fast full-wavefield seismic inversion using encoded sources
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GEOPHYSICS, VOL. 74, NO. 6 �NOVEMBER-DECEMBER 2009�; P. WCC177–WCC188, 14 FIGS., 1 TABLE.10.1190/1.3230502
ast full-wavefield seismic inversion using encoded sources
erome R. Krebs1, John E. Anderson1, David Hinkley1, Ramesh Neelamani2, Sunwoong Lee1,natoly Baumstein1, and Martin-Daniel Lacasse3
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ABSTRACT
Full-wavefield seismic inversion �FWI� estimates a sub-surface elastic model by iteratively minimizing the differ-ence between observed and simulated data. This process isextremely computationally intensive, with a cost comparableto at least hundreds of prestack reverse-time depth migra-tions. When FWI is applied using explicit time-domain orfrequency-domain iterative-solver-based methods, the seis-mic simulations are performed for each seismic-source con-figuration individually. Therefore, the cost of FWI is propor-tional to the number of sources. We have found that the costof FWI for fixed-spread data can be significantly reduced byapplying it to data formed by encoding and summing datafrom individual sources. The encoding step forms a singlegather from many input source gathers. This gather repre-sents data that would have been acquired from a spatially dis-tributed set of sources operating simultaneously with differ-ent source signatures. The computational cost of FWI usingencoded simultaneous-source gathers is reduced by a factorroughly equal to the number of sources. Further, this efficien-cy is gained without significantly reducing the accuracy ofthe final inverted model. The efficiency gain depends on sub-surface complexity and seismic-acquisition parameters.There is potential for even larger improvements of process-ing speed.
INTRODUCTION
Full-wavefield seismic inversion �FWI� can potentially extracthe maximum amount of information from geophysical data in a
anner consistent with geological and geophysical constraints. FWIttempts to find an earth model �i.e., density, P-wave velocity,
Manuscript received by the Editor 29 December 2008; revised manuscriptished online 17 December 2009.
1ExxonMobil Upstream Research Company, Integrated Seismic [email protected]; [email protected]; sunw
2ExxonMobil Exploration Company, Houston, Texas, U.S.A. E-mail: ram3ExxonMobil Research and Engineering, Corporate Strategic Laboratory,2009 Society of Exploration Geophysicists.All rights reserved.
WCC17
-wave velocity, etc.� that best explains the measured seismic datand also satisfies known constraints. The method directly comparesimulated and measured seismic data. Consequently, it is very im-ortant to use a seismic simulator that reproduces the leading orderffects �e.g., elasticity, anisotropy� in seismic data. Computing evensingle shot gather over the full seismic band with such a degree ofccuracy could take many hours on today’s fastest computers.
Given the large number of model parameters estimated by FWI,he only practical optimization techniques are iterative gradientearch methods �Nocedal and Wright, 2006�. Gradient-based opti-ization techniques require evaluation of the objective function and
ts gradient with respect to model parameters. The number of seismicimulations needed to compute the objective function is equal to theumber of sources �Ns� needed to perform FWI. Most ongoing FWIesearch is based on the work of Tarantola �1984�, which shows thathe practical computation of the gradient can be achieved using thedjoint method for roughly the cost of 2Ns time-domain simulations.hus, 2D FWI is practical, but 3D elastic FWI is still far from practi-al — even on today’s fastest computers.
Most FWI methods are based on finite-difference simulators. Weategorize these simulators as source-configuration independentSCI� or source-configuration dependent �SCD�. SCI simulatorsave the advantage that most of their computational effort does notepend on the source configuration and therefore all source configu-ations can be computed for about the cost of one simulation. Onexample of SCI simulation is the frequency-domain direct-solverechnique, which simulates all sources essentially for the cost of oneomplex LU decomposition or matrix inversion and the cost of a ma-rix-vector multiply for each source �Marfurt, 1984; Pratt and Wor-hington, 1990�. To date, SCI simulators have only been practical forD and small-scale 3D calculations.
SCD simulators, on the other hand, can be run on large-scale 3Droblems at the cost of performing one simulation for each sourceonfiguration. Common SCD simulators include explicit time-do-ain methods �Tarantola, 1987� and iterative solver-based frequen-
d 16 April 2009; published online 3 December 2009; corrected version pub-
nversion, Houston, Texas, U.S.A. E-mail: [email protected];[email protected]; [email protected].
[email protected], New Jersey, U.S.A. E-mail: [email protected].
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y-domain methods �Erlanga et al., 2006; Operto et al., 2006; Riy-nti et al., 2006�. Irrespective of whether SCI or SCD solvers aresed, the many simulations required for FWI limits its 3D applica-ion to small-scale, low-frequency, or less accurate �acoustic� phys-cs.
