e cient seismic forward modeling and acquisition using ......used for vibroseis and marine...

Efficient Seismic Forward Modeling and Acquisition using

Simultaneous Random Sources and Sparsity

Ramesh (Neelsh) Neelamani∗, Christine E. Krohn†, Jerry R. Krebs†, Justin K.

Romberg‡, Max Deffenbaugh§, and John E. Anderson†

∗ExxonMobil Exploration Company, Houston, TX, USA

†ExxonMobil Upstream Research Company, Houston, TX, USA

‡Georgia Institute of Technology, Atlanta, GA, USA

§ExxonMobil Research and Engineering Company, Clinton, NJ, USA

(February 22, 2010)

Running head: Simultaneous Source Modeling and Acquisition

ABSTRACT

The high cost of densely sampled simulated model data or field data results from activating

sources one at a time in sequence. Simultaneous-source acquisition, which has recently been

used for vibroseis and marine acquisition, involves activating 2 to 6 sources simultaneously

to moderately increase efficiency, but data quality is degraded since the responses to the

multiple sources interfere. We present two new simultaneous-source approaches that exploit

random source waveforms to greatly increase the efficiency of seismic forward modeling (or

field acquisition) when model or field data possess sparseness. In each approach, the first

step is to measure the cumulative model (or Earth) response with all sources activated

simultaneously using either randomly scaled band-limited impulses or continuous band-

limited random-noise waveforms. In the second step, the response to each individual source

is separated from the cumulative receiver measurement by exploiting knowledge of the

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random source waveforms and the sparsity of the response in a suitable domain.

By invoking results recently developed in the field of compressive sensing, we provide

theoretical bounds on the maximum efficiency our approaches can achieve for almost noise-

free separation of individual source responses. Our approaches’ efficiency is bounded by the

sparsity of the responses; that is, our approaches need less modeling/acquisition time for

simpler or more structured model/Earth responses. We show with a simulated modeling

example that data collected simultaneously with as many as 8192 sources can be separated

into the 8192 individual source gathers with data quality comparable to that obtained when

the sources were activated sequentially.

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INTRODUCTION

The high cost of conventional seismic data acquisition, for both field data and simulated

model data, is due to the time required to sequentially record the Earth or model response

to each source. To record the entire response, each source activation is followed by a

listening time. When the delay between successive source activations is too small, their

responses interfere with each other. In the limit of simultaneous or nearly simultaneous

source activations, the interference noise or cross-talk between sources can be substantial

and cannot be ignored. Our goal in this paper is super-efficient acquisition or modeling with

the use of many (thousands) simultaneous sources followed by separation into individual

records with noise quality comparable to that of sequentially acquired data.

In this paper, we specifically address the efficiency of forward modeling, in which the

response of a source is calculated at a receiver placed arbitrarily in an acoustic or elastic

model. This desired response, termed the Green’s function, is typically a 5-dimensional

(5-D) quantity (2-D receiver location × 2-D source location × 1-D time) when receivers

and sources are constrained to lie on the surface of a 3-D model. Conventionally, the 5-D

Green’s function for a given model is computed by running a finite-difference (FD) setup

several times. During each sequential-source run, each source is activated with a band-

limited wavelet (for example, a Ricker wavelet). The response to the source wavelet is

computed at all receiver locations simultaneously by running FD modeling for some time,

say Tseq. For Ns sources, the total computational effort for the sequential-source method is

proportional to Ns ∗ Tseq.

The complexity of FD modeling makes the sequential-source approach computationally

expensive. In fact, it is currently impractical to model the elastic response of a realistic 3-D

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model using the sequential-source approach. Furthermore, for conventional full waveform

inversion (FWI), the Green’s function must be computed for many different models in

order to match the field data. Hence, large gains in modeling efficiency are desirable. In

addition, the efficiency of the sequential-source modeling approach is almost independent of

the complexity of the target Green’s function. We would rather have a modeling technique

whose computational expense scales with the complexity of the target Green’s function

(that is, one with fewer computations for structured Green’s functions).

We propose to perform seismic modeling (or field acquisition) by simultaneously acti-

vating all sources with different random noise waveforms. We describe two approaches that

differ in the choice of source waveforms. In the first approach, all sources emit different

randomly scaled (for example, with ±1) band-limited impulse waveforms. In this case, the

FD modeling is run Nsim < Ns times, with each run lasting for a duration Tseq. Therefore

the total FD modeling time is Tsim = Nsim ∗ Tseq. In the second approach, FD modeling is

performed just once with all sources activated simultaneously, but for a duration Tsim > Tseq

longer than each sequential-source modeling run. In this case, each source emits a different

band-limited random-noise waveform continuously throughout the modeling duration Tsim.

In both cases, the modeling or acquisition is more efficient because Tsim is less than Ns∗Tseq.

The second step in each of our approaches is to separate the interfering Green’s functions

from the many receiver measurements. We formulate this key separation step as a linear

inverse problem. In high-efficiency cases (our focus), the separation problem is clearly ill-

posed because the number of measured samples Tsim is less than the total samples Ns ∗ Tseq

associated with all of the separated Green’s functions. We propose to estimate the Green’s

function for each source-receiver pair by exploiting the sparsity (that is, structure) of the

Green’s function in a suitable transform domain [for example, the curvelet domain (Candes

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et al., 2006a)] in tandem with our knowledge of the random source waveforms. Our approach

draws inspiration from vibroseis acquisition in geophysics and from the recently developed

compressive sensing (CS) field (Candes et al., 2006c; Donoho, 2006). Neelamani et al. (2008)

presented a preliminary version of this paper for modeling, and Neelamani and Krohn (2008)

specifically addressed the vibroseis acquisition problem.

