later times than the compressional-wave image counterpart. Reflectivities are different

for each wave mode and therefore, multicomponent images are not related simply by time

shifts. These complications historically required that the alignment, also called registration

of corresponding image features be done manually, a tedious process. This thesis devel-

ops an approach to automatically register multicomponent images using a smooth dynamic

warping algorithm that can be accurate with respect to problems unrelated to time shifts.

Interval VP/VS ratios can be estimated from derivatives of time shifts that align reflections

in multicomponent images, and these VP/VS ratios may be used to assess the accuracy of the

automatic registration process. To improve the accuracy of estimated time shifts and VP/VS

ratios, we automatically construct a coarse lattice of points located on reflections with high

amplitudes, and then estimate time shifts at only those points. By adjusting the coarseness

of the lattice, we trade off resolution of changes in VP/VS with increased accuracy in VP/VS

estimates. The result is an efficient, robust, and automatic method for multicomponent

image registration.

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TABLE OF CONTENTS

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

CHAPTER 2 SMOOTH DYNAMIC WARPING . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Dynamic time warping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Smooth dynamic time warping . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Smooth dynamic image warping . . . . . . . . . . . . . . . . . . . . . . . . . . 17

CHAPTER 3 MULTICOMPONENT IMAGE REGISTRATION . . . . . . . . . . . . 20

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Traveltime relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Geophysical constraints on time shifts . . . . . . . . . . . . . . . . . . . . . . . 22

3.4 SDIW versus DIW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.5 Coarse sample locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.6 Interpolation of time shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

CHAPTER 4 SIMULTANEOUS MULTICOMPONENT REGISTRATION . . . . . . . 37

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2 Traveltime relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.3 Simultaneous registration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

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4.4 Reducing computational cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.5 Independent versus simultaneous registration . . . . . . . . . . . . . . . . . . . 46

CHAPTER 5 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

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LIST OF FIGURES

Figure 2.1 Synthetic PP trace in PP time and synthetic PS trace in PS time. . . . . . . 9

Figure 2.2 Alignment errors computed from synthetic traces in Figure 2.1. Thecorrect time-shift function is shown as the dashed white curves, and thecorrect VP/VS ratios are shown as the dashed red curves. The solid whitecurve is the time-shift function computed using Algorithm 1, with thecorresponding VP/VS ratios shown in black. . . . . . . . . . . . . . . . . . 10

Figure 2.3 Smoothed version of the computed time shifts in Figure 2.2, shown asthe solid white curve in the top panel, and corresponding VP/VS ratiosshown in black in the bottom panel. . . . . . . . . . . . . . . . . . . . . . 11

Figure 2.4 Zoomed views of alignment errors computed from synthetic traces inFigure 2.1 and time shifts computed using Algorithm 1 (a) andAlgorithm 2 (b) with a coarse sampling interval h = 10∆t. Thecomputational stencils for each algorithm are shown in red. Correctvalues are the dashed curves and computed values are the solid curves. . . 15

Figure 2.5 Time shifts computed from smooth dynamic warping (Algorithm 2), fordifferent coarse sampling intervals h. White dots show the coarse samplelocations where time shifts are estimated. For h = ∆t and h = 10∆t dotsare not shown because they would hide the time-shift curves completely.Dashed curves represent correct values and solid curves representcomputed values. The solid curves are obtained by piecewise-linearinterpolation of computed values. As the coarse sample interval increasesto h = 200∆t, time shifts and VP/VS ratios are more accuratelyestimated. When the coarse sample interval is too large (h = 400∆t)estimates are less accurate. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Figure 3.1 Time-migrated PP and PS images before image registration. Note thatreflectors are approximately aligned, but the PP and PS images areplotted on two different time scales. . . . . . . . . . . . . . . . . . . . . . . 21

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Figure 3.2 Alignment errors and time shifts computed from a single pair of tracesfrom the PP and PS images shown in Figure 3.1. Only the alignmenterrors for times 0 to 1 second are shown. Overlaid curves are the 1Dtime shifts extracted from the 3D time shifts u(x, y, tpp) at thecorresponding trace location. The solid white curve shows rough timeshifts computed from DIW and the dashed blue curve shows these shiftsafter smoothing. The dashed yellow curve shows the time shiftscomputed from SDIW with a minimum subsampling interval h = 100 ms. . 25

Figure 3.3 Slices through three 3D volumes with the same PP time window shownin the alignment errors of Figure 3.2. The location for those alignmenterrors corresponds to the vertical black lines in the inline and crosslinesections. The strong events at 0.7 s are well aligned in each warpedimage, but the time slice through the event at 0.38 s shows that timeshifts computed using SDIW are more accurate. The differences in timeshifts can be seen clearly in Figure 3.2. . . . . . . . . . . . . . . . . . . . 26

Figure 3.4 Interval VP/VS ratios estimated from smoothed DIW time shifts andfrom SDIW time shifts. The VP/VS ratios from smoothed DIW timeshifts exhibit fine detail, but those smoothed shifts are less accurate, asshown in Figure 3.2 and Figure 3.3, than shifts estimated using SDIW. . . 27

Figure 3.5 Alignment errors (a) computed from a single trace from the PP and PSimages shown in Figure 3.1, with time shifts (b) computed from 1Dsmooth dynamic time warping using two different subsampling grids. Anearly-uniform coarse grid was used for the blue curve, with subsamplinglocations represented by blue dots, and a reflector-aligned grid was usedfor the yellow curve with subsampling locations represented by yellowsquares. The PP trace (c) overlaid with the envelope of the trace showsthat strong PP events correlate with strong vertical features in thealignment errors above. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Figure 3.6 The PP image (a) and time shifts (b) used to flatten the image so that aconstant time corresponds to a single seismic horizon. The coarse latticeof points on the flattened image (c) is shown by the white dots. Thislattice can be mapped back to the original (unflattened) coordinatespace to obtain a coarse lattice of points that is aligned with dippingreflectors (d). We use this lattice in SDIW to compute time shifts atlocations with strong reflectors in the PP image. . . . . . . . . . . . . . . . 32

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Figure 3.7 White dots indicate coarse lattice points used to compute time shiftswith SDIW. Both nearly-uniform and reflector-aligned grids haveminimum intervals of 500 m laterally. The nearly-uniform points are aminimum of 160 ms apart in time, while reflector-aligned points are aminimum of 100 ms apart but are aligned with strong reflectors in theimage. Points in the nearly-uniform grid cross strong reflectorshorizontally, resulting in a noticeable stair step effect in thecorresponding VP/VS ratios. . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Figure 3.8 Interval VP/VS estimates resulting from two different interpolationmethods for time shifts, piecewise cubic with monotonic splines andpiecewise linear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Figure 3.9 Interpolated time shifts and interval VP/VS extracted from 3D SDIWshift and VP/VS volumes. The three different curves represent threedifferent methods of interpolating time shifts: linear (black), cubicmonotonic spline (red), and cubic spline (blue). . . . . . . . . . . . . . . . 36

Figure 4.1 3D alignment errors for simultaneous warping of synthetic traces shownin Figure 4.3. The computed solution is shown as the white curve. . . . . . 44

Figure 4.2 Alignment errors for synthetic PP and PS1 traces with no initialsqueezing of the PS trace (a). Yellow triangles indicate large areas wherecomputation time is wasted. Alignment errors for the same traces, butafter squeezing the PS1 trace by a constant factor c = 1.7 (b). Thedashed white curves show the known time-shift function. . . . . . . . . . . 45

Figure 4.3 Synthetic PP, PS1, and PS2 traces with a signal-to-noise (S/N) ratio of1:1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Figure 4.4 Time shifts estimated by individually warping three pairs of synthetictraces shown in Figure 4.3, PP-PS1 (red), PP-PS2 (blue), PS1-PS2(green). Correct time shifts are shown as dashed white curves. Resultsfor noise-free synthetic traces are shown on the left and results forsynthetic traces with S/N = 1:1 are shown on the right. The bottompanels show errors in γ(s) estimated from equations 4.15 (magenta), 4.16(yellow), and 4.17 (cyan). . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

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Figure 4.5 Time shifts estimated by simultaneously warping the three synthetictraces shown in Figure 4.3. PP-PS1 (red) and PS1-PS2 (green) aredirectly computed from the simultaneous warping, PP-PS2 (blue) iscomputed from equation 4.7. Correct time shifts are shown as dashedwhite curves. Results for noise-free synthetic traces are shown on the leftand results for synthetic traces with S/N = 1:1 are shown on the right.The bottom panels show errors in γ(s) estimated from equations 4.15(magenta), 4.16 (yellow), and 4.17 (cyan). All of which are identical. . . . 50

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ACKNOWLEDGMENTS

I would like to thank my advisor, Dr. Dave Hale, for his guidance and support throughout

my graduate studies. His valuable insights and enthusiasm were important factors in this

research. Thanks to Jim Gaiser and Lee Bell for their helpful discussions and thanks to

Geokinetics Inc. and Geophysical Pursuit, Inc. for providing the seismic data that was

invaluable to this research. I would also like to acknowledge my colleagues in CWP for

making my time here enjoyable.

Special thanks to my family, especially my wife Mara for her patience and support

throughout this process.

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CHAPTER 1

INTRODUCTION

Multicomponent seismic technology is a valuable tool in exploration geophysics. Histor-

ically, much seismic analysis focused on compressional-wave (P-wave) seismic data which

provide only partial information about the subsurface. Shear-wave (S-wave) data provide

additional information about the same subsurface, and each wave mode contains unique in-

formation about elastic constants, grain cementation, pore geometry, anisotropy axes, and

lateral variations in rock and fluid types (Hardage et al., 2011).

Data acquisition options for multicomponent seismic surveys correspond to the types

of wave modes recorded. Common acquisition options are referred to as 3C, 4C, 6C, or

9C, depending on the orientations of displacement generated by seismic sources and the

orientations of displacement recorded by receivers. All possible wave modes are contained

in 9C data, collected when a source generates three orthogonal displacements recorded by

three orthogonal sensors.

In the simplest (3C) multicomponent survey the downgoing wavefield is generated by a P-

wave source and the upgoing wavefield is recorded by a receiver containing three orthogonally

oriented sensors. For an isotropic earth, 3C seismic surveys would record predominantly

PP and PSV modes. The acronyms PP and PSV denote a downgoing P-wave, and an

upgoing P- or SV-wave. The downgoing P-wave is reflected at a subsurface interface and

travels upward as both a P-wave and mode converted SV-wave. The P- and SV-wave have

orthogonal polarizations within the vertical plane containing the source and receiver. The

SV mode is distinguished from the shear-wave SH mode, with polarization orthogonal to

the vertical plane containing the source and receiver. The SH modes have significantly less

energy than SV modes in 3C surveys. Therefore, the more complete mode description, PSV,

is often abbreviated as simply PS. Because converted-wave surveys do not involve shear-wave

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sources, converted-wave seismic data can also be acquired in the marine environment with

the use of 4C (three orthogonal sensors plus a hydrophone) sensors on the seafloor.