Several authors demonstrate methods for reducing the computa-ional cost of FWI. Particularly noteworthy is work which demon-trates that frequency-domain FWI can yield very good results by in-erting only a few frequencies �Pratt, 1999; Sirgue and Pratt, 2004�.ther authors suggest that the efficiency of SCD-based FWI coulde increased by inverting coherent sums of sources �Berkhout, 1992;arner et al., 2008� or by inverting sums of widely spaced sources
Mora, 1987; Capdeville et al., 2005�.Another way to increase efficiency is by using incoherent sums.
ncoherent source sums have been used to increase the speed of sev-ral geophysical tasks. Incoherent sums are achieved in the field byring a number of sources simultaneously, each of which has a spa-
ially incoherent source signature. Incoherent sums also can beormed during processing by first encoding source gathers with a dif-erent code for each source and then summing traces with the sameeceiver location. Such methods have increased the efficiency ofeismic data acquisition �Neelamani and Krohn, 2008�, wave-equa-ion migration �Romero et al., 2000�, and seismic simulation �Ikelle,007; Neelamani et al., 2008�. Popular encoding methods includehase reversal, phase shifting, time shifting, and convolution withandom sequences �Romero et al., 2000�. Most methods that exploitncoherent source sums suffer from large amounts of noise resultingrom crosstalk between the encoded sources. This noise can only beitigated by limiting the number of sources summed, thus limiting
peed-up factors to an order of 10 �Romero et al., 2000�.In this paper, we demonstrate that a modified version of these in-
oherent summation methods can achieve large efficiency gains forWI without significant crosstalk noise. The key modification ishanging the incoherent sum between iterations of FWI. In our im-lementation, this change is made by altering the random-numbereed used to generate the source-encoding functions between itera-ions. The computational time needed to change the coding is insig-ificant compared to the time needed to evaluate the objective func-ion and its gradient. Changing the coding between iterations chang-s the crosstalk noise between iterations such that it is incoherentrom iteration to iteration. Thus, crosstalk noise essentially stacksut of the inverted earth model.
We call this method encoded simultaneous-source FWI �ESS-WI�. ESSFWI is limited to fixed-spread data �i.e., land or oceanottom cable �OBC��; only with modification can it be applied toarine streamer data. In effect, ESSFWI reduces the cost of an itera-
ion of FWI by a factor roughly proportional to Ns because only oneimulation is needed to compute the data corresponding to the simul-aneous encoded sources by linearity of the wave equation.
During testing, we made the following observations, which weemonstrate in the examples that follow:
Short codes show greater efficiency gain than long codes.ESSFWI is more sensitive to ambient noise in the measured datathan is FWI.The efficiency gain of ESSFWI is relatively insensitive to the ac-curacy of the starting model.
e begin with a theoretical discussion of ESSFWI and then demon-trate the value and sensitivity of ESSFWI by comparing applica-
ions of FWI and ESSFWI to 2D synthetic test data. Details of the ap-lication of ESSFWI and of the model and data used in the inversionests are also included.
THEORY
The mathematical formulation of FWI can be stated generally as
minc
h�u�c�,c� such that F�u,c��0, �1�
here h is the objective function, F�u,c� is the wave equation �e.g.,�u,c���2u� �1 /c2��� 2u /� t2��s, where s is the source func-
ion�, c is the subsurface viscoelastic property model �e.g., densitynd coefficient of stiffness C33. . .�, and u�c� is the simulated seismicavefield.The simplest objective function commonly used for FWI is the
east-squares objective function:
h�u�c�,c��1
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here the sum over time samples and receivers is implied by theorm and where u�c,sn� is the simulated wavefield for source sn andodel c, sn is a source function �for simplicity, we assume that all
ource functions have the same number of time samples Ts�, dn iseasured seismic data for source sn, and Ns is the number of source
athers in the seismic survey.FWI could use more sophisticated objective functions that in-
lude regularization terms and weighting. Also, there is no assump-ion that the sources sn are point sources. They could be more generalource types such as plane waves. In FWI, the solution of equation 1s usually approximated by iteratively minimizing h using a gradientearch.
A typical flow for gradient-search FWI is shown in Figure 1a. Werst compute the objective function value and its gradient using an
nitial model, the measured data, and estimated or measured sourceignatures. The gradient then determines a search direction �e.g., byonjugate gradients; Nocedal and Wright, 2006�, and the initial mod-l is perturbed in a direction determined by the search direction �e.g.,y line search; Nocedal and Wright, 2006�. The objective functionnd its gradient are then evaluated at the perturbed model, and thesealues are checked to determine if some search criteria are satisfiede.g., Wolfe conditions; Nocedal and Wright, 2006�. If the search cri-eria are not satisfied, then the search process is iterated until theearch conditions are satisfied. At this point, convergence criteria,uch as the objective function value, are checked. If they are not sat-sfied, then the process is iterated by computing a new search direc-ion using the perturbed model as the initial model.
For ESSFWI, we replace the objective function in equation 2 with
h�u�c�,c��� �n�1
Ns
en � u�c,sn�� �n�1
Ns
en � dn�2
, �3�
here en is the encoding sequence in time and � represents convolu-ion with respect to time. For simplicity, we assume that all encodingequences en have the same number of time samples Tc. Note that, ineneral, en�em for n�m.
Because u is a linear function of the sources, equation 3 can be re-ritten as
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. �4�
quation 4 is only appropriate if receivers are fixed and all receiversecord data from all sources. Although this is true for many types ofand and ocean-bottom acquisition, it is not true for marine-streamercquisition.