The computational efficiency of each of our proposed approaches scales with the sparsity

of the target Green’s function. The total computational effort for simultaneous-source

modeling or acquisition is proportional to Tsim, assuming that the computations necessary

to solve the separation problem are negligible compared to FD computations. We prove that

the duration Tsim necessary for satisfactory separation is controlled by the Green’s-function’s

sparsity. For cases in which the Green’s function has an extremely sparse representation,

the simultaneous-source method is significantly more efficient than the sequential-source

method (that is, Tsim << Ns ∗ Tseq). Our results obtained by employing 8192 simultaneous

sources substantiate our contention that the proposed simultaneous-source approaches to

modeling and field acquisition indeed holds promise.

RELATED WORK

The use of simultaneous sources for field acquisition is not new. It has been commonly

employed in vibroseis acquisition (Bagaini, 2006). Vibroseis-based simultaneous-source ac-

quisition conventionally uses multiple sweeps and phase rotations to suppress harmonic

noise and facilitate better separation into individual source gathers. For example, High

Fidelity Vibratory Seismic (HFVS) (Allen et al., 1998; Krohn and Johnson, 2006) solves

a well-posed linear inverse problem using least-squares inversion to separate the source

gathers. After separation, the interference noise level can be almost 60 dB lower than the

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signal. However, multiple sweeps (that are necessary to make the separation well-posed)

limit the efficiency of HFVS. Krohn et al. (2006) show that more efficient acquisition can

be achieved using a single cascaded sweep but the efficiency comes at the expense of in-

creased interference noise at later times in the separated source gathers. Recently, Sallas

et al. (2008) proposed activating simultaneous sources with pseudo-random waveforms, and

separating the individual gathers by cross-correlation. However, the gathers obtained by

cross-correlation would also suffer from high interference noise, especially as the number

of sources is increased (For example, see Figure 7c). Howe et al. (2008) proposed the use

of continuous vibroseis acquisition with several simultaneously operating sources, but their

proposal does not solve the problem of separating the source gathers.

More recently, simultaneous sources have received increased attention for reducing the

cost of obtaining wide-azimuth marine data (Beasley et al., 1998; Beasley, 2008; Berkhout,

2008). Moore et al. (2008) and Hampson et al. (2008) describe combining marine-source

gathers by introducing random time-shifts in the source waveform activations. Hampson

et al. (2008) advocate suppressing the interference noise from simultaneous sources by ex-

ploiting the incoherency of the interference in an aptly chosen domain, while Moore et al.

(2008) further extend the method by exploiting the known time-shifts. Huo et al. (2009)

use multi-directional vector-median filters to separate gathers obtained with randomly time-

shifted simultaneous sources. Akerberg et al. (2008) use Radon-domain sparsity to separate

simultaneously acquired source gathers. Spitz et al. (2008) focused on using dip and related

information to separate source gathers.

Ikelle (2007) proposed the use of simultaneous sources followed by separation for both

modeling and field acquisition but their work focused on estimating the unknown source

waveforms by exploiting principal/independent component analysis (PCA/ICA). Recently,

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Herrmann et al. (2009) also described a simultaneous-source approach that is similar to our

work. Their paper also improves forward modeling efficiency by employing random source

waveforms and sparsity. The difference is that their method is tailored for frequency-domain

forward modeling and employs random waveforms designed in the frequency domain. Fur-

thermore, in contrast to our method, their method is not suitable for field acquisition since

it operates in the frequency domain.

In contrast to some of the work cited above (Beasley et al., 1998; Beasley, 2008;

Berkhout, 2008; Moore et al., 2008; Hampson et al., 2008; Huo et al., 2009; Akerberg et al.,

2008; Spitz et al., 2008), our approach can separate thousands of sources rather than just a

few (ten or less) sources. When many sources are employed, we believe that simultaneous

sources must be activated with randomized waveforms, such as randomly scaled impulsive

or continuous-time waveforms, to facilitate optimal separation of the gathers. Furthermore,

unlike existing literature, we provide a rigorous framework (based on CS) to analyze the

performance limits of simultaneous-source techniques.

Our paper is focused on obtaining separated modeled or field gathers that can be input

to conventional imaging or inversion methods. Recently, several papers have proposed di-

rectly using the raw unseparated gathers that result from simultaneous-source acquisition.

Berkhout (2008) uses unseparated gathers obtained from employing simultaneous sources

with random time-shifts (termed source blending) as input to conventional migration, and

Berkhout et al. (2009) extend the approach to encompass receiver blending as well. Since

the migration step does not account for the interference between the different simultaneous

sources, the resulting migrated images are degraded by the interference noise. Dai and

Schuster (2009), Tang and Biondi (2009), and Verschuur and Berkhout (2009) demonstrate

lower levels of interference noise by employing least-squares migration on simultaneous-

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source gathers. Krebs et al. (2009a) and Krebs et al. (2009b) exploit simultaneous sources

activated with random waveforms (similar to our paper) to directly invert for model pa-

rameters from the unseparated gathers, and thereby speed up FWI.