The goal of converted-wave seismic data processing is to preserve and exploit the infor-

mation provided by both PP and PS reflections. Joint PP and PS analysis has applications

in structural imaging, lithology estimation, anisotropy analysis, fluid description, and reser-

voir monitoring (Stewart et al., 2003). A typical first step is to generate PP and PS seismic

images. Because traveltimes differ for seismic reflectors in PP images and corresponding

reflectors in PS images, a crucial first step in this analysis is the alignment, or registration,

of these reflectors.

The vertical shifts that align PP and PS images can be used to quantify the ratio (VP/VS)

of P- and S-wave velocities, an important subsurface property (Stewart et al., 2003). How-

ever, PP and PS image registration is complicated because the PS image is not simply a

time-shifted version of the PP image. PP and PS reflectivities differ, leading to amplitude

and phase differences between the two images that can be significant. In fact, it is possible

for a reflector in one image to not be apparent in the other image. For these reasons, image

registration is often done by manually selecting corresponding events in each image. Never-

theless, many techniques have been developed to automate the registration process and to

estimate VP/VS ratios (Gaiser, 1996; Fomel and Backus, 2003; Nickel and Sonneland, 2004;

Yuan et al., 2008; Liang and Hale, 2012).

Liang and Hale (2012) used dynamic image warping (Hale, 2013) for this purpose. Dy-

namic image warping computes vertical time shifts that align events in PP and PS images;

these shifts are a globally optimal solution of a minimization problem with many possible

local minima. The dynamic warping algorithm enforces strict bounds on both time shifts

and derivatives of those time shifts. For these reasons, the dynamic warping algorithm may

be preferable when compared to other methods.

For instance, Fomel and Backus (2003) describe a method that solves a similar minimiza-

tion problem, and uses regularization terms that enhance, but do not strictly constrain, the

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smoothness of estimated time shifts. This method also requires a good initial guess of the

warping function, to avoid convergence to local minima. Constraints in dynamic warping

are strict, hard bounds, and the solution is globally optimal, even though no starting model

is required.

Yuan et al. (2008) use a global search method, simulated annealing, which solves problems

with local maxima and allows for a poor initial model. However, the maximized cost function

is the normalized crosscorrelation of PP and warped PS images. Hale (2013) shows that the

success of local crosscorrelation methods depends on how rapidly shifts vary within windows

used to compute correlation coefficients. If shifts vary rapidly (as for PP and PS images,

especially near the surface) then a suitable correlation window may not exist. Dynamic

warping does not require windows and is accurate even when shifts vary rapidly.

A common goal of the previously cited methods is to obtain, in conjunction with regis-

tered PP and PS images, a high-resolution VP/VS volume. The term high-resolution implies

fine detail, but what level of detail can be extracted from traveltimes between PP and PS

images, in which differences in noise and reflection waveforms are significant? To obtain

high-resolution information, more expensive and thorough techniques are required, such as

joint PP-PS inversion of seismic amplitudes. For PP and PS image registration, time shifts

are related to time-variant averages of VP/VS ratios, and thus, are often smoothly varying.

The dynamic warping algorithm can be modified to exploit this inherent smoothness and,

by sacrificing temporal resolution of time shifts, more accurately estimate the change in time

shift.

Chapter 2 develops a smooth dynamic warping algorithm, that is quite general and has

applications beyond multicomponent image registration. Chapter 3 discusses the practical

application of smooth dynamic warping to registration of multicomponent images. Smooth

dynamic warping is shown to be a robust and accurate method for automatic image regis-

tration through an application to 3D PP and PS time-migrated images. Chapter 4 extends

dynamic warping beyond the alignment of two images, for in the presence of anisotropy, the

3

PS mode can split into PS1 (fast-S) and PS2 (slow-S) modes. With three related seismograms

(PP, PS1, and PS2), the algorithm and application developed in Chapter 2 and Chapter 3,

respectively, could be applied independently to each pair of seismograms. However, time

shifts estimated with that approach would likely be inconsistent. Therefore, in Chapter 4,

the dynamic warping optimization problem is reformulated to simultaneously align PP, PS1,

and PS2 seismograms with consistent time shifts.

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CHAPTER 2

SMOOTH DYNAMIC WARPING

2.1 Introduction

Dynamic image warping is a robust method for computing shifts that align features in

one image with corresponding features in another image. The computed shifts are a globally

optimal solution to a non-linear error minimization problem that satisfies linear inequality

constraints. For many problems, these linear inequality constraints can be directly related

to physical parameters. Dynamic warping is a dynamic programming algorithm that solves

a collection of nested subproblems to find the globally optimal solution. In the context of

aligning features in seismic images, the error function may have many local minima, and

thus the ability to find the globally optimal solution is a key advantage of this method.

A dynamic programming algorithm for spoken word recognition was described by Sakoe

and Chiba (1978). They were the first to introduce strict constraints on the rate of change

of time shifts. Their algorithm has become known as dynamic time warping (DTW) (e.g.,

Muller (2007) chapter 4). Anderson and Gaby (1983) propose applications of DTW to

estimate time shifts in geophysical time series, and refer to this process as “dynamic waveform

matching”. The use of DTW in geophysics is not new and has many potential applications.

Time shifts that align seismic traces can increase or decrease rapidly with time, but the

time derivative of shifts often varies slowly. We therefore introduce a modified version of the

dynamic warping algorithm, called smooth dynamic warping (Hale and Compton, 2013), to

exploit this inherent smoothness and estimate shifts at coarsely sampled times. The resulting

decrease in temporal resolution yields more accurate estimates of the change in shift between

the coarsely sampled times. This resolution and accuracy tradeoff is especially significant

when differences between traces are not related to time shifts alone, but include differences

in noise and reflection waveforms.

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The application of DTW to 1D signals is useful for certain geophysical problems, but we

must often estimate shifts between multidimensional images. DTW can produce highly in-

accurate results when applied independently to pairs of corresponding traces in such images.

Depending on the geometry of the geophysical problem, it is often reasonable to assume that

neighboring traces should have similar time shifts, but there are no constraints for the rate

at which shifts vary laterally if DTW is applied independently to pairs of traces. The exact

extension of DTW to higher dimensions, with strict lateral constraints on shifts, has been

shown to be computationally infeasible (Keysers and Unger, 2003). The time to compute an

exact solution grows exponentially with image size. Therefore, an approximate solution to

this multidimensional problem was proposed by Mottl et al. (2002) and extended by Hale

(2013). These solutions are referred to as dynamic image warping (DIW) algorithms. The

smooth dynamic time warping algorithm can be extended to multidimensional images in the

same fashion.

2.2 Dynamic time warping

For two sampled sequences f and g, dynamic time warping will find a sequence of time

shifts u[0 : ni − 1] ≡ {u[0], u[1], ..., u[ni − 1]} so that

f [i] ≈ g[i+ u[i]], i = 0, 1, ..., ni − 1. (2.1)

The time shifts u[i] are the solution to the optimization problem:

u[0 : ni − 1] ≡ arg minl[0:ni−1]

ni−1∑i=0

e[i, l[i]], (2.2)

subject to constraints

ul ≤ u[i] ≤ uu, rl ≤ u[i]− u[i− 1] ≤ ru, (2.3)

where

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e[i, l] ≡ (f [i]− g[i+ l])2. (2.4)

From the inequality constraints 2.3, time shifts u[i] are bounded by specified ul and uu;

and the rate of change of those time shifts (time strains) are bounded by specified rl and

ru. For some problems these time strains can be directly related to geophysical parameters.

This is indeed the case for PP-PS registration, as discussed in Chapter 3.

For the 1D sequences f and g we first compute alignment errors e[i, l] for all sample

indices i and lag indices l (equation 2.4). Alternative measures of alignment could be used,

but here we use the squared difference between amplitudes in f and g. The lag l satisfies the

constraint ul ≤ l ≤ uu. In practice, we use the bounds 0 ≤ l ≤ nl− 1, where nl = uu−ul + 1

is the number of lags for which alignment errors are computed. This change simplifies array

indexing, but the lower bound ul must now be added to l in equation 2.4.

The program to compute the time shifts is simple to implement, and the pseudocode for

this program is shown below in Algorithm 1. The inputs to the procedure FindShifts are

the bounds on strain rl and ru, a 2D array of alignment errors e[i, l], and an array u in which

to store the computed shifts.

In lines 2-3 a 2D array of accumulated errors d[i, l] is initialized. At sample index i = 0,

the accumulated errors are equal to the alignment errors.

Lines 4-15 recursively compute accumulated errors at sample index i from accumulated

errors at i − 1. Lines 7 and 8 compute lower and upper slopes that satisfy strain con-

straints (the notation drle denotes the smallest integer not less than rl, and bruc denotes

the largest integer not greater than ru). The accumulation is nonlinear because all possible

slopes from ql to qu are queried to find the error minimizing move, ml (lines 9-13). The

error minimizing moves are stored in the 2D array m[i, l], to simplify the final step of the

procedure. It is important to note that the slope q between samples i and i− 1 is an integer

and therefore coarsely quantized, so that the computed time shifts u will be rough. This

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shortcoming will be revisited later.

Algorithm 1 Find shifts u[i]

1: procedure FindShifts(rl, ru, e, u)

2: for l = 0 to nl − 1 . initialize

3: d[0, l] = e[0, l]

4: for i = 1 to ni − 1 . accumulate

5: for l = 0 to nl − 1

6: dl =∞7: ql = max(drle, l − nl + 1)

8: qu = min(bruc, l)9: for q = ql to qu10: dq = d[i− 1, l − q] + e[i, l]

11: if dq < dl12: dl = dq13: ml = q

14: d[i, l] = dl15: m[i, l] = ml

16: i = ni − 1 . minimize d

17: di =∞18: for l = 0 to nl − 1

19: if d[i, l] < di20: di = d[i, l]

21: u[i] = l

22: while i > 0 . backtrack

23: u[i− 1] = u[i]−m[i, u[i]]

24: i = i− 1

When the accumulation step is complete, lines 16-21 find the lag l that minimizes d[ni−

1, l]. This lag is the optimal shift u[ni − 1], and the accumulated error d[ni − 1, u[ni − 1]] is

the minimized sum in equation 2.2.

Finally, lines 22-24 compute the optimal shifts for all indices i. After each optimal shift

at index i is found and stored in u[i], we can use the error minimizing moves stored in array

m[i, l] to compute the optimal shift u[i− 1] at index i− 1. This last step simply follows the

optimal path backwards from index i = ni − 1 to index i = 0.