Equation 4 has the advantage that only one seismic simulation iseeded to compute u, as opposed to the Ns simulations needed for theonventional FWI objective function in equation 2. This is becausehe first term of equation 4 is computed by running the simulator oneime with all sources acting simultaneously and with each source in-ecting its encoded signature into the model. On the other hand, theew encoded source signatures have Ts�Tc�1 time samples.herefore, to evaluate equation 4, the simulator must run for Tc more
ime steps than are necessary in equation 2. The computational effi-iency gain � eval for evaluating equation 4 is then given by
� eval�NsTs
�Ts�Tc�1�. �5�
mportantly, equation 5 also gives the computational efficiency gainor evaluating gradients.
The ESSFWI objective function can be evalu-ted very efficiently. However, inversion usinghis objective function is more efficient only if thefficiency gain in objective function evaluation isot offset by a reduced convergence rate relativeo the FWI objective function. This can be sum-
arized by the following equation for the overallfficiency gain � resulting from using ESSFWI:
� �� conv� eval�� convNsTs
�Ts�Tc�1�,
�6�
here � conv is the convergence efficiency of ESS-WI relative to conventional FWI. Convergencefficiency can be evaluated by first determininghe number of objective function and gradientvaluations needed to achieve some level of mod-l fit for FWI and ESSFWI. The convergence effi-iency � conv is then the number of FWI evalua-ions divided by the number of ESSFWI evalua-ions. An important aspect of our implementationf ESSFWI is that we change the encoding func-ions between iterations to increase convergencefficiency greatly.
Encoded simultaneous-source methods haveeen proposed to improve the efficiency of seis-ic acquisition and processing by acquiring data
rom simultaneous sources and then processinghose data to separate them into the data thatould have been acquired from individual sourc-
s �Romero et al., 2000; Ikelle, 2007; Neelamanit al., 2008b�. The efficiency factor of encoded si-ultaneous-source separation, which can be
osed as a linear optimization problem, can beigorously analyzed and predicted by invokingesults from the area of compressed sensing in
Directioniteration
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ignal processing �Hennenfent and Hermann, 2008; Neelamani etl., 2008�. In contrast, ESSFWI is a nonlinear optimization problem.he existing compressed-sensing framework is not directly applica-le to predicting the efficiency limits of such nonlinear problems andontains no concept of changing encoding between iterations. Here,e provide some physical understanding of why ESSFWI greatly
ncreases computational efficiency.Incoherently encoded gathers contain much more information to
onstrain the inversion model than do a single point-source gather orcoherently encoded gather �e.g., a plane-wave gather�. In particu-
ar, an incoherently encoded gather illuminates more of the modelhan a point-source gather. On the other hand, an incoherently en-oded gather has a much broader spectrum of wave-propagation di-ections �i.e., a much broader f-k spectrum� than a coherently encod-d gather. This broad range of propagation directions is critical foronstraining the large-scale spatial variations in the model.
Although an incoherently encoded gather contains much more in-ormation than a single point-source gather, it does not contain asuch information as a distributed set of point-source gathers. Thus,e speculate that the dimensionality of the null space for the encod-
d simultaneous-source gather is larger than that for its parent set ofoint-source gathers. This may explain the artifacts in the invertedodel. However, these artifacts are dependent on the particular en-
Measured dataand sourcesignatures
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wcharts for �a� gradient-search FWI and �b� ESSFWI. Note the op-the data and signatures between iterations of ESSFWI.
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oding applied to make the simultaneous-source gather. Changinghe encoding between iterations produces artifacts that do not addoherently and are therefore suppressed.
A high-level flowchart for our implementation of ESSFWI ishown in Figure 1b. The only difference from the flowchart for FWIFigure 1a� is that the measured data and source signatures are en-oded before the objective function and gradient evaluation. Fur-hermore, the data and source signatures are reencoded using differ-nt encoding sequences between iterations. The encoded data coulde acquired in the field as opposed to encoding by processing, ashown in Figure 1b. If encoded data are measured directly during ac-uisition, then implementation of code changes between iterationsequires acquisition of several realizations of the encoded data. Thisould increase acquisition efficiency if the number of encoding real-zations required is less than the number of sources. This area needsurther investigation.
As we move to 3D simulations and inversions, there is an addi-ional computational advantage of the multisource feature of ESS-WI. A single-source time-domain forward simulation typicallyains efficiency by using an expanding computational grid centeredbout the source location — one that grows with simulation time andperates only on grid points with nonzero wavefields. A multisourceimulation may not need to honor this implementation detail with theame level of attention because sources are activated in more loca-ions and an active wavefield is propagated to all grid locations moreuickly.
For large 3D inversions, forward simulation on a parallel comput-r usually is implemented via a domain decomposition of the modelnd wavefield across computer nodes. The parallel implementationequires clever coding for changing the domain decomposition withimulation time to obtain full advantage of an expanding grid for aingle source location: otherwise, some computer nodes will sit idle,aiting for the wavefield to reach their part of the model domain.his can be inefficient for large 3D simulation models that must bepread across many distributed-memory nodes. However, ESSFWIaturally puts sources on many parallel nodes, yielding a more effi-ient parallel forward simulation where all nodes are active even atarly simulation times and without the need to address some of theoftware complexities associated with expanding computationalrids.