MATHEMATICAL FORMULATION

We formulate all expressions in terms of forward modeling, but the same equations explain

field acquisition as well. In practice, the key advantage of applying our approach to forward

modeling over field acquisition is that the source signatures in forward modeling can be

controlled precisely.

Let G(si, rj , tk) denote the Green’s function for an arbitrary model with si and ri denot-

ing the source and receiver locations, respectively, and tk denoting discrete time samples.

Let u(tk)~Ψ(si, tk) represent a band-limited waveform emitted by a source placed at si, with

~ denoting convolution, u(tk) a benign band-limiting operator, and Ψ(si, tk) the full band-

width source waveform. Let R(rj, tk) denote the measurement made by a receiver placed at

rj . We use Gu(si, rj , tk) to denote the band-limited Green’s function G(si, rj , tk)~ u(tk).

Sequential sources

In the sequential-source scenario, with a single source placed at si, the receiver measures

Rsi(rj , tk) = Gu(si, rj , tk)~Ψ(si, tk) =∑

`

Gu(si, rj , t`) Ψ(si, tk−`), (1)

where tk ∈ [0, Tseq]. If Ψ(si, tk) is chosen to be an impulse function, then the receiver directly

measures Gu(si, rj , tk).

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Simultaneous sources

In the simultaneous-source scenario, sources at several shot locations si are activated si-

multaneously. The receivers measure the linear superposition of all of the Green’s functions

convolved with the respective source waveforms.

R(rj , tk) =∑

i,`

Gu(si, rj , t`)Ψ(si, tk−`),

=∑

i

Rsi(rj , tk). (2)

Impulsive-source case

In this case, we run FD modeling Nsim < Ns times, with each FD run lasting for a duration

Tseq. During each run (indexed by n), each source emits a different scaled band-limited

impulse

u(tk)~Ψ(si, tk) = ψ(n)(si) u (tk)~ δ(tk) = ψ(n)(si) u (tk),

with ψ(n)(si) denoting the scaling factors and δ(tk) denoting an impulse function. Each

receiver measures

R(n)(rj , tk) =∑

i

ψ(n)(si)Gu(si, rj , tk), (3)

which is simply a weighted linear combination of the Green’s functions. The FD modeling

is run for a total duration of Tsim = Nsim ∗ Tseq.

Continuous-source case

In this case, we run FD modeling only once continuously for a duration Tsim > Tseq, with

each source emitting a different band-limited waveform u(tk)~Ψ(si, tk), tk ∈ [0, Tsim]. Each

receiver measures one long trace R(rj , tk), as described in equation (2), with tk ∈ [0, Tsim].

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Problem

In both impulsive-source and continuous-source cases, the key problem is that the individual

band-limited Green’s function Gu(si, rj , tk) needs to be separated from the simultaneous-

source receiver measurements. Separating the Green’s functions from equation (2) is clearly

an ill-posed linear inverse problem because the number of measurements (Tsim) is less than

the number of inversion variables (Ns ∗ Tseq).

The Green’s-function separation problem encountered in both impulsive- and

continuous-source cases naturally raises three fundamental questions:

1. Measurement: What source waveforms Ψ’s would facilitate better separation?

2. Efficiency: What is the minimum number of samples (controlled by Tsim) needed to

accurately estimate the Green’s functions?

3. Algorithm: What algorithm can be used to efficiently separate the Green’s functions?

COMPRESSIVE SENSING (CS) OVERVIEW

The recently developed CS field contains theoretical insights that are useful in rigorously

answering the three questions raised in the previous section. This section distills the essential

results presented of Candes et al. (2006b), Candes et al. (2006c), Donoho (2006), and Candes

(2006). In-depth tutorials on the fundamentals of CS can be found in the IEEE Signal

Processing Magazine (2008) and Baraniuk (2007).∗

CS theory describes a general framework to sense and reconstruct signals that enjoy

sparse representations. In geophysics, the use of sparsity to solve ill-posed linear inverse

∗Also see www.dsp.rice.edu/cs for a comprehensive list of software and papers on this topic.

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problems is common. However, as is well known, the use of sparsity can yield unstable

results. The strength and contribution of CS is that it identifies a coherent measurement-

to-reconstruction framework that facilitates robust exploitation of sparsity.

Assume that we wish to sense a discrete-time signal g that is M samples long and that

we are allowed to measure the projections λk of g onto measurement vectors ψk of our

choice.

λk = 〈g, ψk〉 , (4)

where 〈., .〉 denote inner products. The measurement formulation in equation (4) generalizes

the concept of sampling. For example, if ψk are impulse functions in time, then λk are simply

different time samples of g. If ψk are complex exponentials (sinusoids), then λk comprise

the Fourier coefficients of g. In the modeling problem, as described below, ψk comprise all

time-shifts of a waveform (due to convolution).

The CS framework contains rigorous answers to the three questions (mentioned above)

that arise when one tries to reconstruct a signal g from linear measurements λk. Clearly, for

arbitrary g, standard linear algebra dictates that at least M measurements are necessary to

recover g exactly. However, if g is known to be sparse in an orthonormal transform domain

H, then we can use the CS methodology to accurately reconstruct g with measurements

that are far fewer than M . A signal g is said to be sparse in H if most of its energy is

captured by a few large coefficients of its transform-domain representation H(g).