As an example, I apply this algorithm to a synthetic PP trace and a synthetic PS trace

shown in Figure 2.1. These traces are plotted on two different time scales as seen in the

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Figure 2.1: Synthetic PP trace in PP time and synthetic PS trace in PS time.

time axis labels of Figure 2.1. I denote the PP trace as f and the PS trace as g. The

trace g was generated by convolving a random reflectivity series with a Ricker wavelet. The

random reflectivity series used to construct g was warped by a time varying shift function

and then convolved with a Ricker wavelet to form the trace f . All events in g appear at

times not less than the corresponding times in f , because S-wave velocities cannot exceed

P-wave velocities. Finally, different bandlimited random noise sequences were added to f

and g.

The top panels of Figure 2.2 show alignment errors computed from the synthetic traces

in Figure 2.1. The dashed white curve in each top panel is the correct time-shift function.

The dashed red curve in each bottom panel is the corresponding correct VP/VS function:

VPVS

(tpp) = 1 + 2du(tpp)

dtpp. (2.5)

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This relationship will be derived in Chapter 3, but for this first synthetic example it is

important only to observe that the VP/VS ratios are related to the derivative of the time

shifts u. This relationship is exhibited in Figure 2.2 by the decrease in VP/VS values as the

slope of the time-shift function decreases. Time shifts computed using Algorithm 1 are shown

as the solid white curve. The VP/VS ratios computed from those time shifts (equation 2.5)

are shown in black.

Figure 2.2: Alignment errors computed from synthetic traces in Figure 2.1. The correcttime-shift function is shown as the dashed white curves, and the correct VP/VS ratios areshown as the dashed red curves. The solid white curve is the time-shift function computedusing Algorithm 1, with the corresponding VP/VS ratios shown in black.

Recall that Algorithm 1 returns a sequence of integer shifts subject to inequality con-

straints 2.3. The phrase “integer shifts” may be confusing here since Figure 2.2 shows time

shifts with units of seconds that are clearly not integer values. However, remember that the

2D array of alignment errors, e[i, l], is a sampled function of PP time and time shift, with

ni PP-time samples and nl time-shift samples. The uniform time sampling interval in this

example is ∆t = 2 ms, for both PP time and time shift, but Algorithm 1 operates with only

integer sample indices. Thus the phrase “integer shifts” implies that computed time shifts

can be only integer multiples of the time sampling interval ∆t. In this example, the bounds

in the constraints 2.3 are the integers ul = 0, uu = 1331, rl = 0 and ru = 1. Therefore,

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for each sample index i, the computed time shift can only increase by one or stay the same.

In other words, there are only two possible slopes, 0 and 1. From equation 2.5, this coarse

sampling of slope is the reason why estimated VP/VS values in Figure 2.2 oscillate wildly

between only two values, 1 and 3.

Figure 2.3: Smoothed version of the computed time shifts in Figure 2.2, shown as the solidwhite curve in the top panel, and corresponding VP/VS ratios shown in black in the bottompanel.

Of course, in a practical application the estimated VP/VS ratios shown in Figure 2.2 are

worthless. Algorithm 1’s estimation of time shifts at every PP-time sample limits the possible

values of slope and, hence, VP/VS ratios. To obtain more values, the integer shifts can be

smoothed, better approximating fractional changes in shifts. Figure 2.3 shows the result of

smoothing the solid white time shifts from Figure 2.2. The smoothed estimated time shifts

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better approximate the correct shifts at PP times greater than 1.2 seconds, so that the black

curve in the bottom panel of Figure 2.3 more accurately approximates the correct VP/VS

ratios. However, the shifts estimated at earlier times remain significantly different from the

correct shifts. This example demonstrates that we cannot obtain the correct shifts by simply

smoothing incorrect shifts.

Another way to obtain more slope values, while still satisfying the slope constraints

0 ≤ u[i]−u[i−1] ≤ 1, would be to sample the lag axis more finely. That is, sample alignment

errors at some fraction of the time sampling interval to obtain more possible changes in slope

between each time sample. Hale (2013) makes this observation, but notes that this approach

can greatly increase computation time and memory, especially when estimating shifts for

multidimensional image warping.

In fact, there is another reason why finely sampling the lag axis is not a good solution.

This approach would imply that between two consecutive time samples we could accurately

estimate a change in time shift equal to a fraction of the sampling interval. When sequences

f and g are not simply shifted versions of one another, it is unrealistic to think such minute

changes in shift can be detected. The algorithm will find the globally optimal solution, but

are the estimated changes in shift meaningful at that fine scale? For that matter, is it even

reasonable to estimate time shifts between each consecutive time sample? For typical seismic

data, the time sampling interval can be several times smaller than the dominant period of

the data. This observation leads us to a modified version of the dynamic warping algorithm.

2.3 Smooth dynamic time warping

The smooth dynamic warping algorithm is a simple modification of Algorithm 1. Instead

of estimating shifts at every time sample, the smooth warping algorithm estimates shifts at

only coarsely sampled locations. This approach makes smooth dynamic warping more robust

in the presence of noise and other differences between the sampled sequences. The coarse

sampling sacrifices temporal resolution of time shifts to more accurately estimate changes in

those time shifts. Such resolution and accuracy tradeoffs are common in signal processing.

12

Algorithm 2 shows how Algorithm 1 is modified to implement smooth dynamic warping.

The algorithms are similar, but there are several key differences. The first difference is that

Algorithm 2 computes shifts ui[j] ≡ u[i[j]] for only a subset of sample indices i[0 : nj − 1] ≡

i[0], i[1], ..., i[nj − 1]. Therefore, note that an array of nj subsampled indices i[0 : nj − 1] are

input to the procedure. Shifts u[i] for all sample indices i can be later interpolated from the

subsampled shifts ui[j].

Algorithm 2 Find subsampled shifts ui[j] ≡ u[i[j]]

1: procedure FindShiftsI(rl, ru, e, i, ui)

2: for l = 0 to nl − 1 . initialize

3: d[0, l] = e[0, l]

4: for j = 1 to nj − 1 . accumulate

5: h = i[j]− i[j − 1]

6: for l = 0 to nl − 1

7: dl =∞8: ql = max(dhrle, l − nl + 1)

9: qu = min(bhruc, l)10: for q = ql to qu11: dq = d[j − 1, l − q]12: for p = 0 to h− 1

13: dq = dq + e[i[j]− p, l − pq/h]

14: if dq < dl15: dl = dq16: ml = q

17: d[j, l] = dl18: m[j, l] = ml

19: j = nj − 1 . minimize d

20: dj =∞21: for l = 0 to nl − 1

22: if d[j, l] < dj23: dj = d[j, l]

24: ui[j] = l

25: while j > 0 . backtrack

26: ui[j − 1] = ui[j]−m[j, ui[j]]

27: j = j − 1

The next significant difference is in the computation of slope bounds on lines 8 and 9.

The integers ql and qu depend on h, the difference between subsampled indices i. Thus, this

13

algorithm satisfies the same slope constraints as Algorithm 1, but with more slopes possible

between the bounds ql and qu. Lines 11-13 replace line 10 in Algorithm 1. In Algorithm 2 we

find the q with minimum accumulated error, where q defines a linear trajectory from index

i[j − 1] to i[j]. Finding the minimum accumulated error requires summing the alignment

errors along each linear trajectory (lines 12 and 13) for each slope q between ql and qu.

Note that the term l − pq/h in line 13 may not have integer value, and thus interpolation

of alignment errors e[i, l] at non-integer lags is required. Finally, note that if h = 1, then

Algorithm 2 is equivalent to Algorithm 1.

Figure 2.4a shows a zoomed view of the correct and computed time shifts in Figure 2.2.

Again, the dashed white curve is the correct time-shift function and the solid white curve is

computed with Algorithm 1. The computational stencil is shown in red, and as previously

discussed, has only two possible values of slope, 0 and 1. Figure 2.4b shows the smooth-

dynamic-warping stencil for a coarse interval h = 10∆t, where ∆t is the time sampling

interval (2 ms). Given this stencil, smooth dynamic warping estimates time shifts at only

coarsely sampled locations approximately 10∆t apart. These locations are computed to be

at least h samples apart, but some intervals may be larger to ensure that the first and last

samples are included. The solid white curve was obtained by piecewise-linear interpolation

of time shifts ui[j].

The smooth dynamic warping stencil shown in Figure 2.4b satisfies the same constraints

as the stencil in Figure 2.4a, but has 11 possible values of slope. Therefore, smooth dynamic

warping can more accurately estimate the change in time shift between coarse samples. The

improvement is apparent in the bottom panel of Figure 2.4b, which shows estimated VP/VS

ratios in black. These values are still incorrect, but what happens if we increase the coarse

sampling interval h even more?

Figure 2.5 shows a comparison for a range of possible values of h. In all Figures 2.5a-

f, the dashed white curve shows the correct time shifts, the solid white curve shows the

computed time shifts, the dashed red curve shows the correct VP/VS ratios, and the black

14

Figure 2.4: Zoomed views of alignment errors computed from synthetic traces in Figure 2.1and time shifts computed using Algorithm 1 (a) and Algorithm 2 (b) with a coarse samplinginterval h = 10∆t. The computational stencils for each algorithm are shown in red. Correctvalues are the dashed curves and computed values are the solid curves.

curve shows the computed VP/VS ratios. Figure 2.5a is the same as Figure 2.2b, but for

coarse sampling interval h equal to the time sampling interval ∆t, and Figure 2.5b shows the

zoomed out version of Figure 2.4b that was already discussed. Figures 2.5c-f show results for

larger intervals h. These figures also display white dots that indicate locations along the shift

curve where time shift was actually computed by the smooth dynamic warping algorithm.

The curves were obtained by piecewise-linear interpolation of those computed time shifts. In

Figure 2.5a and Figure 2.5b these dots are not shown because they would be so dense that

the time-shift curves would be invisible.

Figure 2.5 clearly shows that, as the interval h is increased, we are better able to recover

the correct time shifts. Also, notice the improved accuracy of the estimated VP/VS ratios.

Shifts in Figure 2.5f are estimated at only 5 locations, and while this interval h is perhaps too

coarse, the estimates are more accurate than those for intervals h < 200∆t. This example

shows that, by increasing h, and thereby decreasing the temporal resolution of shifts, we are

able to more accurately estimate the change in shift.

15

Figure 2.5: Time shifts computed from smooth dynamic warping (Algorithm 2), for differentcoarse sampling intervals h. White dots show the coarse sample locations where time shiftsare estimated. For h = ∆t and h = 10∆t dots are not shown because they would hidethe time-shift curves completely. Dashed curves represent correct values and solid curvesrepresent computed values. The solid curves are obtained by piecewise-linear interpolationof computed values. As the coarse sample interval increases to h = 200∆t, time shifts andVP/VS ratios are more accurately estimated. When the coarse sample interval is too large(h = 400∆t) estimates are less accurate.

16

2.4 Smooth dynamic image warping

Hale (2013) describes how dynamic time warping can be extended to multidimensional

image warping. This dynamic image warping (DIW) algorithm still estimates only vertical

shifts. In fact, the last step of DIW is an application of DTW, via Algorithm 1, for each pair

of traces in two multidimensional images. However, prior to using Algorithm 1, alignment

errors are first smoothed vertically and horizontally so that strains (changes in shifts) are

limited in each image dimension. Without this smoothing step, time-shift estimates can vary

wildly from trace to trace where the two images are not related simply by time shifts.