Along with increased compute efficiency, ESSFWI also greatlyeduces input/output �I/O� and memory requirements. This is be-ause the size of the measured data is reduced by a factor roughlyqual to Ns. These additional efficiency gains were not factored intour results.
METHODS
eismic simulator
All computations were performed using a 2D finite-differenceime-domain acoustic simulator. The simulator is second order inime and fourteenth order spatially. Perfectly matched layer bound-ry conditions �Marcinkovich and Olsen, 2003� were used at theides and bottom of the model. A realistic free-surface boundaryondition was used at the top of the model, so our inversions use notnly primary reflections and interbed multiples but also surface-re-ated multiples. Of course, direct arrivals help to constrain these in-ersions.
Simulations and gradient computations were performed on aodel that was padded by seven cells at the top for the free-surface
oundary and by 60 cells on the sides and bottom. The padding wasdded for the boundary condition computations and replicated thedge values of the model.
ncoding
The measured data were encoded by convolving all traces in eachource gather with the code for that source gather. The encodedource gathers were then stacked over receivers to form one simulta-eous encoded gather. The source signatures also were convolvedith their individual encoding functions. When calculating objec-
ive functions and gradients, the encoded signatures were injected si-ultaneously into a single run of the simulator.We tested several encoding schemes but found that random phase
ncoding �Romero et al., 2000� provided the best convergence rates.e normalized the random phase codes so they all had the same total
ower. Then we tested several lengths for the encoding function andound that a code with only one sample in time gave the most effi-ient inversion. Note that using a normalized random phase codeith only one sample is equivalent to randomly multiplying the shotathers by �1 or �1.
We changed the value of en �equation 4� between iterations byhanging the seed of the uniform random-number generator used toenerate the random phase code. This choice of an optimal encodingethod is very different from that reported in previous work on in-
reasing the speed of wave-equation migration �WEM� by applyingt to incoherent sums of sources �Romero et al., 2000�. Romero et al.2000� show that the optimal encoding method to increase the speedf WEM is to use random phase codes with the same number of sam-les as the data. We found that a code that is one time sample longielded the largest efficiency gains. Note that short codes imply larg-r values of the Ts / �Ts�Tc�1� factor of equation 6, yielding largerverall efficiency gains. When Tc�1, this factor achieves the maxi-um value of one.Because ESSFWI involves random encoding, the inverted result
an depend on the encoding functions randomly chosen during an in-ersion run. In particular, because we generate our encoding func-ions using a random-number generator, the inversion result can de-end on the seed provided to the random-number generator. There-ore, in all of the ESSFWI examples below, we ran five test inver-ions using different seeds for the random-number generator. To beonservative in our conclusions, all ESSFWI inversion resultshown below correspond to the test that gave the largest error be-ween the inverted model and the true model. When ESSFWI con-erges, there are only very subtle differences between inversions us-ng different random-number generator seeds.
nversion method
Following the FWI and ESSFWI algorithms shown in Figure 1,e computed the gradient of the objective function using the adjoint-
tate method �Tarantola, 1984�. The search direction was computedsing the Hestenes-Stiefel conjugate gradient algorithm �Nocedalnd Wright, 2006�. The search direction was multiplied by thequare root of depth to partially account for spherical divergence.
We used a line-search method to find an updated model. The lineearch evaluated the objective function for five models that wereroduced by adding uniformly scaled versions of the search direc-ion to the current model. We then selected the search model thatave the lowest objective-function value as the updated model of theurrent iteration. Finally, we adjusted the line-search scale such that
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he picked model would have been in the center of the line search.his updated scale was used to perform the line search during theext iteration.
To simplify our comparisons, we limited our testing to the firsti.e., lowest frequency� stage of a multiscale inversion �Bunks et al.,995�. We hypothesize that ESSFWI will be at least as effective inater stages of multiscale inversion as it is in the first stage, but veri-ying this is the subject of future research.
TEST DATA
odel
For simplicity, all examples are 2D constant-density acoustic in-ersions of a modified Marmousi II model �Martin, 2004; see Figurea�, with density set equal to one and S-wave velocity to zero. Theodel was first subsampled from 3400 � 700 �horizontal � vertical�
ells at 5-m cell size to 850 � 175 cells at 20-m cell size. The top 400of the model was then stripped off, which is equivalent to remov-
ng most of the water layer. This left a model with 850 � 155 cells.inally, a variable near-surface interval was added to the top 100 mf the model to simulate an earth that would require application oftatic corrections. The near-surface perturbation was created bymoothing a random model using a smoother measuring 200 � 60. The velocity perturbation range of this model was �500 to 0 m/s.hese modifications make the model more challenging for inversionecause they introduce significant errors near the sources and receiv-rs in the starting model.