CS theory states the following solutions to the three questions raised above:

1. Measurement: For sensing sparse signals, ψk can be chosen randomly from a basis

whose functions are maximally uncorrelated or incoherent with the basis function of

its sparsifying domain H. For example, impulse functions in time are maximally inco-

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herent with complex exponentials. Therefore, if g comprises a sparse set of sinusoids,

then g should be measured by randomly sampling it in the time domain.

Intuitively, while the information in g is concentrated in a few unknown H-domain

coefficients, the information gets uniformly distributed over all coefficients in the inco-

herent measurement domain. Therefore, the samples λk obtained by using incoherent

ψk contain equally important information about g, which in turn facilitate its accurate

reconstruction. (See (Romberg, 2008) for an in-depth discussion.)

Given a fixed basis H, it is easy to find measurement vectors ψk that are incoher-

ent with H. Random-noise functions can be chosen as ψk in equation (4) because

they are incoherent with all fixed basis functions with overwhelming probability (see

Candes and Tao (2006); Baraniuk et al. (2008) for a rigorous statement). Indepen-

dent Gaussian-distributed zero-mean random variables or uniformly distributed ±1

binary random variables are both good choices to construct ψk.†. These choices are

also universal in the sense that they provide robust reconstruction of signals that are

sparsely represented by any orthonormal transform H.

2. Algorithm: Computationally tractable algorithms (for example, convex optimization

methods) exist to accurately reconstruct g from the the measurements λk in equa-

tion (4). If the λk are noise-free, then g can be reconstructed by solving the following

linear-programming problem:

g = arg ming‖H(g)‖1, such that λk = 〈g, ψk〉 , for all k. (5)

In equation (5), ‖H(g)‖1 denotes the `1 norm of H(g). The `1 norm of a signal is the

†Many different types of random variables can be used instead of Gaussian as long as they have an

appropriate variance, and the support of the probability distribution is either finite or the tails decay at

least as fast as a Gaussian

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sum of its entry’s absolute values. The `1 norm provides a good measure of a signal’s

sparsity. Equation (5) conveys that we should seek a signal that not only satisfies the

measurements, but also has a sparse representation in the domain H.

If the λk are corrupted with noise, then g can be reconstructed by solving the following

convex-programming problem (a second-order cone-programming problem):

g = arg ming‖H(g)‖1, such that

∑

k

(λk − 〈g, ψk〉)2 ≤ ε2, (6)

where ε is determined by the noise contaminating λk. Equation (6) seeks a signal that

satisfies the measurements up to a noise tolerance factor and has a sparse representa-

tion in H.

3. Efficiency: The number of measurements necessary to reconstruct g from λk by solving

the linear-programming problem in equation (5) is predominantly controlled not by

the size of the signal but by it’s complexity (sparsity). Furthermore, the reconstruction

is stable even when λk are corrupted with noise. This assertion can be formalized as

follows (see Candes et al. (2006c) and Candes and Wakin (2008) for details).

Theorem 1. Assume that g is estimated by solving the convex-programming problem

in equation (6) from approximately K log(M) measurements λk that are obtained using

random-noise-based ψk. Let HM−K(g) denote the smallest M−K coefficients of H(g).

Then with overwhelming probability, the estimation error can be bounded as follows

‖g − g‖2 ≤ C0‖HM−K(g)‖1√

K+ C1ε, (7)

for some constants C0 and C1, and with ‖g−g‖2 denoting the `2 norm of the estimation

error.

In other words, the estimation error is essentially bounded by the `1 norm of the

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M −K smallest coefficients of H(g) and the noise level ε that corrupts λk. The first

term becomes negligible if g has a sparse representation in the domain H. The second

term assures stability in the presence of noise; it says that the measurement noise

does not get unduly amplified while reconstructing g.

Consider the case when H(g) has precisely K non-zero coefficients and that the λk

are noise-free (ε = 0). In this case, according to Theorem 1, g can be exactly re-

constructed from approximately K log(M) random measurements. Note that at least

K measurements are necessary to recover any K-sparse signal, even if the locations

of the K non-zero H(g) coefficients are known. Thus, with random measurement

vectors, the number of measurements necessary to reconstruct g is guaranteed to

be near-optimal (up to logM factors); no other combination of measurement and

reconstruction strategies can obtain a comparable reconstruction of a sparse g with

substantially fewer measurements! In fact, even when g is not K-sparse, and ε 6= 0, no

technique can substantially improve the estimation error in equation (7). (See Candes

and Wakin (2008) for an in-depth discussion.)

EFFICIENT SEISMIC FORWARD MODELING USING

SIMULTANEOUS SOURCES

We are now in a position to fine tune the two proposed simultaneous-source modeling

approaches and answer the three questions raised earlier by invoking CS.

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Approach 1: Random Impulsive Sources

In this case, recollect that we propose to activate all sources during modeling with scaled

band-limited impulses Ψ(si, tk−`) = ψ(n)(si) u (tk). This case resembles a simultaneous-

source marine acquisition scenario in which we allow the source locations to be revisited so

that multiple simultaneous-source gathers can be collected.

1. Measurement: The receiver measurements in equation (3) indeed conform to the CS

measurement setup in equation (4). That is,

R(n)(rj , tk) =∑

i

ψ(n)(si)Gu(si, rj , tk) =⟨Gu(si, rj , tk), ψ

(n)(si)⟩i, (8)

where the inner product operates along the source axes (denoted by 〈., .〉i). In words,

the receiver measurement at time tk is obtained by projecting Gu(si, rj , tk) onto the

measurement vector comprising the source scaling factors ψ(n)(si).