Let f and g denote two 3D images with ni2 × ni3 traces and ni1 samples per trace. With

DIW we can find vertical shifts u[i1, i2, i3] such that

f [i1, i2, i3] ≈ g[i1, i2, i3 + u[i1, i2, i3]]. (2.6)

The shifts u[i1, i2, i3] are constrained by

ul ≤ u[i1, i2, i3] ≤ uu, (2.7)

and

rl1 ≤ u[i1, i2, i3]− u[i1 − 1, i2, i3] ≤ ru1 , (2.8)

rl2 ≤ u[i1, i2, i3]− u[i1, i2 − 1, i3] ≤ ru2 , (2.9)

rl3 ≤ u[i1, i2, i3]− u[i1, i2, i3 − 1] ≤ ru3 . (2.10)

The bounds rl1 and ru1 are analogous to the slope constraints in equation 2.3, and these

control the change in vertical shift within each trace in the 3D image. The bounds rl2 , ru2 ,

rl3 , and ru3 constrain the rates at which the vertical shifts u[i1, i2, i3] vary laterally. These

slope constraints are used in the vertical and horizontal smoothing of all alignment errors

computed from f and g.

The key to this smoothing step lies in recognizing that lines 4-14 in Algorithm 1 perform

a recursive non-linear smoothing of alignment errors e[i, l]. For 3D images f and g, align-

17

ment errors are computed for all pairs of traces, resulting in a 4D array of alignment errors

e[i1, i2, i3, l]. 2D alignment error subsets, e[ik, l], can be extracted from this 4D array in each

k dimension, where k = 1, 2, 3. To smooth alignment errors in the kth dimension, apply the

recursive non-linear smoothing in both forward and reverse directions. The forward direction

is that in which errors are accumulated in order of increasing index ik, as is the case in line

4 of Algorithm 1. The reverse direction means that errors are accumulated in decreasing

order of index ik, that is, from nik − 1 to 0. The sum

e[ik, l] = ef [ik, l] + er[ik, l]− e[ik, l] (2.11)

is an array of alignment errors smoothed in the kth dimension. The 2D array ef [ik, l] is the

result of smoothing e[ik, l] in the forward direction, and the 2D array er[ik, l] is the result

of smoothing e[ik, l] in the reverse direction. Subtraction of the original alignment errors

e[ik, l] is necessary so that this error is not included twice in the sum. This smoothing is

first applied in the vertical direction (k = 1) and then in each horizontal direction (k = 2, 3).

Smoothing in each dimension is constrained by the corresponding rlk and ruk. The final step

of DIW applies Algorithm 1 to e[i1, i2, i3, l] for all traces indexed by i2 and i3.

A similar method applies to extend smooth dynamic time warping (Algorithm 2) to

smooth dynamic image warping (SDIW). In Algorithm 2 it is lines 4-17 that perform a

recursive non-linear smoothing as well as a subsampling of alignment errors. Alignment

errors are summed along piecewise-linear paths (Figure 2.4b) and stored for subsampled

indices ik[jk]. Therefore, the smoothed and subsampled alignment errors in dimension k can

be written

e[jk, l] = ef [jk, l] + er[jk, l]− e[ik[jk], l]. (2.12)

The 2D array ef [jk, l] is the result of smoothing and subsampling e[ik[jk], l] in the forward

direction, and the 2D array er[jk, l] is the result of smoothing and subsampling e[ik[jk], l] in

the reverse direction. This smoothing and subsampling is first applied in the vertical direction

18

(k = 1) and then in each horizontal direction (k = 2, 3). Smoothing in each dimension is

constrained by the corresponding bounds on slope rlk and ruk, but the subsampling interval

hk = ik[jk] − ik[jk − 1] now allows slopes to be more accurately estimated between coarse

samples in the kth dimension. The final step applies a simple dynamic time warping (with no

further subsampling) to e[j1, j2, j3, l] for all traces indexed by j2 and j3. This last dynamic

time warping is similar to Algorithm 1, but accounts for the distance between coarse samples,

hk, when computing lower and upper slopes ql and qu (lines 7 and 8). The result is an array

of coarsely sampled vertical shifts u[j1, j2, j3] that can be interpolated to obtain the desired

shifts u[i1, i2, i3].

The subsampling for multidimensional images yields significant savings in computer mem-

ory. The 4D array of alignment errors computed in DIW requires memory proportional to

ni1ni2ni3nl; and, as image dimensions grow, this array can quickly become too large to fit in

computer memory. SDIW must also compute alignment errors for all pairs of traces in the

two images f and g, but these alignment errors are, for each pair of traces, smoothed and

subsampled vertically (k = 1).

Consider for example the alignment errors e[i1, l] in Figure 2.5e, where ni1 = 1941. With

subsampling interval h1 = 200∆t, nj1 = 10 and the array of smoothed alignment errors

e[j1, l] is approximately 200 times smaller. For multidimensional images this smoothing and

subsampling is performed for all ni2×ni3 traces. Before smoothing horizontally (k = 2, 3), the

number of alignment errors is now proportional to nj1ni2ni3nl. Smoothing and subsampling in

horizontal directions reduces this number further to nj1nj2nj3nl, where nj2 and nj3 depend on

horizontal subsampling intervals h2 and h3, respectively. When hk = 1, SDIW is equivalent

to DIW. Increasing h1 allows more accurate estimation of changes in shifts between vertical

coarse samples, and the same is true for h2 and h3. Examples of dynamic image warping are

discussed in the next chapter..

19

CHAPTER 3

MULTICOMPONENT IMAGE REGISTRATION

3.1 Introduction

We can use dynamic image warping to compute time shifts that align reflectors in a

PP image with corresponding reflectors in a PS image. This alignment process, known as

image registration, is complicated by the fact that a PS image is not simply a time shifted

version of a PP image due to differences in noise and reflection waveforms. Because of these

difference, we use the algorithm described in Chapter 2, smooth dynamic image warping

(Hale and Compton, 2013), to estimate image-registration time shifts. This method can

more accurately estimate changes in time shifts which are directly related to interval VP/VS

ratios, an important subsurface property. In this chapter I discuss the application of smooth

dynamic image warping to 3D time-migrated PP and PS images shown in Figure 3.1. These

data are licensed and provided courtesy of Geokinetics Inc. and Geophysical Pursuit, Inc.

3.2 Traveltime relations

Let tpp denote the PP traveltime for a reflector apparent in a PP image, and let tps denote

the PS traveltime for the same reflector in the corresponding PS image. For each tpp, we

have a corresponding tps defined by the function

tps(tpp) = tpp + u(tpp), (3.1)

where u(tpp) denotes the time shift, which is a function of tpp.

The time shift u(tpp) is related to the ratio VP/VS of P- and S-wave velocities. For an

infinitesimal interval dz at depth z, the P- and S-wave velocities are

VP = 2dz

dtpp, VS = 2

dz

dtss, (3.2)

where tss denotes the SS time corresponding to the PP time tpp. The factors of 2 are necessary

because tpp and tss are two-way traveltimes; the infinitesimal time intervals dtpp and dtss are

20

Figure 3.1: Time-migrated PP and PS images before image registration. Note that reflectorsare approximately aligned, but the PP and PS images are plotted on two different time scales.

the times required for P- and S-waves to travel twice through the depth interval dz. From

these definitions of interval velocities we define the interval VP/VS ratio by

γi(tpp) ≡VPVS

=dtssdtpp

. (3.3)

The time tss is that at which we would observe a reflector in an SS image, if we had one.

If we instead have a converted-wave PS image, we must express the interval VP/VS ratio

γi(tpp) in terms of the time tps(tpp) or, equivalently, the time-shift function u(tpp). To do

this, we recall that PS time is the average of PP and SS times

tps(tpp) =tpp + tss(tpp)

2. (3.4)

Substituting this expression into equation 3.3, we obtain

γi(tpp) = 2dtpsdtpp− 1. (3.5)

Then, using equation 3.1, we have

21

γi(tpp) = 1 + 2du(tpp)

dtpp. (3.6)

3.3 Geophysical constraints on time shifts

The linear inequality constraints 2.3 define lower and upper bounds, ul and uu, respec-

tively, on the time shifts u(tpp). To specify a lower bound ul, we may use the fact that

VP ≥ VS, which implies that tps ≥ tpp and that all time shifts u(tpp) must be non-negative,

so that the lower bound is ul = 0. Indeed, at the surface, where tpp = tps = 0, we should

theoretically find that u(0) = 0. In practice, we might relax this lower bound to permit

negative shifts, because uncertainties in near-surface variations of VP and VS may lead to in-

consistencies in datums and statics corrections used to obtain PP and PS images. However,

near the surface we are also likely to find ratios VP/VS � 1, so that shifts u(tpp) will increase

rapidly with time tpp, and therefore quickly become positive. For all examples shown in this

chapter, we use the lower bound ul = 0.

We calculate the upper bound uu from an estimate of the average VP/VS ratio and a

maximum time of interest. Analogous to equation 3.5, the average VP/VS ratio is

γa(tpp) = 2tpstpp− 1. (3.7)

Therefore, if Tps is the maximum time of interest in a PS image, then the corresponding time

in the PP image is

Tpp =2Tps

1 + γa(Tpp). (3.8)

Note that Tpp is not the maximum recording time in the PP image. Rather, Tpp is the

maximum time in the PP image for which a corresponding event of interest appears in the

PS image at the maximum PS time, Tps. We will ignore times tpp > Tpp in the PP image for

which there are no corresponding events of interest in the PS image.

If we could solve equation 3.8 for Tpp, then we could simply compute the upper bound on

shift as uu = Tps− Tpp. Although we are unlikely to know the function γa(tpp) until after we

22

have estimated time shifts u(tpp), a precise upper bound uu is unnecessary. In practice, we

need only ensure that we do not set this bound too low, so that the correct shifts u(tpp) will

not exceed uu. This requirement means that we should generally overestimate the average

VP/VS ratio γa(Tpp) at the maximum PP time Tpp. For the images shown in Figure 3.1, we

find that γa(Tpp) ≈ 2, so that uu ≈ Tps/3 is a safe choice.

In addition to the bounds ul and uu on estimated time shifts u(tpp), the constraints 2.3

define lower and upper bounds on time strain, rl and ru, respectively. According to equa-

tion 3.6, these time-shift derivatives are directly related to interval VP/VS ratios γi(tpp).

Therefore, rl and ru can be expressed in terms of lower and upper bounds on interval VP/VS,

γiMin≤ γi ≤ γiMax

, from which we compute

rl =γiMin

− 1

2, (3.9)

ru =γiMax

− 1

2. (3.10)

Because γi ≥ 1, a safe lower bound is rl = 0. Since rl ≥ 0, the derivative of the time shifts

cannot be negative, and thus the time-shift curve must be monotonically increasing. The

upper bound ru depends on the maximum interval VP/VS ratio γiMax. Because this ratio is

seldom known in advance, an overestimate should generally be used.