In our testing, the initial model was created by smoothing the trueodel, but excluding the near-surface perturbation. The smoothingas performed in the slowness domain to preserve traveltimes. The
ize of the smoothing operator was increased from 0.5 � 1.0 kmvertical � horizontal� at zero depth to 2.0 � 4.0 km at the bottom ofhe model. The size of the smoothing operator was chosen to be the
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igure 2. True and initial models used in FWI and ESSFWI tests. �a�-wave velocity of the modified version of the Marmousi II model;olor scale is absolute velocity. �b� Initial model used for most testsn this paper; color scale is absolute velocity. �c� True model and �d�nitial model; color scale is relative velocity with respect to a verymooth background model.
argest that would produce good convergence given the signal-to-oise ratio �S/N� of the low frequencies in the measured data. Thisnitial model is displayed in Figure 2b.
To improve the level of detail of the inversion images, we sub-racted a very smooth background model from the remaining dis-lays �see Figure 2c and d�. The background model is the modelhown in Figure 2a, smoothed with a 2�106 km �vertical � hori-ontal� smoothing operator, yielding a model that is almost laterallyonstant and vertically very smooth. This background model is usedor display purposes only and is not used in the inversion calcula-ions.
cquisition parameters
The data to be inverted in these examples were generated usinghe same simulator that was used to perform the inversion.Auniformxed-spread geometry was simulated with parameters in Table 1.ote that a fixed-spread acquisition geometry, such as used to ac-uire these data, is necessary for ESSFWI. Also, the source and re-eiver spreads cover the entire model, giving a maximum offset of6,900 m. An 8-s trace length was chosen so that direct-arrival ener-y was recorded at the longest offset. Because we do not invert veryow frequencies �see below�, these long offsets are necessary to re-over the longest-wavelength parts of the model, largely from infor-ation in the direct arrivals. The cost of conventional FWI is propor-
ional to the number of sources inverted, so fair efficiency compari-ons require that conventional FWI be run with as few sources asossible. For conventional FWI in inversion tests run at variousource intervals, we found that only 50 sources �340-m source inter-al� were necessary to avoid operator aliasing artifacts in the final in-erted model. Therefore, all tests were performed with 50 sources at340-m interval. This unusually large source interval is valid for two
easons. First, all of our tests were performed on low-frequency datapeak frequency of roughly 5 Hz�. Second, although there are signif-cant operator aliasing artifacts in the early iterations of FWI, thesertifacts are greatly reduced by subsequent iterations and are insig-ificant in the final inverted model.
oise
If multiscale techniques �Bunks et al., 1995� are used in FWI tovoid local minima, then the measured data must have high S/N atery low frequencies �i.e., below 5 Hz� or the initial model must ac-urately predict seismic traveltimes. If low-frequency S/N is high, aess accurate initial model can be tolerated. Conversely, if low-fre-uency S/N is low, then a more accurate starting model is required.
able 1. Parameters for the uniform fixed-spread geometry.
arameter Measurement
umber of receivers 850
eceiver interval 20 m
eceiver depth 50 m
umber of sources 50
ource interval 340 m
ource depth 30 m
ource wavelet 15-Hz Ricker wavelet centered at 1 s
race length 8 s
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herefore, realistic testing of FWI and ESSFWI on synthetic data re-uires that ambient noise be added to the measured data, especiallyt low frequencies.
Another reason for testing in the presence of noise is that ESSFWIay be more sensitive to ambient noise than is conventional FWI.he S/N of the inverted model from conventional FWI should in-rease with the fold of the input seismic data. On the other hand, ES-FWI may not achieve this scaling because encoding randomizes
he signal before summation of the shot gathers. Therefore, the sig-al and noise will increase roughly by the square root of the numberf sources summed into an encoded simultaneous-source �ESS�ather. Thus, the S/N of an ESS gather will be close to the S/N of onef the individual source gathers summed into the ESS gather. Inver-ion of an ESS gather, at least if the coding is not changed between it-rations, should yield a result having the S/N reduced relative to thatf a conventional inversion by roughly the square root of the numberf sources.
In the following examples, white random noise was added to theeasured data before encoding or inversion. For all of the examples,
xcept those investigating sensitivity to noise, the noise level washosen to mimic the lower-frequency portion of the S/N spectrumwith respect to ambient noise� of an OBC seismic survey from theulf of Mexico �we attempted to match the low-frequency end of the
ed curve to the black curve in Figure 3a�. We are not concerned withource-generated noises because our goal is to treat all source-gener-ted noise as signal that can better constrain the inverted model.
reprocessing
After adding white random noise to the data, the S/N below 5 Hznd above 40 Hz was very low. We applied a Butterworth bandpasslter with high and low cuts of 5 Hz and high and low slopes of 36
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igure 3. Synthetic data for a shot near the center of the model. �a�omparison of �S�N� /N spectra �with respect to ambient noise� offield gather from a Gulf of Mexico OBC survey and the syntheticata with added noise. �b� Spectra of the synthetic data before and af-er filtering. �c� Unfiltered and �d� filtered synthetic shot record withoise added. Inversion tests were performed with the data shown ind� but with traces muted above the first arrival.