Invoking CS, during each FD run n, we employ scaling factors ψ(n)(si) that are

Gaussian-distributed, zero-mean, independent random variables or independent, uni-

formly distributed, ±1 binary random variables. Such a choice ensures that we are

compressively sampling each time slice of a common-receiver Green’s-function vol-

ume. (The common-receiver volume is the dataset obtained by holding rj constant

in Gu(si, rj , tk), and a common-receiver volume time slice is the dataset obtained by

holding both rj and tk constant in Gu(si, rj , tk)).

2. Algorithm: Assume that the transform domain H that sparsifies the band-limited

common-receiver Green’s-function volume is known. We propose estimating the

common-receiver volume from each receiver measurement by solving the following

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convex-programming problem:

Gu = arg minGu‖H (Gu)‖1 , such that

Nsim∑

n=1

(R(n)(rj , tk)−

∑

i

ψ(n)(si)Gu(si, rj , tk)

)2

≤ ε2 for each rj and tk, (9)

In equation (9), ε accounts for the presence of the noise (for example, computational

dispersion noise or round-off errors) in the receiver measurements.

3. Efficiency: For special choices of H, we can determine the total number of FD runs

Nsim necessary to extract an acceptable Gu by solving equation (9).

Theorem 2. Let H denote an orthonormal transform that operates only along a

Green’s-function’s source axes. Assume that Nsim receiver measurements R(n)(rj , tk)

are obtained using simultaneous randomly scaled band-limited impulses, and that the

total energy of the noise in R(n)(rj, tk), for any fixed rj and tk but across all n, is ε. Let

Ns denote the total number of sources. Assume that Gu is estimated from R(n)(rj , tk)

by solving equation (9). Then with overwhelming probability, the estimation error for

each common-receiver time slice (that is, for a given rj and tk) can be bounded as

follows:

∥∥∥Gu(si, rj , tk)− Gu(si, rj , tk)∥∥∥

2,j,k≤ C2

‖HNs−Ksim (Gu(si, rj , tk))‖1√Ksim

+ C3 ε, (10)

with C2 and C3 denoting constants. In inequality (10), Ksim = C4∗Nsimlog(Ns)

, with C4

denoting a constant, is almost as big as Nsim; HNs−Ksim (Gu(si, rj , tk)) denotes the

Ksim smallest coefficients of H (Gu(si, rj , tk)) at a given rj and tk; and


2,j,k=

(∑

i

(Gu(si, rj, tk)− Gu(si, rj , tk)

)2)0.5

.

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Theorem 2 is stated without proof; it is an immediate consequence of Theorem 1.

It formalizes a powerful, desirable property of the simultaneous impulsive-source ap-

proach. It guarantees that the estimation error measured at each time slice of the

common-receiver Green’s-function volume is bounded by the sum of two terms. The

first term measures the `1 norm (sparsity) of the Ksim smallest coefficients of that time

slice in the H domain. The second term ensures stability to noise. Thus, if all time

slices of the common-receiver Green’s-function volume are sparsely represented in the

domain H, and if the receiver measurements are noise-free, then the estimation error

becomes negligible for Nsim � Ns (implying modeling efficiency).

Since the total FD modeling run time Tsim = Nsim ∗ Tseq is controlled by the sparsity

of H (G(si, rj , tk)), the choice of an appropriate H is crucial to maximize efficiency.

Note that Theorem 2 bounds the estimation error only if we estimate each time slice

of the common-receiver Green’s-function volume separately and H only acts along the

source axes. In practice, the estimation error can be improved substantially by solving

for the entire Green’s function and employing an H that exploits sparsity of the entire

Green’s function by acting along the source, receiver, and time axes. We expect that

high-dimensional extensions of the curvelet transform (Candes et al., 2006a) make

suitable choices for H because they would provide sparse representations for typical

5-D Green’s functions.‡

Clearly, if G(si, rj , tk) is too complex, then no domain H would be able to provide a

sparse representation. In such a case, the modeling time Tsim necessary to perform

the separation would be very large (≈ Ns ∗ Tseq). Consequently, the simultaneous

‡Though the curvelet transform is not an orthonormal transform (it is a tight frame), it enjoys several

desirable properties of orthonormal transforms.

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impulsive-source approach (like all other methods), would be unable to provide much

efficiency improvement over the conventional sequential-source method. On the other

hand, there would be nothing lost; the method would simply take nearly the same

amount of time as the sequential-source method.

Approach 2: Continuous Random-Noise Sources

In this case, recollect that we proposed activating each source during modeling with a

continuous band-limited waveform u(tk)~Ψ(si, tk), tk ∈ [0, Tsim]. Note that in the impulsive-

source case, we randomly mix only across sources, but in this case we mix across both time

and sources. Such an approach resembles a vibroseis-based land acquisition scenario. We

show here that if the Green’s function is appropriately sparse, then Tsim can be reduced

considerably.

1. Measurement: In this case too, the receiver measurements (equation (2)) conform to

the CS measurement setup (equation (4)).

R(rj , tk) =∑

i,`

Gu(si, rj , t`)Ψ(si, tk−`) = 〈Gu(si, rj , tk),Ψ(si, tk−`)〉i,` , (11)

where the inner product operates along the source si and time-shift tk−` axes (denoted

by 〈., .〉i,`). From a CS perspective, the receiver measurement at time tk is obtained by

projecting Gu(si, rj , tk) onto a measurement vector comprising time-shifted versions

of all the source waveforms.