For the PP and PS images shown in Figure 3.1 we must compute time shifts u(tpp) at

all x, y locations in the 3D image, thus the time-shift volume u(x, y, tpp) is the solution we

must compute with smooth dynamic image warping. The bounds 3.9 and 3.10 apply to only

derivatives of time shifts with respect to time tpp. Although the time shifts u(x, y, tpp) can

never decrease with increasing time tpp, they may decrease or increase with increasing x or

y coordinates. The lateral constraints defined in section 2.4 control rates at which the time

shifts u(x, y, tpp) may vary laterally. These lateral time-strain constraints are related to the

rates at which corresponding estimates of VP/VS ratios, γi(x, y, tpp), can vary laterally. In

practice, we do not know in advance what these rates will be, so we typically begin with loose

bounds on lateral variations and tighten them as necessary to reduce suspicious oscillations

23

in estimated VP/VS ratios.

3.4 SDIW versus DIW

Chapter 2 showed how smooth dynamic time warping (Algorithm 2) could estimate

changes in time shift more accurately than dynamic time warping (Algorithm 1), for a

synthetic 1D example. In this section I compare those two algorithms applied to the real

data shown in Figure 3.1. Algorithms 1 and 2, extended to multidimensional images, will be

referred to as dynamic image warping (DIW) and smooth dynamic image warping (SDIW),

respectively.

DIW was previously used by Liang and Hale (2012) to register 2D PP and PS images and

to estimate VP/VS ratios. As discussed in Chapter 2, DIW returns a sequence of integer shifts,

and the number of possible shift derivatives is small; thus from equation 3.6, the number

of possible computed γi(tpp) values is correspondingly small. Therefore, an important step

in Liang and Hale’s (2012) application is to smooth the optimally computed sequences of

integer shifts before estimating interval VP/VS. Since time shifts are related to the integral

of VP/VS, we expect u to be a smooth function, compared to γi; however, estimating VP/VS

by smoothing a rough sequence of integer shifts may be inaccurate. SDIW directly computes

a smooth and more accurate time-shift function (Hale and Compton, 2013).

Figure 3.2 shows the alignment errors for a single pair of traces in the PP and PS images

from Figure 3.1. Overlaid on the alignment errors are time-shift curves extracted from the

3D DIW and SDIW algorithms. For a more detailed comparison, these curves and the

alignment errors are shown for only a 1-second time window.

The solid white curve in Figure 3.2 shows the time shifts computed from DIW. Note the

roughness of the white curve, which results from the fact that DIW computes time shifts

at every time sample. In fact, the white time-shift curve contains only two slopes, 0 and

1. These slopes correspond to the bounds rl and ru from equation 2.3, and thus γiMin= 1

and γiMax= 3 from equations 3.9 and 3.10, respectively. Not only are γiMin

and γiMaxthe

bounds on γi(tpp), but from equation 3.6 we know that these are the only two possible values

24

Figure 3.2: Alignment errors and time shifts computed from a single pair of traces from thePP and PS images shown in Figure 3.1. Only the alignment errors for times 0 to 1 second areshown. Overlaid curves are the 1D time shifts extracted from the 3D time shifts u(x, y, tpp)at the corresponding trace location. The solid white curve shows rough time shifts computedfrom DIW and the dashed blue curve shows these shifts after smoothing. The dashed yellowcurve shows the time shifts computed from SDIW with a minimum subsampling intervalh = 100 ms.

of γi(tpp) we can compute. Hence, the motivation to smooth these shifts, as Liang and Hale

(2012) do, is to obtain more estimates of interval VP/VS. The dashed blue curve in Figure 3.2

shows the white time-shift curve after smoothing.

The yellow curve in Figure 3.2 shows the interpolated time shifts computed from SDIW.

With SDIW, time shifts are computed at only subsampled PP times and the time shifts

u(x, y, tpp), are obtained by interpolation of those subsampled time-shift estimates. The

subsampled locations correspond to high-amplitude reflectors in the PP image that, in this

example, are a minimum of 100 ms apart. This selection process is discussed in a section

below.

Note the large difference between the yellow and blue curves between 0 and 0.6 s. To see

that the yellow curve in Figure 3.2 is more accurate, look at the warped images in Figure 3.3.

Compare the PP image with the PS image warped using smoothed DIW time shifts, and with

the PS image warped using SDIW time shifts. Recall that the curves overlaid on alignment

25

Figure 3.3: Slices through three 3D volumes with the same PP time window shown in thealignment errors of Figure 3.2. The location for those alignment errors corresponds to thevertical black lines in the inline and crossline sections. The strong events at 0.7 s are wellaligned in each warped image, but the time slice through the event at 0.38 s shows that timeshifts computed using SDIW are more accurate. The differences in time shifts can be seenclearly in Figure 3.2.

errors in Figure 3.2 were extracted from 3D shift functions u(x, y, tpp). The location of those

alignment errors corresponds to the vertical black lines in the inline and crossline sections of

Figure 3.3.

The reflectors at 0.7 s are well aligned in both warped PS images in Figure 3.3, and we

can see in Figure 3.2 that the yellow and blue curves are consistent at this time. However,

a large discrepancy between the two shift functions can be seen at 0.38 s in the alignment

errors in Figure 3.2, and correspondingly in the constant-time (0.38 s) slices of the warped

PS images in Figure 3.3. The warped PS image using the SDIW time shifts is a better match

to the PP image, indicating that the yellow curve is more accurate.

The subsampling in SDIW greatly increases the number of possible changes in shift, and,

as was shown in Figure 3.2 and Figure 3.3, increases the accuracy of estimated time shifts.

The white curve in Figure 3.2 demonstrates that trying to resolve changes in time shifts at

every image sample can result in misalignment of image features; furthermore, accurate time

26

shifts cannot always be obtained by smoothing inaccurate time shifts.

Figure 3.4: Interval VP/VS ratios estimated from smoothed DIW time shifts and from SDIWtime shifts. The VP/VS ratios from smoothed DIW time shifts exhibit fine detail, but thosesmoothed shifts are less accurate, as shown in Figure 3.2 and Figure 3.3, than shifts estimatedusing SDIW.

Figure 3.4 shows the interval VP/VS ratios estimated from DIW time shifts and SDIW

time shifts. The two images exhibit some similar trends of high or low VP/VS; however,

the interval VP/VS ratios computed from DIW time shifts show a high level of detail that is

unwarranted given the resolution of seismic reflections and differences in noise and reflection

waveforms between the PP and PS images.

3.5 Coarse sample locations

Coarse sampling improves the accuracy of computed shifts, but the placement of the

coarse lattice of sample points affects the accuracy as well. Increasing the distance between

coarse samples increases the smoothness of estimated time shifts. One simple approach

27

to choosing coarse sample locations is to use a nearly-uniform grid parameterized by a

coarse sampling interval in each dimension. We examine this approach for a 1D example,

where we compute time shifts u(tpp) using smooth dynamic time warping, and focus only on

subsampling the PP-time axis.

Figure 3.5a shows alignment errors computed from a single pair of traces from the 3D PP

and PS images shown in Figure 3.1. In Figure 3.5b the blue curve represents the time shifts

u(tpp) after interpolation and the blue dots represent the coarse sample locations where time

shifts were computed. In this example, we specified a coarse interval of 160 ms and computed

sample locations that are almost uniformly spaced. The spacing is slightly variable so as to

include the first and last time samples; however, all intervals are greater than or equal to

the specified interval of 160 ms.

In Figure 3.5, note that strong events in the PP trace (Figure 3.5c) appear as pre-

dominantly vertical features with high error (white) in the array of alignment errors. The

diagonally crossing features with high error correspond to strong events in the PS trace.

We can estimate shifts best at the locations where these strong events occur. However, by

selecting coarse sample locations on a nearly-uniform grid, as shown by the blue dots in

Figure 3.5b, strong events are sampled only when they happen to coincide with our coarse

sampling grid. Rather than selecting coarse samples on a nearly-uniform grid, we should

instead select coarse sample locations to coincide with high-amplitude events in the PP trace.

Let f(tpp) denote the PP trace and a(tpp) denote the trace envelope computed as

a(tpp) =

√f 2(tpp) + f 2(tpp), (3.11)

where f(tpp) is the Hilbert transform of f(tpp). Figure 3.5c shows the PP trace in black and

the trace envelope in red. An easy way to improve the coarse sample locations is to use

the envelope a(tpp) to preferentially select locations with the highest amplitudes, while also

satisfying a specified minimum interval. We search for coarse grid points in descending order

of envelope amplitudes in a(tpp). A coarse sample is chosen when its location is at a distance

greater than or equal to the specified minimum interval from all previously selected coarse

28

Figure 3.5: Alignment errors (a) computed from a single trace from the PP and PS imagesshown in Figure 3.1, with time shifts (b) computed from 1D smooth dynamic time warpingusing two different subsampling grids. A nearly-uniform coarse grid was used for the bluecurve, with subsampling locations represented by blue dots, and a reflector-aligned grid wasused for the yellow curve with subsampling locations represented by yellow squares. The PPtrace (c) overlaid with the envelope of the trace shows that strong PP events correlate withstrong vertical features in the alignment errors above.

29

samples. As with the nearly-uniform-grid scheme, we include the endpoints of the original

sampling grid.

The yellow squares in Figure 3.5b represent the coarse sample locations where time shifts

were computed along the yellow time-shift curve. These squares were preferentially selected

to correspond with locations of strong events, while maintaining a minimum separation

interval of 100 ms. We reduce the minimum interval used for the nearly-uniform grid (160

ms) in order to include some of the highest amplitude events that are within that interval

of one another. In this example, we have selected the same number of subsamples for the

nearly-uniform grid and the reflector-aligned grid.

The time-shift curves shown in Figure 3.5b for each grid are quite similar from 0 to 1.6 s

except for a large discrepancy around 1.2 s. The blue dot indicates that a time shift was

computed for that location, but there is no corresponding yellow square nearby. From both

the alignment errors and the envelope of the PP trace in Figure 3.5c we see that there are

weak events at, and around 1.2 s; therefore, we have little confidence in the accuracy of the

computed time shift at the blue dot. After 1.6 s, the two curves are quite different, with a

significant change in shift at approximately 1.6 s. Time shifts shown by the blue curve are

inaccurate, based on 3D time shifts and warping results not shown here. This discrepancy

in the blue curve is largely due to the fact that it is difficult to accurately register PP and

PS events using a single pair of traces. However, the yellow curve corresponds to the correct

shifts, even in the 1D case. Therefore, this example demonstrates that better placement of

coarse sample locations can improve shift estimates.