B/octave to the seismic data and the source signatures �Figure 3b�.he 5-Hz low-cut frequency was chosen to reduce the weight of veryoisy low-frequency components in the inversion. The 5-Hz high-ut frequency was chosen to mitigate cycle-skipping problems in thenversion �Bunks et al., 1995�. Therefore, these examples representnly the initial phase of inversion. Later phases had higher high-cutrequencies. The only other preprocessing applied to the measuredata was to mute the ambient noise preceding the first arrival.
Figure 3c displays a shot gather from the measured data after add-ng random noise, and Figure 3d shows the same gather after apply-ng these preprocessing steps, with the exception of muting. Figure 4hows one realization �i.e., corresponding to a particular seed for theandom-number generator used to make the code� of the encodedource signatures and seismic data with noise for a code length ofne.
RESULTS
The first two examples provided below demonstrate the computa-ional advantages of ESSFWI. The remaining examples examine theensitivity of ESSFWI to code length, ambient noise level, and accu-acy of the initial model.
xample 1: ESSFWI efficiency demonstration
Here, we demonstrate the computational efficiency of ESSFWIsee equation 6�. FWI and ESSFWI were performed using the mea-
a)
b)
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ured data and starting model discussed earlier. The only differenceetween these tests was whether the ESS method was used. The ES-FWI inversion was performed with a code length of one. Figure 5ompares FWI and ESSFWI at iteration 100. Figure 6 shows theame comparison after iteration 5. Figure 7 compares data fits of thenitial iteration and iteration 100.
a)
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∆V
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igure 5. �a� FWI and �b� ESSFWI inversions after 100 iterations.he two models are very similar. �c� Vertical profiles from the centerf the true, initial, and inverted models after 100 iterations.
Figure 8a shows the data fit �the objective-function value dividedy the L2-norm of the measured data� versus iteration for FWI andSSFWI corresponding to the inversions shown in Figure 5. Be-ause this is a synthetic data inversion test, we can display the modelt �square root of the L2-norm of the difference between the true and
nverted models� versus iteration �Figure 8b�.
xample 2: Importance of encoding and changingodes
We performed two other tests to highlight the importance to ESS-WI of changing the encoding between iterations and using differentodes for different sources. Figure 9a shows an ESSFWI invertedodel at iteration 100 when the encoding is not changed between it-
rations. Figure 9b shows the ESSFWI inverted model at iteration00 when the encoding is the same for all sources. The model fit ver-us iteration for these two tests is displayed in Figure 9c.
xample 3: Sensitivity to code length
We tried several types of encoding and found that random phasencoding performed as well as or better than any other type. Romerot al. �2000� reach a similar conclusion for applying source encodingo WEM. In random phase encoding, the code has a uniform ampli-ude spectrum but a random phase spectrum. We expected that theonvergence rate of ESSFWI would depend on the length �numberf samples� of the random phase code.
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igure 6. �a� FWI and �b� ESSFWI after five iterations.
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Recall that efficiency gain decreases with code length �equation�. However, if longer codes result in faster convergence per itera-ion �increasing � conv in equation 6�, then long codes may be betterhan short codes. We expected long codes would converge faster be-ause they are more orthogonal than short codes. We therefore antic-pate that these pseudo-orthogonal codes might produce smaller arti-acts as a result of crosstalk between encoded sources. By pseudo-or-hogonal, we mean that the crosscorrelation between two differentodes and the autocorrelation of the codes at nonzero lag are muchmaller than the zero-lag autocorrelation. Figure 10 compares con-ergence rates for ESSFWI for three choices of code length.
xample 4: Sensitivity to ambient noise
As discussed, we believe that ESSFWI is more sensitive to ambi-nt noise levels than conventional FWI because the S/N of the en-oded simultaneous-source gathers is approximately the same ashat of a single-source gather. Our previous tests were performed onata with an S/N corresponding to that of an OBC survey in the Gulff Mexico. Figure 11 shows the measured data and ESSFWI modelsor ambient noise levels 32 and 64 times higher than the referenceulf of Mexico data. Figure 12 compares convergence rates for
hese two ambient noise levels. In Figure 12, a noise level equal tone corresponds to the ambient noise level of the reference Gulf ofexico survey.
xample 5: Sensitivity to initial modelFWI is strongly sensitive to the starting model. It is therefore im-
ortant to determine whether the sensitivity of ESSFWI is similar tohat of FWI.
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igure 7. Data residuals of �a� FWI after iteration 1 and �b� FWI afterteration 100. �c� ESSFWI after iteration 1 and �d� ESSFWI after it-ration 100. Residuals for both iterations are displayed with theame amplitude scale.