CS theory states that the measurement vectors should be independent random-noise

realizations. However, in this setup, the convolutional constraint placed by the mod-

eling setup prevents the collection of measurement vectors {Ψ(si, tk−`)}k in equa-

tion (11) from being statistically independent. To maximize the incoherence between

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the measurement vectors and the sparsity-domain basis functions, we propose choos-

ing Ψ(si, tk) as random Gaussian white noise vectors.§

2. Algorithm:

If the domain H that provides a sparse representation for Gu is known, then Gu can

be estimated by solving the following convex-programming problem:

Gu = arg minGu‖H (Gu) ‖1, such that

∑

k

(R(rj , tk)− 〈Gu(si, rj , tk),Ψ(si, tk−`)〉i,`

)2≤ ε2 for each rj . (12)

3. Efficiency: We cannot invoke established CS results to easily deduce the efficiency of

this approach because, as discussed above, the {Ψ(si, tk−`)}k cannot be statistically

independent. However, Romberg and Neelamani (2010) recently extended Theorem 1

to provide similar performance bounds (within log factors) for this approach as well.

Theorem 3. Let H denote an orthonormal transform that operates only along a

Green’s function’s Gu(si, rj , tk) source axes si. Assume that the receiver measurements

R(rj , tk) are obtained using simultaneous sources activated with independent white

Gaussian noise waveforms that are Ncont samples long. Further assume that the total

energy of the noise in R(rj, tk) is ε. Let Ns denote the total number of sources,

and Nseq the number of time samples in each trace of Gu(si, rj , tk). Assume that

Gu is estimated from R(n)(rj, tk) by solving equation (12). Then with overwhelming

probability, the estimation error for each common-receiver volume (that is, for fixed

rj) can be bounded as follows.


2,j≤ C5

∥∥HNs∗Nseq−Kcont (Gu(si, rj , tk))∥∥

1√Kcont

+ C6 ε, (13)

§Binary random-noise vectors may also be feasible, but we are currently unable to provide any perfor-

mance bounds for such vectors.

19

with C5, and C6 denoting constants. In inequality (13), Kcont = C7∗Ncont

log5(Ns∗Nseq), with C7

denoting a constant, is almost as big as Ncont, HNs∗Nseq−Kcont (Gu(si, rj , tk)) denotes

the Kcont smallest coefficients of Gu(si, rj , tk) at a given rj, and


2,j=

∑

i,k

(Gu(si, rj , tk)− Gu(si, rj, tk)

)2

0.5

.

Theorem 3 provides a guarantee similar to Theorem 2. It is stated without proof

since it is an immediate consequence of the main result in Romberg and Neelamani

(2010). It assures us that the estimation error for the entire common receiver Green’s-

function volume is bounded by the sum of two terms. The first term in the right side

of inequality (3) measures the `1 norm (sparsity) of the Kcont smallest coefficients of

common-receiver Green’s-function volume in the H domain, whereas the second term

ensures stability to noise. Therefore, if the common-receiver Green’s-function volume

is sparsely represented in the domain H, and if the receiver measurements are noise-

free, then the estimation error becomes negligible for Ncont � Ns ∗ Nseq (implying

modeling efficiency).

Again, the choice of H is crucial to minimizing Tsim, which is given by the product of

Ncont and the sampling interval. Note that Theorem 3 bounds the estimation error only

if we estimate each common-receiver Green’s-function volume separately and H acts

only along the source axes. Similar to the simultaneous impulsive-source approach, in

practice, the estimation error can be improved substantially by solving for the entire

Green’s function and employing an H, such as a high-dimensional extension of the

curvelet transform (Candes et al., 2006a), that exploits sparsity of the entire Green’s

function by acting along the source, receiver, and time axes.

20

RESULTS

We evaluate the efficacy of simultaneous-source modeling by using two models: a simple

model and a more complex 3-D model comprising diffractors and dipping reflectors. As

shown in Figure 1a, we place Ns = 8192 sources on a 128 × 64 rectangular grid, and

1 receiver close to the center of the source grid. The desired Green’s function is a 3-D

common-receiver volume (2-D source locations × 1-D time). Figures 2a and 2b illustrate

the true band-limited Green’s-function volumes obtained by sequential-source modeling for

the simple and the more complex model, respectively. In both cases, the Green’s-function

amplitudes are negligible after 1 second (Tseq = 1 second).

[Figure 1 about here.]


To simulate a simultaneous-source modeling setup, we convolve the Green’s function

associated with each source and the receiver with different band-limited random waveforms,

and then stack them (in accordance with equation (2)). Given the cumulative receiver

response and the random source signatures, we seek the Green’s function associated with

each source and the single receiver (a 3-D volume).

Table 1 summarizes the signal-to-noise ratios (SNRs) of all separation results

described below. The SNR of an estimate Gu is computed using the formula

10 log10

(‖Gu‖22/‖Gu − Gu‖22

). Table 1 confirms that as desired, the efficiency of both

impulsive- and continuous-source approaches scale with the complexity of the target Green’s

function. Furthermore, as expected, the SNR of the Green’s-function estimate improves

with increased modeling effort (Tsim).

21

[Table 1 about here.]