When smoothing and subsampling alignment errors in 3D, we require the same number

of coarse-grid time samples for all coarse inline and crossline sample locations. Therefore,

we wish to select the highest amplitudes that are consistent throughout the image, i.e., the

strongest reflectors. We cannot directly use the method described above because reflectors in

seismic images are generally not horizontal, but when seismic reflectors are not structurally

complex, we can make them horizontal using seismic image flattening (Lomask, 2006; Parks

30

et al., 2008; Parks, 2010).

Figure 3.6a shows the PP image, which is structurally simple, but contains dipping

reflectors. Therefore, we compute a volume of shifts (Figure 3.6b) that flatten image features

so that a constant time in the flattened image (Figure 3.6c) corresponds to a single seismic

horizon. In this flattened image space we stack the envelopes of all image traces:

as(tpp) =

ni3−1∑

i3=0

ni2−1∑

i2=0

ai2i3(tpp), (3.12)

where ni3 and ni2 are the number of inline and crossline samples, respectively. Now, we can

search for times corresponding to the strongest reflection events in the flattened 3D image

and compute a coarse lattice of points in all three dimensions, indicated by the white dots in

Figure 3.6c. We then map the coordinates of these lattice points from the flattened space to

the original image space to obtain a reflector-aligned 3D lattice on which to compute time

shifts using SDIW.

Figure 3.7 compares a simple nearly-uniform grid, defined by coarse-interval parameters

in each dimension, with a reflector-aligned coarse grid. Both grids have the same nearly-

uniform sampling in the inline and crossline directions. Time shifts are estimated at coarse

sample locations, and we improve the accuracy of these estimates by aligning the coarse grid

with strong reflectors. Figure 3.7 shows interval VP/VS ratios estimated from the nearly-

uniform grid and the reflector-aligned grid. Note the step in VP/VS at 1.7 and 1.8 s in

the crossline section in Figure 3.7. This step compensates for the nearly-uniform coarse

grid that samples across a dipping layer in which VP/VS is consistently low throughout the

image. This layer is more apparent at the same location in the reflector-aligned VP/VS ratios

because coarse grid points are aligned with reflectors in the image.

Note that this automated approach to picking sample locations is independent of our

smooth image warping algorithm. Thus, a more interpretive approach of selecting specific

seismic horizons could be used instead, especially for a more complex image that might be

more difficult to flatten.

31

Figure 3.6: The PP image (a) and time shifts (b) used to flatten the image so that aconstant time corresponds to a single seismic horizon. The coarse lattice of points on theflattened image (c) is shown by the white dots. This lattice can be mapped back to theoriginal (unflattened) coordinate space to obtain a coarse lattice of points that is alignedwith dipping reflectors (d). We use this lattice in SDIW to compute time shifts at locationswith strong reflectors in the PP image.

32

Figure 3.7: White dots indicate coarse lattice points used to compute time shifts with SDIW.Both nearly-uniform and reflector-aligned grids have minimum intervals of 500 m laterally.The nearly-uniform points are a minimum of 160 ms apart in time, while reflector-alignedpoints are a minimum of 100 ms apart but are aligned with strong reflectors in the image.Points in the nearly-uniform grid cross strong reflectors horizontally, resulting in a noticeablestair step effect in the corresponding VP/VS ratios.

33

3.6 Interpolation of time shifts

The smooth dynamic warping algorithm computes time shifts only at specified subsam-

pled locations to improve accuracy, but we ultimately need time shifts for every image sample

to align PP and PS images. Therefore, time shifts for all image samples not specified by the

coarse grid are interpolated from the computed coarse lattice of time shifts.

For the 3D examples shown here, we do this interpolation in two steps. The interpolation

is first done laterally using bilinear interpolation. With bilinear interpolation we ensure that

the bounds on shifts ul and uu and our assumption of monotonically increasing time shifts

are satisfied. For vertical interpolation of our time shifts, we must consider the effect on the

VP/VS ratios we are trying to estimate.

Figure 3.8: Interval VP/VS estimates resulting from two different interpolation methods fortime shifts, piecewise cubic with monotonic splines and piecewise linear.

As described in Chapter 2, the SDIW algorithm accumulates alignment errors at coarse

sample locations by summing errors along piecewise-linear trajectories. Therefore, the most

34

consistent approach for interpolating time shifts is piecewise-linear interpolation. Figure 3.8

shows the estimated interval VP/VS ratios when time shifts are interpolated using piecewise-

cubic interpolation with monotonic splines and using piecewise-linear interpolation. Both

results were obtained from time shifts computed at coarse sample locations shown in the

reflector-aligned grid of Figure 3.7.

The VP/VS ratios from piecewise-linear interpolation of time shifts in Figure 3.8 are

piecewise-constant between subsampled time locations and are more consistent with the

SDIW algorithm than VP/VS ratios from piecewise-cubic interpolation of time shifts. We

computed time shifts at only coarse sample locations, and thus estimated the change in

shift between only these locations. When smoother estimates of VP/VS are required, we can

interpolate time shifts using a smoother method. However, smoother VP/VS ratios will be

less consistent with the SDIW method used to compute time shifts.

Figure 3.9 shows 1D time shifts u(tpp) and corresponding interval VP/VS ratios γi(tpp)

extracted from 3D functions u(x, y, tpp) and γi(x, y, tpp), respectively. The three curves in

each image show results from three methods of interpolation. The black curves show linearly-

interpolated time shifts and the corresponding piecewise-constant VP/VS ratios. The other

two interpolation methods use piecewise-cubic polynomials. The red time-shift curve was

interpolated using cubic polynomials that preserve monotonicity, while the blue curve was

interpolated using the classic cubic spline with continuous first and second derivatives.

Linear interpolation is the only method that guarantees time shifts will be both monotonic

and bounded by [rl, ru]. Monotonicity is still guaranteed for the red time-shift curve, but

the bounds [rl, ru] are no longer strictly satisfied. For example, at 1.7 s the red curve dips

slightly below the lower bound, rl = 1.5, used in this example. Cubic spline interpolation

is shown for completeness, but this method is worse still. Overshoot with this method will

generally be more severe than with piecewise-cubic monotonic interpolation. For this method

we can guarantee neither monotonicity of interpolated time shifts nor derivatives bounded

by [rl, ru]. Therefore, it is possible to compute VP/VS < 1, but for the example shown in

35

Figure 3.9: Interpolated time shifts and interval VP/VS extracted from 3D SDIW shift andVP/VS volumes. The three different curves represent three different methods of interpolatingtime shifts: linear (black), cubic monotonic spline (red), and cubic spline (blue).

Figure 3.9 this does not occur.

If we want interval VP/VS estimates that are continuous, then cubic monotonic interpo-

lation might be satisfactory. Otherwise, the piecewise-constant VP/VS ratios are the most

consistent estimates we can obtain from time shifts computed using SDIW.

36

CHAPTER 4

SIMULTANEOUS MULTICOMPONENT REGISTRATION

4.1 Introduction

The previous chapter discussed the application of smooth dynamic image warping to

automatically register a pair of multicomponent PP and PS images. However, in multi-

component surveys more image pairs can exist. For example, a 9C multicomponent seismic

survey can capture nine different wave modes, yielding nine different images. Even the sim-

plest 3C multicomponent seismic survey can result in more than the PP and PS images

previously considered.

In the presence of azimuthal anisotropy, which is associated with fractures or differen-

tial stress, the mode-converted S-wave can split into fast and slow waves with orthogonal

polarizations. This phenomenon is known as shear-wave splitting or seismic birefringence.

The fast-S mode is commonly referred to as S1 and is polarized parallel to fracture strike

or maximum stress. The slow-S mode, S2, is polarized parallel to fracture normal or min-

imum stress. The S1 polarization angle and the time lag between S1 and S2 give valuable

information about anisotropy and fracture characterization.

Thus the simple 3C multicomponent seismic survey can record three combinations of

downgoing and upgoing waves: PP, PS1, and PS2. With multicomponent seismic data

processing techniques, these different wave modes can be separated into three corresponding

seismic images.

The automatic image registration method discussed in the previous chapter could be

used to register independently the three image pairs: PP-PS1, PP-PS2, and PS1-PS2. This

straightforward approach yields an estimated time-shift volume for each image pair and

these three time-shift volumes should be related. However, independently warping image

pairs will result in inconsistent time-shift estimates. Instead, the time-shift relations can be

37

incorporated into a single minimization problem involving all three images simultaneously.

This chapter formulates the simultaneous dynamic warping problem and demonstrates its

solution using synthetic PP, PS1, and PS2 seismograms.

4.2 Traveltime relations

Section 3.2 describes traveltime relations for PP and PS seismic traces, and shows how

time shifts that align these traces can be used to estimate VP/VS ratios. This section derives

similar traveltime relations for PP, PS1, and PS2 traces. From these three traces we can

estimate not only VP/VS ratios, but also anisotropy, which causes time shifts between PS1

and PS2 traces.

Let tpp denote the PP traveltime for an event in a PP trace, and let tps1 and tps2 denote

the traveltimes for the same event in PS1 and PS2 traces, respectively. For each tpp, we have

a corresponding tps1 and tps2 given by

tps1(tpp) = tpp + u1(tpp), (4.1)

tps2(tpp) = tpp + u2(tpp), (4.2)

where u1(tpp) denotes time shifts between PP and PS1 traces and u2(tpp) denotes time shifts

between PP and PS2 traces, and both time shifts are functions of tpp. Additionally, time

shifts can be found between PS1 and PS2 traces. Therefore, each tps1 has a corresponding

tps2 given by

tps2(tps1) = tps1 + uS(tps1), (4.3)

where uS(tps1) denotes time shifts between PS1 and PS2 traces and is a function of tps1 .

To see the relationship between time shifts u1(tpp), u2(tpp), and uS(tps1), we use equa-

tion 4.1 to rewrite equation 4.3 in terms of tpp, as

tps2(tpp) = tpp + u1(tpp) + uS(tpp + u1(tpp)). (4.4)

38

Equations 4.4 and 4.2 give the relationship between the three time shifts

u2(tpp) = u1(tpp) + uS(tpp + u1(tpp)). (4.5)

The time shifts uS(tpp+u1(tpp)) are specified at PS1 times given by tpp+u1(tpp), but ultimately

we want a function of PP time

uS(tpp) ≡ uS(tpp + u1(tpp)), (4.6)

which can be computed from the time shifts uS(tps1) and u1(tpp). Of course, the accuracy of

uS(tpp) will depend partly on the accuracy of u1(tpp). Equation 4.6 simplifies equation 4.5 to

u2(tpp) = u1(tpp) + uS(tpp). (4.7)

This relationship is intuitive because of the relative traveltimes for PP, PS1, and PS2 traces.

The longest traveltimes correspond to the PS2 trace and the shortest traveltimes correspond

to the PP trace. Thus the largest time shifts are between PP and PS2 traces (u2(tpp)), and

are a combination of the smaller time shifts.