We compared ESSFWI and FWI inversions for two starting mod-ls that differ from that used for the earlier tests �Figure 2d�. One ofhese was significantly closer to the true model, and the other wasignificantly farther from the true model �top panels, Figure 13�. TheWI and ESSFWI models yielded at iteration 100 by these startingodels are shown in the middle and bottom panels of Figure 13. Fig-
re 14 compares convergence rates for the three starting models. Inigure 14, the starting model from Figure 2d is labeled Intermediateccuracy.As was the case for the starting model of Figure 2d, these initialodels were created by smoothing the true model in the slowness
omain and excluding near-surface perturbation. The smoother usedo create the “more accurate” starting model increased in size from.25 � 0.5 km �vertical � horizontal� at zero depth to 1.0 � 2.0 kmt the bottom of the model. The smoother used to make the “less ac-urate” starting model increased from 1.0 � 2.0 km �vertical � hori-ontal� at zero depth to 4.0 � 8.0 km at the bottom of the model.
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igure 8. Comparison of measured data and model convergence ver-us iteration for FWI and ESSFWI. The data fit comparison �a�hows the value of the objective function �normalized by the energyn the measured data� versus iteration. The model fit comparison �b�hows the square root of the L2-norm of the difference between thenverted and true models as a function of iteration. The error bars onhe ESSFWI curve indicate the range of model fits for five separateuns of ESSFWI using different random-number seeds to generateandom phase codes.
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DISCUSSION
As shown by Figure 5, ESSFWI yields an inverted image that isery close to that produced by FWI. Both images are close to the trueodel �Figure 2c�, but neither resolves the smallest features of the
rue model because of the limited bandwidth of the measured data.oth inversions are considerably less accurate below 2.5 km. Somef this inaccuracy can be explained by poor illumination of the steep
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igure 9. Comparison of �a� ESSFWI run with encoded sourcesithout changing the code between iterations and �b� ESSFWI runithout encoding the sources before summation. �c� Model conver-ence rates versus iteration for tests compared with those from FWInd ESSFWI �from Figure 8b�. The error bars on the ESSFWI curvendicate the range of model fits for five separate runs of ESSFWI us-ng different random-number seeds to generate the random phaseodes.
ips in that region. However, the inaccuracy of the inversion near theenter of the model is more likely caused by inaccuracy of the start-ng model.
The main difference between conventional FWI and ESSFWI im-ges is the presence of low-amplitude, short-wavelength noise in theSSFWI images. This noise is caused by crosstalk between the en-oded sources, and it is much stronger in early iterations �see Figure�. We do not believe that this noise would significantly affect geo-ogic interpretation of the ESSFWI image. Furthermore, this noise
ight be mitigated by imposing smoothness constraints during in-ersion. Note that some signal remains in the iteration 100 data re-idual �Figure 7�, indicating that further iteration might improve theWI and ESSFWI images.
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igure 10. Model convergence rates for ESSFWI runs with randomhase encoding functions of different lengths. A code length of oneas used in the previous tests �Figures 4–9�. The error bars indicate
he range of model fits for five separate runs of ESSFWI using differ-nt random-number seeds to generate the random phase codes.
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igure 11. ESSFWI runs for ambient noise levels 32 and 64 timeshat of the previous tests �Figures 4–9�. �a, b� Shot records �afterandpass filtering but before encoding� from the input data for ESS-WI. �c, d� The resultant ESSFWI inverted models. For noise levelt 32�, the iteration 100 model is shown. For noise level at 64�, ES-FWI diverged, so the last iteration that yielded a model that did notause the simulator to fail �iteration 22� is shown.
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The data fit for FWI is a monotonically decreasing function of it-ration �Figure 8a�. In contrast, the objective function for ESSFWIoes not decrease monotonically �Figure 8a�. We believe this isaused by our modification of the encoding functions between itera-ions, implying that the objective function changes from one itera-ion to the next. The model convergence �as opposed to the data fit�or ESSFWI is smoother than that for FWI �Figure 8b�. Of more im-ortance, the ESSFWI and FWI model-convergence curves areearly equivalent, implying that � conv in equation 6 is approximatelyqual to one. Because Tc�1 for this example, equation 6 yields anfficiency gain � �Ns�50.
Comparing Figure 5b �ESSFWI changing encoding between iter-tions� and Figure 9a �ESSFWI without changing encoding betweenterations� clearly demonstrates the value of changing the encodingetween iterations when performing ESSFWI. The high level ofoise in Figure 9a is the result of crosstalk between the encodedources in the encoded source gather. Randomly changing the codeetween iterations randomly changes this crosstalk noise. Thus, theoise does not sum constructively in Figure 5b, resulting in a moreccurate inversion.
Comparing Figure 9b with Figure 5b demonstrates the impor-ance of phase encoding in ESSFWI. For the ESSFWI inversion ofigure 9b, the source gathers were not phase encoded, implying that
he “encoded” source approximates a vertically traveling plane-ave source. Because a vertically traveling plane wave propagatesver a very limited angular range in the subsurface, such data poorlyonstrain the large-scale variations of the model. To mitigate thisroblem, we tried plane-wave inversions for which the propagationirection of the plane wave varied between iterations �not shownere� to better constrain the background model. These plane-wavenversions perform poorly when compared with inversion of a verti-ally traveling plane wave and even diverge in some cases.As showny Figure 9c, the convergence rate of ESSFWI performed withoutncoding or without changing codes between iterations is very poor.