Approach 1: Random Impulsive Sources

Figures 3a and 3c illustrate sample band-limited source impulses scaled randomly by ±1

for two different modeling runs respectively. Each simulated modeling run uses a different

set of randomly scaled band-limited source impulses. The band-limiting frequencies were

8 − 55 Hz. Figures 3b and 3d displays the corresponding receiver measurements for each

run.


To estimate the simple Green’s function in Figure 2a, we simulated 256 modeling runs

(Nsim = 256). Therefore, Tsim = 256 seconds. Figure 4a shows the band-limited Green’s

function estimated by solving equation (9) using Berg and Friedlander (2007)’s SPGL1

method (also see Berg and Friedlander (2008)). The estimate not only matches the true

receiver measurements shown in Figure 2a up to 0.01% in energy terms but also possesses a

sparse 3-D curvelet representation (that is, H in equation (9) is the 3-D curvelet transform).

The difference plot Figure 4b confirms that our method captures most of the coherent

features in the actual Green’s function reliably (SNR = 10.3 dB).

For reference, Figure 4c shows the Green’s function estimated by a cross-correlation

approach. Each source gather is estimated by first cross-correlating each receiver measure-

ment with the corresponding random scalar, and then summing across all Nsim receiver

22

measurements as follows:

Gu,xcorr(sp, tk) =1

Nsim

Nsim∑

n=1

R(n)(rj , tk)ψ(n)(sp), (14)

= Gu(si, rj , tk) +1

Nsim

∑

i6=pGu(si, rj , tk)

(Nsim∑

n=1

ψ(n)(si)ψ(n)(sp)

).(15)

Such an approach would be feasible if the second term in equation (15) were negligible. The

second term can go to zero only whenNsim = Ns (that is, no efficiency) and when the random

scaling factors are chosen to be orthogonal across modeling runs. In the simultaneous-source

setup, since Nsim << Ns, the cross-terms cannot be neglected, as illustrated by the poor

result in Figure 4c.


In contrast to the simple Green’s-function case, for the more complex Green’s function

in Figure 2b, we need to increase the modeling time Tsim to 512 seconds to get an estimate

that is as good as that of Figure 4a. This increased modeling effort is to be expected because

the Green’s function enjoys a less sparse curvelet domain representation. See Figures 5a

and 5b for the estimated Green’s function and difference (between the true and estimated

Green’s function) plot respectively; the SNR for this result was 9.6 dB. Even with longer

Tsim, the cross-correlation based estimate (Figure 5c) is unsatisfactory.

For both the simple and the more complex cases, the sequential approach would take

a total of 8192 seconds to obtain the results of Figures 2a and 2b . Since Tsim = 256

and 512 seconds respectively for the simple and the more complex Green’s-function cases,

the simultaneous-source modeling effort would be less than the sequential-source effort by

a factor of 32 times and 16 times respectively (assuming that the separation step takes

negligible computations).

23


Approach 2: Continuous Random-Noise Sources

Figure 6a illustrates sample continuous band-limited random-noise source waveforms that

we employ during the experiment. We choose the source waveforms u(tk) ~Ψ(si, tk) such

that u(tk) is band-limited between 8 − 55 Hz and Ψ(si, tk) has a flat Fourier spectrum,

with phases chosen at random. The phase-encoded Ψ(si, tk) provide better separation than

white Gaussian noise. Figure 6b displays the receiver measurement.


The pattern of results is very similar to that for the random impulsive source above. See

Table 1 for the results summary. Figures 7a and 8a, respectively, depict estimates obtained

in the simple and more complex Green’s-function cases. These estimates result from solving

equation (12) with H set to the 3-D curvelet transform, and noise level set equal to 0.01%

in energy terms. The total modeling effort for the two cases are Tsim = 256 and 512 seconds

respectively.

Figures 7c and 8c shows the Green’s function estimated using cross-correlations. That

is,

Gu(sp, rj , tk) = R(rj , tk)~Ψ(sp,−tk)

=∑

i

Gu(si, rj , tk)~Ψ(si, tk)~Ψ(sp,−tk)

= Gu(sn, rj , tk) +∑

i6=pGu(si, rj , tk)~Ψ(si, tk)~Ψ(sp,−tk). (16)

In this approach too, the cross-terms due to Ψ(si, tk)~Ψ(sp,−tk) in equation (16) cannot

be neglected, as illustrated by the poor results in Figures 7c and 8c. Note that the cross-

24

correlation between two random functions is zero. However, the cross-correlations go to

zero only in the statistical sense (that is, averaged over several noise realizations).



DISCUSSION

We propose two simultaneous-source approaches that can dramatically increase the effi-

ciency of seismic forward modeling when the target Green’s function is structured (that

is, sparse in some known domain). The use of these approaches can also increase the effi-

ciency of field acquisition when the source signatures can be controlled appropriately. Both

approaches involve activating all sources simultaneously with random waveforms and mea-

suring the cumulative model response at the receiver. Invoking techniques from the recently

developed CS field, we then separate the individual source-to-receiver Green’s functions from

the cumulative receiver response by solving a convex-programming problem that exploits

the sparsity of the Green’s function. The two proposed approaches differ in their choices of

random source waveforms.

Both approaches are guaranteed to enjoy the following desirable property: the total

modeling time necessary to estimate the Green’s function accurately is predominantly con-

trolled by the complexity (sparsity) of the Green’s function and not by the number of

sources (as in sequential-source modeling). Thus, more structured Green’s functions can be

generated with less modeling effort. Our experimental results confirm that our approach

indeed is promising.