The time delay between two shear waves is conventionally described by the fractional

difference

γ(s) ≡ VS1 − VS2VS2

, (4.8)

where VS1 and VS2 represent the fast and slow shear mode velocities, respectively (e.g.,

Tsvankin (2012)). Equation 4.8 provides a measure of anisotropy, but requires the velocities

VS1 and VS2. Alternatively, γ(s) can be computed from time shifts that align PP, PS1 and

PS2 traces. In fact, a combination of any two of the three time shifts u1(tpp), u2(tpp), and

uS(tpp), can be used to compute γ(s)(tpp).

For an infinitesimal interval dz at depth z, the P-, S1-, and S2-wave velocities are

39

VP = 2dz

dtpp, VS1 = 2

dz

dtss1, VS2 = 2

dz

dtss2(4.9)

where tss1 and tss2 denote the SS1 and SS2 times corresponding to the PP time tpp. The

factors of 2 are necessary because tpp, tss1 , and tss2 are two-way traveltimes; the infinitesimal

time intervals dtpp, dtss1 , and dtss2 are the times required for P-, S1-, and S2-waves to

travel twice through the depth interval dz. Using the velocity equations 4.9 we can rewrite

equation 4.8 as

γ(s) ≡2 dzdtss1− 2 dz

dtss2

2 dzdtss2

, (4.10)

which simplifies to

γ(s) ≡ dtss2 − dtss1dtss1

. (4.11)

If we multiply the numerator and denominator of equation 4.11 by the factor 1/dtpp, then for

each term we recognize the definition of interval VP/VS from equation 3.3, and equation 4.11

can be expressed as

γ(s)(tpp) ≡γi2(tpp)− γi1(tpp)

γi1(tpp), (4.12)

where

γi1(tpp) = 1 + 2du1(tpp)

dtpp, (4.13)

γi2(tpp) = 1 + 2du2(tpp)

dtpp. (4.14)

Equations 4.13 and 4.14 follow from equation 3.6 by simply replacing VS and u(tpp) (from

the derivation of equation 3.6) with VS1 and u1(tpp), and with VS2 and u2(tpp).

40

To relate γ(s)(tpp) to time shifts u1(tpp), u2(tpp), and uS(tpp), we use equations 4.13

and 4.14 in equation 4.12 to obtain

γ(s)(tpp) ≡2(u2(tpp)− u1(tpp))

1 + 2u1(tpp), (4.15)

with the shorthand notation x(t) ≡ dx(t)/dt. Equation 4.15 requires only u1(tpp) and u2(tpp)

time shifts. Using the time-shift relations in equation 4.7, this can be written equivalently

as

γ(s)(tpp) ≡2uS(tpp)

1 + 2u1(tpp), (4.16)

which requires only u1(tpp) and uS(tpp) time shifts, and

γ(s)(tpp) ≡2uS(tpp)

1 + 2(u2(tpp)− uS(tpp)), (4.17)

which requires only u2(tpp) and uS(tpp) time shifts. However, recall that u1(tpp) is needed to

compute uS(tpp), and thus equation 4.17 implicitly requires all three time shifts.

In summary, γ(s) can be computed from different combinations of time-shift estimates.

Equations 4.15, 4.16, and 4.17 provide a consistency check for estimated γ(s) values. If

the estimated time shifts satisfy equation 4.7, then equations 4.15, 4.16, and 4.17 must be

equivalent. To strictly satisfy this condition, we reformulate the dynamic warping problem

using equation 4.7.

4.3 Simultaneous registration

Let f , g, and h represent sampled sequences for PP, PS1, and PS2 traces, respectively,

and let u[0 : ni − 1] ≡ {u[0], u[1], ..., u[ni − 1]} define a sequence of time shifts. Using

equation 4.7, the sampled sequences of shifts can be written as

u2[0 : ni − 1] = u1[0 : ni − 1] + uS[0 : ni − 1]. (4.18)

41

Using equation 4.18 I choose two time shifts that will be computed simultaneously with

dynamic warping. From these two time shifts, the third can be easily computed. I choose

registration of f , g, and h, in terms of u1 and uS, as follows:

f [i] ≈ g[i+ u1[i]],

f [i] ≈ h[i+ u1[i] + uS[i]],

g[i+ u1[i]] ≈ h[i+ u1[i] + uS[i]], i = 0, 1, ..., ni − 1. (4.19)

The time shifts u1[i] and uS[i] are the solution to the optimization problem:

u1[0 : ni − 1], uS[0 : ni − 1] ≡ arg minl1[0:ni−1],lS [0:ni−1]

ni−1∑i=0

e[i, l1[i], lS[i]], (4.20)

subject to constraints

u1l ≤u1[i] ≤ u1u , r1l ≤ u1[i]− u1[i− 1] ≤ r1u , (4.21)

uSl≤uS[i] ≤ uSu , rSl

≤ uS[i]− uS[i− 1] ≤ rSu , (4.22)

where

e[i, l1, lS] ≡ w1 (f [i]− g[i+ l1])2

+ w2 (f [i]− h[i+ l1 + lS])2

+ wS (g[i+ l1]− h[i+ l1 + lS])2 . (4.23)

Weights w1, w2, and wS enable us to specify the relative importance of each trace pair. For

all examples that follow, w1 = w2 = wS = 1. Alternative difference measures could be used,

but here I use the squared differences between amplitudes in each trace pair. The lag indices

l1 and lS satisfy constraints u1l ≤ l1 ≤ u1u and uSl≤ lS ≤ uSu , respectively.

This formulation for the simultaneous dynamic warping problem is similar to dynamic

warping from section 2.2, but here the optimization problem is better constrained, is guar-

anteed to yield a consistent set of shifts, and takes advantage of information in all three

sequences f , g, and h. However, with these advantages comes increased computational cost.

42

Algorithm 2 is the foundation for simultaneous smooth dynamic warping, and the exten-

sion is conceptually simple. Whereas Algorithm 2 loops over all lags l, simultaneous smooth

dynamic warping must loop over two lag dimensions, l1 and lS. Solving for shifts uS helps

to reduce this cost because the range of time shifts uS should generally be much smaller

than the time-shift range for either u1 or u2. Alignment errors e[i, l1, lS] for simultaneous

warping are shown in Figure 4.1. The vertical axis labeled “Time shift 1” represents the

time shifts u1 and the horizontal axis labeled “Time shift S” represents the time shifts uS.

In Figure 4.1 the uS axis is stretched for better visibility, but note that the range in time

shifts is much smaller than the range of time shifts for the u1 axis. The white curve is the

solution computed with simultaneous smooth dynamic warping. The shape of the white

curve is different from previous examples, and this difference will be explained in the next

section. The u1 and uS coordinates at each PP time minimize the sum in equation 4.20.

The choice of time shifts uS in the formulation of the simultaneous registration problem

reduces computational cost, but another significant reduction in cost comes from a simple

observation of the 2D alignment errors previously shown.

4.4 Reducing computational cost

Chapter 3 states that the lower bound for time shifts ul, is zero, and the upper bound uu

can be estimated from guessing an average VP/VS ratio. These bounds imply that alignment

errors need not be computed for time shifts out-of-bounds of input data arrays. That is,

alignment errors were computed for only PP times where all time shifts, from ul to uu,

corresponded to times within the PS trace.

This familiar array of alignment errors is shown in Figure 4.2a with the correct time shifts

shown as the dashed white curve. Note the large range of time-shift values on the vertical

axis, and the large yellow triangular areas above and below the correct time shifts. Because

the time shift should theoretically be zero at tpp = 0, a solution through the triangular areas

would be unphysical given time-strain constraints based on typical minimum and maximum

interval VP/VS ratios.

43

Figure 4.1: 3D alignment errors for simultaneous warping of synthetic traces shown in Fig-ure 4.3. The computed solution is shown as the white curve.

Algorithms 1 and 2 from Chapter 2 show that the accumulation step requires a loop over

all lags, l. Since the cost of the accumulation process is proportional to the number of lags, nl,

a large reduction in computational cost can come from applying an initial squeezing of the PS

trace before computing alignment errors. This initial squeezing is a rough alignment of PP

and PS traces that greatly reduces the number of lags in the accumulation step. Figure 4.2b

shows alignment errors computed for a PP trace and a PS trace that has been squeezed by

a constant factor c = 1.7. The time-shift axis in Figure 4.2b is vertically exaggerated, but

the time-shift range is significantly smaller than the time-shift range in Figure 4.2a. In fact,

44

Figure 4.2: Alignment errors for synthetic PP and PS1 traces with no initial squeezing ofthe PS trace (a). Yellow triangles indicate large areas where computation time is wasted.Alignment errors for the same traces, but after squeezing the PS1 trace by a constant factorc = 1.7 (b). The dashed white curves show the known time-shift function.

nl = 1362 and nl = 176 in Figure 4.2a and Figure 4.2b, respectively. This initial squeezing

of the PS trace helps to reduce computational cost, but also introduces some previously

avoided complexities.

One such complexity is alignment errors for time shifts that are out-of-bounds of the

input data arrays. These problematic time shifts occur at early PP times where negative

time shifts might correspond to negative PS times, and at late PP times where positive

time shifts might correspond to unrecorded PS times. Because no alignment error can be

computed, a suitable replacement value must be used instead. One solution that seems to

work well is, for each PP time, to reflect the valid alignment errors where time shifts are

out-of-bounds of the squeezed PS data. This is the method used for alignment errors shown

in Figure 4.2b. The mirroring of alignment errors occurs in the lower left and upper right

corners, but is only apparent upon close inspection.

Another complexity arises from the slope constraints, rl and ru, computed from minimum

and maximum interval VP/VS ratios (equations 3.9 and 3.10). Physically, because γi ≥ 1,

the natural choice for the lower bound on slope is rl ≥ 0. Without the initial squeezing, this

constraint meant that the computed time-shift function had to be monotonically increasing,

45

and thus the computational stencil (Figure 2.4) was only one sided. However, after the

initial squeezing, the assumption of monotonically increasing time shifts is no longer valid.

Clearly, the correct time shifts in Figure 4.2b are not monotonically increasing after initially

squeezing the PS trace. Still, it is important that the constraints, in terms of interval VP/VS

ratios, remain valid. Those constraints can be rederived by accounting for the scaling factor

c in the traveltime relations. For instance, equation 4.1 becomes

tps1(tpp) = c [tpp + u1(tpp)] . (4.24)

From equation 3.5 we know that interval VP/VS ratios are related to the derivative of equa-

tion 4.24,

tps1(tpp) = c [1 + u1(tpp)] . (4.25)

Using equation 4.25 in equation 3.5, we obtain

γi1(tpp) = (2c− 1) + 2cu1(tpp), (4.26)

and thus slope constraints can still be defined in terms of minimum and maximum interval

VP/VS, γi1Minand γi1Max

. Using equation 4.26, slope constraints 3.9 and 3.10 become

rl =γi1Min

− 2c+ 1

2c, (4.27)

ru =γi1Max

− 2c+ 1

2c. (4.28)

Note that equation 4.26 is equal to equation 4.13 when c = 1, that is, when no initial

squeezing is applied to the PS1 trace.