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igure 12. Sensitivity of ESSFWI model convergence to the level ofmbient noise added to the measured data.Anoise level of 1� corre-ponds to a S/N spectrum similar to that of the OBC field data set ac-uired in the Gulf of Mexico that was used in the tests shown in Fig-res 4–9. The error bars indicate the range of model fits for five sepa-ate runs of ESSFWI using different random-number seeds to gener-te the random phase codes.
Figure 10 shows that code length has very little effect on modelonvergence rate, so short codes yield higher efficiency gains �seequation 6�. In fact, a random phase code of length one is as good as,r better than, any other random phase code. Note that using a ran-om phase code of length one means that we randomly multiply theources by �1 or �1.
Figures 11 and 12 demonstrate that even if ambient noise level is2 times that of the Gulf of Mexico data, the efficiency gains of ESS-WI over conventional FWI remain. However, for high ambientoise levels, ESSFWI converges for a few iterations until the encod-d simultaneous-source data are matched to within S/N. From there,SSFWI tries to match the ambient noise in the encoded data and
herefore starts to diverge from the true model �see Figure 11b and�. The conventional FWI inversions �not displayed here� are virtu-lly unaffected by these levels of ambient noise because coherenttacking of information for different sources in conventional FWIrovides a significant increase in the S/N.
As shown by Figure 13, FWI and ESSFWI converge to very simi-ar models after 100 iterations, irrespective of the accuracy of thetarting model. This is true even for the region �dashed boxes in Fig-re 13� where the less accurate starting model is probably stuck in aocal minimum. This may indicate that the equivalence of the global
inimum for the FWI and ESSFWI objective functions also extendso local minima. Note that the model convergence rates for FWI andSSFWI �Figure 14� are similar for all three starting models.
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igure 13. FWI and ESSFWI for greater and less accurate startingodels than the initial model used for the examples in Figures 4–9.he upper row shows the initial models, the middle row shows theWI model after 100 iterations, and the lower row shows the ESS-WI model after 100 iterations. The dashed boxes enclose regions of
he model that are probably stuck in a local minimum.
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CONCLUSIONS
We have used 2D acoustic synthetic data inversion to demonstratehat the efficiency of ESSFWI over conventional FWI increases by aactor that is approximately equal to the number of sources encoded50 in the examples presented here�. Our testing indicates that en-oding the sources and changing the code between iterations areery important steps — not only to increase efficiency but also tochieve a satisfactory inversion from ESSFWI. Short, random phaseodes perform slightly better than long codes. Therefore, we choseo encode by multiplying the sources randomly by �1 or �1. ESS-WI is significantly more sensitive to ambient noise levels than isWI, so we must be careful to limit the number of sources encoded
nto a simultaneous-source gather if ambient noise levels are high.e also show that ESSFWI efficiency gains are relatively insensi-
ive to the accuracy of the starting model.Beyond the conclusions drawn from the tests presented here, we
ypothesize the following about the ESSFWI efficiency gain:
� Probably increases with increasing complexity of the true mod-el, not because model complexity affects ESSFWI conver-gence but because the number of sources required for conven-tional FWI increases with increasing model complexity, espe-cially for near-surface complexity.
� May increase for data with high maximum frequencies becausethe number of sources required for conventional FWI should beproportional to the maximum frequency and because the S/Nwith respect to ambient noise should increase for high frequen-cies.
� May be larger for 3D inversion than for 2D inversion because3D data provide more sources for summation into an encodedsimultaneous-source gather.
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igure 14. Model convergence versus iteration for three startingodels with different levels of smoothness. The intermediate accu-
acy curves represent the initial model used in the tests shown in Fig-res 4–9. The more and less accurate curves correspond to the initialodels in Figure 13. The error bars on the ESSFWI curves indicate
he range of model fits for five separate runs of ESSFWI using differ-nt random-number seeds to generate the random phase codes.
In addition, single-sample codes work as well as longer, more or-hogonal codes. This may indicate that a more deterministic incoher-nt coding would be advantageous, especially when the S/N is low.
Our understanding of the sensitivity of ESSFWI to inversiontrategy is incomplete. Our testing has included the steepest descentlgorithm and three versions of the conjugate gradient algorithm.ur ESSFWI results show some sensitivity to the conjugate gradient
lgorithm used. There may be similar sensitivities to more sophisti-ated methods, such as Newton and quasi-Newton methods. Alter-ative search strategies may affect the efficiency gains achieved bysing ESSFWI.
We believe that ESSFWI can be used to achieve accurate 3D FWIor large data sets. Although we have used an explicit time-domainimulator for our examples, our technique should be applicable tony FWI that is based on a source-configuration-dependent simula-or, such as those that use iterative frequency-domain methods.owever, ESSFWI alone does not make FWI practical in all instanc-
s. In particular, the problem of local minima must be overcome.
ACKNOWLEDGMENTS
We thank ExxonMobil for allowing us to publish these results andingbo Wang, Max Deffenbaugh, and Dong Sun for stimulating dis-ussions. We also thank Tom Dickens and Jean Krebs for construc-ive comments on the manuscript and acknowledge the GEOPHYSICS
ditors and reviewers for their many helpful suggestions.
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