25

This paper ignored the computational expense of solving the separation problem asso-

ciated with the proposed approach. For extremely large problems (for example, when a

large number of sources are simultaneously activated with very long random waveforms),

the computational cost to solve the separation problem using conventional optimization

techniques may not be negligible, thereby adversely affecting the overall efficiency of the

proposed approach. According to Herrmann et al. (2009), the computational expense of

solving convex-programming problems is theoretically expected to be an order of magnitude

smaller than that of conventional FD modeling. Furthermore, motivated by the popular-

ity of CS, the convex-optimization community has recently made tremendous progress in

designing efficient algorithms to solve the types of convex-programming problems that we

encounter in the proposed approaches (see www.dsp.rice.edu/cs). Nevertheless, the practi-

cal issue of balancing the efficiency obtained by running multiple random sources simulta-

neously with the computations required to solve the associated separation problem needs

further investigation.

This paper also ignores our method’s ability to efficiently use all available processors

during an FD simulation. An FD simulation for one source location typically employs

an expanding computational grid around the source location. For parallel algorithms, this

may leave some processors idle until the wave-field reaches their domain-decomposed region.

In contrast, the simultaneous-source approaches would have a larger zone of active wave

propagation at early simulation times, thereby utilizing the processors more efficiently.

Neelamani and Krohn (2008) demonstrated substantial improvement in the quality of

vibroseis data by employing this paper’s sparsity-based separation algorithm. The appli-

cation of the proposed framework to further improve acquisition efficiency is an interesting

topic for future investigation. We envision not just a modest increase in acquisition effi-

26

ciency, but a major change in the way acquisition is performed. Instead of acquiring data

with a few strong sources, we could potentially acquire data with many weak impulsive

sources operated repeatedly or with many weak continuous sources that operate for a long

time. As the sources are operated more often or for longer times, the ratio of signal to

environmental noise would be enhanced along with improvements in the ratio of signal to

interference noise. We do require an estimate of the source waveforms. These can be es-

timated more accurately for marine sources (impulsive or vibrator sources) than for land

sources.

In conclusion, we show in this paper that simultaneous sources indeed have an important

role to play in seismic modeling, acquisition, and inversion.

27

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LIST OF FIGURES

1 Source and receiver geometry. We use 8192 (128 × 64) sources and 1 receiver.2 Desired band-limited Greens’s functions obtained by sequential-source modeling:

(a) Simple case and (b) More complex case.3 (a) and (c) Sample of 4 out of 8192 randomly scaled (±1) band-limited source

impulses for two different modeling runs. Nsim different modeling runs are conducted. (b)and (d) Corresponding receiver measurements for the two modeling runs.

4 (a) Simulation results for the simple Green’s function using the random impulsive-source approach. (b) Error when compared to the true Green’s function (Figure 2a)

5 Simulation results for the more complex Green’s function and the randomimpulsive-source approach

6 (a) Sample of 4 out of 8192 continuous band-limited random noise source wave-forms (512 seconds). (b) Corresponding receiver measurement (512 seconds).

7 Simulation results for the simple Green’s function and the continuous random-noise-source approach

8 Simulation results for the more complex Green’s function and the continuousrandom-noise-source approach

32

Figure 1: Source and receiver geometry. We use 8192 (128 × 64) sources and 1 receiver.

33

(a) Simple case. (b) More complex case.

Figure 2: Desired band-limited Greens’s functions obtained by sequential-source modeling:(a) Simple case and (b) More complex case.

34

(a)

(c)

(b)

(d)

Figure 3: (a) and (c) Sample of 4 out of 8192 randomly scaled (±1) band-limited sourceimpulses for two different modeling runs. Nsim different modeling runs are conducted. (b)and (d) Corresponding receiver measurements for the two modeling runs.

35

(a) Estimated(32x faster, SNR=10.3 dB).

(b) Estimation error(Figure 2a minus 4a)

(c) Cross-correlationestimate.

Figure 4: (a) Simulation results for the simple Green’s function using the random impulsive-source approach. (b) Error when compared to the true Green’s function (Figure 2a)

36


(b) Estimation error(Figure 2b minus 5(a))


Figure 5: Simulation results for the more complex Green’s function and the randomimpulsive-source approach

37

(a) (b)

Figure 6: (a) Sample of 4 out of 8192 continuous band-limited random noise source wave-forms (512 seconds). (b) Corresponding receiver measurement (512 seconds).

38


(b) Estimation error(Figure 2a minus 4a)


Figure 7: Simulation results for the simple Green’s function and the continuous random-noise-source approach

39


(b) Estimation error(Figure 2b minus 5a)


Figure 8: Simulation results for the more complex Green’s function and the continuousrandom-noise-source approach

40

LIST OF TABLES

1 Signal to noise ratios (SNRs) for different Tsim

41

Table 1: Signal to noise ratios (SNRs) for different Tsim

Simultaneous-Source Approach

ModelGreen’s function

Tsim (seconds) Efficiency SNR (dB)

Random (±1)impulsivesources

Simple256 32x 10.3512 16x 14.41024 8x 20.1

Complex256 32x 5.4512 16x 9.61024 8x 15.3

Continuousrandom noise

sources

Simple256 32x 10.6512 16x 13.81024 8x 20.3

Complex256 32x 7.5512 16x 11.31024 8x 17.6

42

e cient seismic forward modeling and acquisition using ......used for vibroseis and marine...

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