4.5 Independent versus simultaneous registration

Figure 4.3 shows synthetic PP, PS1, and PS2 traces. To generate these synthetic traces,

a random reflectivity series was created in PS2 time. That random reflectivity series was

warped to PS1 time by a time-varying shift function ucS(tps1). That warped reflectivity series

46

Figure 4.3: Synthetic PP, PS1, and PS2 traces with a signal-to-noise (S/N) ratio of 1:1.

in PS1 time was again warped to PP time by a different time-varying shift function uc1(tpp).

Each reflectivity series was convolved with a Ricker wavelet to form PP, PS1, and PS2 seismic

traces. Different bandlimited noise was added to each trace such that the rms amplitude

of the signal divided by the rms amplitude of the noise was 1. From ucS(tps1) and uc1(tpp),

ucS(tpp) is computed (equation 4.6). With ucS(tpp), the time shifts uc2(tpp) can be computed

(equation 4.7). The notation uc denotes correct time shifts that relate the three synthetic

traces, and that we want to estimate using dynamic warping.

To test the accuracy of warping pairs of traces independently, I use smooth dynamic time

warping to compute time shifts for all three trace pairs. That is, I compute u1(tpp) from PP

and PS1 traces, u2(tpp) from PP and PS2 traces, and uS(tps1) from PS1 and PS2 traces.

The results are shown in Figure 4.4 for two different signal-to-noise (S/N) ratios. The

left column shows results for noise-free synthetic traces. Results for synthetic traces with

S/N 1:1 (Figure 4.3) are shown in the right column. The correct time-shift functions are

shown as dashed white curves. All time shifts are functions of tpp.

47

The u1 (red), u2 (blue), and uS (green) are well recovered in the noise-free case. Note

that uS (green) is not directly computed from warping the PS1-PS2 trace pair, but rather

is computed from uS(tps1) (not shown) and u1 (red) time shifts (equation 4.6). From the

three time-shift functions, u1 (red), u2 (blue), and uS (green) we compute γ(s) in three

different ways using equations 4.15, 4.16, and 4.17. The bottom panels of Figure 4.4 show

the difference between each computed γ(s) and the correct γ(s) (not shown). The color of each

γ(s) error curve represents the combination of time-shift curves used to compute γ(s); the

magenta curve from u1 (red) and u2 (blue), the yellow curve from u1 (red) and uS (green),

and the cyan curve from u2 (blue) and uS (green).

Figure 4.4: Time shifts estimated by individually warping three pairs of synthetic tracesshown in Figure 4.3, PP-PS1 (red), PP-PS2 (blue), PS1-PS2 (green). Correct time shifts areshown as dashed white curves. Results for noise-free synthetic traces are shown on the leftand results for synthetic traces with S/N = 1:1 are shown on the right. The bottom panelsshow errors in γ(s) estimated from equations 4.15 (magenta), 4.16 (yellow), and 4.17 (cyan).

48

In the bottom panel of the left column, the magenta curve is different from yellow and

cyan curves, which are the same. The yellow and cyan curves are both estimated from

derivatives of the uS (green) curve, that has been converted from time shifts computed in

PS1 time (uS(tps1)). The coarse sampling times used for the PS1-PS2 trace pair in smooth

dynamic warping are different from those for the PP-PS1 and PP-PS2 trace pairs. The

coarse-grid differences affect the interpolated shifts and explain why the yellow and cyan

curves are identical to one another but differ from the magenta curve. The coarse samples

in PP and PS1 time could be made consistent if the time shifts u1 were known, which is

never the case in practice. It could be advantageous to have different coarse samples for PP

and PS1 times since the time-shift range between the PS1-PS2 trace pair is much smaller

than the either the PP-PS1 or PP-PS2 time-shift range. Having independent coarse sample

times could allow the PS1-PS2 time shifts to be more accurately determined. The drawback

is the inconsistency shown in the γ(s) error values. If the coarse sampling interval was one,

then the errors should be consistent, provided that the time-shift curves could be accurately

estimated. However, coarse sampling is key to the accuracy of the recovered shifts, and

when the coarse sampling interval is one, the estimated time shifts are incorrect (example

not shown), even in this noise-free case.

In the right column where S/N is 1:1, not all time shifts are well estimated. The u1 (red)

time shifts are well estimated after 0.5 s, while the u2 (blue) time shifts are poorly estimated

between 0.5 s and 2.2 s. The uS (green) time shifts have two sources of error. First, the

estimate of uS from the PS1-PS2 trace pair, and second, the conversion to uS (green), which

depends on the accuracy of the u1 (red) time shifts. The γ(s) errors for time shifts in the

right column are significant, and the difference between coarse PP and PS1 time sampling

grids is irrelevant. The time shifts are not accurately estimated and the differences in the

γ(s) errors cannot be explained simply by coarse-grid inconsistencies.

In warping pairs of traces independently, there are no constraints to ensure that computed

time shifts will be consistent and satisfy equation 4.7. In practice, the correct γ(s) is unknown,

49

and it is problematic that the estimated γ(s) values depend on which time-shift combination

is used to compute them.

Figure 4.5: Time shifts estimated by simultaneously warping the three synthetic tracesshown in Figure 4.3. PP-PS1 (red) and PS1-PS2 (green) are directly computed from thesimultaneous warping, PP-PS2 (blue) is computed from equation 4.7. Correct time shiftsare shown as dashed white curves. Results for noise-free synthetic traces are shown on theleft and results for synthetic traces with S/N = 1:1 are shown on the right. The bottompanels show errors in γ(s) estimated from equations 4.15 (magenta), 4.16 (yellow), and 4.17(cyan). All of which are identical.

Figure 4.5 shows results for simultaneous dynamic warping. The noise-free case in the

left column of Figure 4.5 shows that the time shifts are accurately recovered, just as they

were in Figure 4.4. However, simultaneous warping directly computes uS(tpp) (green) as

opposed to the independent case where uS(tpp) was computed from the shifts uS(tps1) and

u1(tpp). The time shifts u2(tpp) (blue) are not directly computed from simultaneous dynamic

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warping, but u2 is easily computed from u1 and uS. The simultaneous warping formulation is

based on the time-shift relations 4.7, and thus the time-shift solutions should be consistent.

Equations 4.15, 4.16, and 4.17 are all equivalent when the time shifts are consistent with

equation 4.7 and the three curves in the γ(s) error plot are all identical. This consistency

does not occur when independently warping the three trace pairs, even in the noise-free case.

More significant are the results in the right column, where the S/N ratio is 1:1, and all

three time shifts are accurately recovered. Independent warping of three trace pairs failed to

accurately estimate those time shifts (Figure 4.4). By using all three traces simultaneously,

the solution to the minimization problem is better constrained, and the warping algorithm

is more robust in the presence of noise.

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CHAPTER 5

CONCLUSIONS

This thesis presents a modification to the dynamic warping algorithm described by Hale

(2013). This modified algorithm is called smooth dynamic warping because it computes

time shifts (vertical shifts) that are smoothly varying, by estimating time shifts at only

coarsely sampled times. The smoothness is controlled by the interval between coarse sam-

ples, enabling a tradeoff between accuracy and resolution. Estimating time shifts at coarsely

sampled locations decreases the resolution of rapid variations in time strain, while allowing

that strain to be more accurately estimated. This modification makes time-shift estimates

more accurate when differences between seismic traces are not limited to time shifts, but

include differences in noise and reflection waveforms. The extension of this algorithm to im-

age warping provides a robust method for automatic registration of multicomponent seismic

images.

Smooth dynamic warping (like dynamic warping) computes a globally optimal solution

to a non-linear optimization problem with linear inequality constraints. For multicompo-

nent image registration, the linear inequality constraints can be directly related to interval

VP/VS ratios. Interval VP/VS ratios are an important attribute for interpreting lithology

and fluid properties and are an important by-product of image registration. Therefore, the

goal of multicomponent image registration should be to align corresponding events in PP

and PS images while accurately estimating VP/VS ratios. In other words, some differences

in multicomponent images are unrelated to VP/VS ratios, and if the only goal is to minimize

image differences, then the image registration process can lead to unrealistic and inaccurate

estimates of VP/VS.

Smooth dynamic image warping requires a coarse lattice of samples at which time shifts

are estimated. The locations of those samples should coincide with strong reflectors, to best

52

resolve changes in time shifts. Those locations can be determined automatically using trace

envelopes and image flattening as described in Chapter 3. However, this approach is limited

by the complexity of seismic reflectors and automatic selection of reflectors. Alternatively,

coarse sample locations can be chosen from interpreted horizons in the PP image. Horizons

can be picked quickly and even automatically in many interpretation systems, giving direct

control over the coarse lattice of points used for smooth dynamic image warping.

In addition to the improved accuracy of the smooth dynamic warping algorithm, there is

also a significant reduction in required computer memory. The alignment errors computed

for two 3D images require memory for an array of ni1ni2ni3nl floating-point numbers, where

ni1 , ni2 , and ni3 are the number of samples in each image dimension and nl is the number

of lags for which errors must be computed. Since smooth dynamic warping computes time

shifts at only coarsely sampled locations, this large array is never constructed. Dynamic

image warping requires a smoothing of alignment errors to constrain the lateral variation

of time strain. In the case of smooth dynamic image warping, this smoothing of alignment

errors enables a subsampling of alignment errors, yielding significant savings in computer

memory. For example, the 3D images shown in Chapter 3 are approximately 120 Mb each.

For dynamic image warping, alignment error array size for these rather small 3D images is

approximately 70 Gb, while the maximum memory required by the smooth dynamic image

warping algorithm is reduced by a factor of 100 to only 700 Mb.

In Chapter 4 I extend smooth dynamic image warping to simultaneous registration of

more than two multicomponent seismic traces. With PP, PS1, and PS2 seismic traces we

can estimate VP/VS ratios as well as γ(s) from the time delay between shear waves. It is

straightforward to use the methods described in Chapter 2 and Chapter 3 to independently

warp PP-PS1, PP-PS2, and PS1-PS2 trace pairs. However, as was demonstrated with a 1D

synthetic example, independently warping trace pairs does not produce consistent time-shift

estimates. Thus different combinations of computed time shifts give different estimates of

γ(s).

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The dynamic warping optimization problem was reformulated such that computed time

shifts are consistent and PP, PS1, and PS2 traces are aligned simultaneously. For the 1D

synthetic example, the results for this method were promising and simultaneous warping

succeeded where independent warping failed. By using all three traces, the simultaneous

warping algorithm is better constrained and more robust. This improvement comes at a much

higher computational cost. However, initially squeezing PS1 and PS2 data is a simple and

effective way to reduce unnecessary computations. The extension of simultaneous warping

to 3D PP, PS1, and PS2 images is future work that could be pursued.

54

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