Iterative inversion of data from simultaneous geophysical sources

0001This application claims the benefit of <patcit id="pcit0001" dnum="US84769606P" dnum-type="L"><text>U.S. Provisional application 60/847,696 which was filed on September 28, 2006</text></patcit>.

<u>FIELD OF THE INVENTION</u>

0002The invention relates generally to the field of geophysical prospecting, and more particularly to geophysical data processing. Specifically, the invention is a method for inversion of data acquired from multiple geophysical sources such as seismic sources, involving geophysical simulation that computes the data from many simultaneously-active geophysical sources in one execution of the simulation.

<u>BACKGROUND OF THE INVENTION</u>

0003Geophysical inversion [1,2] attempts to find a model of subsurface properties that optimally explains observed data and satisfies geological and geophysical constraints. There are a large number of well known methods of geophysical inversion. These well known methods fall into one of two categories, iterative inversion and non-iterative inversion. The following are definitions of what is commonly meant by each of the two categories: <ul id="ul0001" list-style="bullet" compact="compact"><li>Non-iterative inversion - inversion that is accomplished by assuming some simple background model and updating the model based on the input data. This method does not use the updated model as input to another step of inversion. For the case of seismic data these methods are commonly referred to as imaging, migration, diffraction tomography or Born inversion.</li><li>Iterative inversion - inversion involving repetitious improvement of the subsurface properties model such that a model is found that satisfactorily explains the observed data. If the inversion converges, then the final model will better explain the observed data and will more closely approximate the actual subsurface properties. Iterative inversion usually produces a more accurate model than non-iterative inversion, but is much more expensive to compute.</li></ul>

0004Two iterative inversion methods commonly employed in geophysics are cost function optimization and series methods. Cost function optimization involves iterative minimization or maximization of the value, with respect to the model <i>M</i>, of a cost function <i>S</i>(<i>M</i>) which is a measure of the misfit between the calculated and observed data (this is also sometimes referred to as the objective function), where the calculated data is simulated with a computer using the current geophysical properties model and the physics governing propagation of the source signal in a medium represented by a given geophysical properties model. The simulation computations may be done by any of several numerical methods including but not limited to finite difference, finite element or ray tracing. Series methods involve inversion by iterative series solution of the scattering equation (Weglein [3]). The solution is written in series form, where each term in the series corresponds to higher orders of scattering. Iterations in this case correspond to adding a higher order term in the series to the solution.

0005Cost function optimization methods are either local or global [4]. Global methods simply involve computing the cost function <i>S</i>(<i>M</i>) for a population of models {<i>M₁, M₂, M₃,...</i>} and selecting a set of one or more models from that population that approximately minimize <i>S(M).</i> If further improvement is desired this new selected set of models can then be used as a basis to generate a new population of models that can be again tested relative to the cost function <i>S</i>(<i>M</i>). For global methods each model in the test population can be considered to be an iteration, or at a higher level each set of populations tested can be considered an iteration. Well known global inversion methods include Monte Carlo, simulated annealing, genetic and evolution algorithms.

0006Local cost function optimization involves: <ol id="ol0001"><li><b>1.</b> selecting a starting model,</li><li><b>2.</b> computing the gradient of the cost function <i>S</i>(<i>M</i>) with respect to the parameters that describe the model,</li><li><b>3.</b> searching for an updated model that is a perturbation of the starting model in the gradient direction that better explains the observed data.</li></ol> This procedure is iterated by using the new updated model as the starting model for another gradient search. The process continues until an updated model is found which satisfactorily explains the observed data. Commonly used local cost function inversion methods include gradient search, conjugate gradients and Newton's method.

0007As discussed above, iterative inversion is preferred over non-iterative inversion, because it yields more accurate subsurface parameter models. Unfortunately, iterative inversion is so computationally expensive that it is impractical to apply it to many problems of interest. This high computational expense is the result of the fact that all inversion techniques require many compute intensive forward and/or reverse simulations. Forward simulation means computation of the data forward in time, and reverse simulation means computation of the data backward in time. The compute time of any individual simulation is proportional to the number of sources to be inverted, and typically there are large numbers of sources in geophysical data. The problem is exacerbated for iterative inversion, because the number of simulations that must be computed is proportional to the number of iterations in the inversion, and the number of iterations required is typically on the order of hundreds to thousands.

0008The compute cost of all categories of inversion can be reduced by inverting data from combinations of the sources, rather than inverting the sources individually. This may be called simultaneous source inversion. Several types of source combination are known including: coherently sum closely spaced sources to produce an effective source that produces a wavefront of some desired shape (e.g. a plane wave), sum widely spaces sources, or fully or partially stacking the data before inversion.

0009The compute cost reduction gained by inverting combined sources is at least partly offset by the fact that inversion of the combined data usually produces a less accurate inverted model. This loss in accuracy is due to the fact that information is lost when the individual sources are summed, and therefore the summed data does not constrain the inverted model as strongly as the unsummed data. This loss of information during summation can be minimized by encoding each shot record before summing. Encoding before combination preserves significantly more information in the simultaneous source data, and therefore better constrains the inversion. Encoding also allows combination of closely spaced sources, thus allowing more sources to be combined for a given computational region. Various encoding schemes can be used with this technique including time shift encoding and random phase encoding. The remainder of this Background section briefly reviews various published geophysical simultaneous source techniques, both encoded and non-encoded.

0010Van Manen [5] suggests using the seismic interferometry method to speedup forward simulation. Seismic interferometry works by placing sources everywhere on the boundary of the region of interest. These sources are modeled individually and the wavefield at all locations for which a Green's function is desired is recorded. The Green's function between any two recorded locations can then be computed by cross-correlating the traces acquired at the two recorded locations and summing over all the boundary sources. If the data to be inverted has a large number of sources and receivers that are within the region of interest (as opposed to having one or the other on the boundary) then this is a very efficient method for computing the desired Green's functions. However, for the seismic data case it is rare that both the source and receiver for the data to be inverted are within the region of interest. Therefore, this improvement has very limited applicability to the seismic inversion problem.

0011Berkhout [6] and Zhang [7] suggest that inversion in general can be improved by inverting non-encoded simultaneous sources that are summed coherently to produce some desired wave front within some region of the subsurface. For example point source data could be summed with time shifts that are a linear function of the source location to produce a down-going plane wave at some particular angle with respect to the surface. This technique could be applied to all categories of inversion. A problem with this method is that coherent summation of the source gathers necessarily reduces the amount of information in the data. So for example, summation to produce a plane wave removes all the information in the seismic data related to travel time versus source-receiver offset. This information is critical for updating the slowly varying background velocity model, and therefore Berkhout's method is not well constrained. To overcome this problem many different coherent sums of the data (e.g. many plane waves with different propagation directions) could be inverted, but then efficiency is lost since the cost of inversion is proportional to the number of different sums inverted. Such coherently summed sources are called generalized sources. Therefore, a generalized source can either be a point source or a sum of point sources that produces a wave front of some desired shape.

0012Van Riel [8] suggests inversion by non-encoded stacking or partial stacking (with respect to source-receiver offset) of the input seismic data, then defining a cost function with respect to this stacked data which will be optimized. Thus, this publication suggests improving cost function based inversion using non-encoded simultaneous sources. As was true of the Berkhout's [6] simultaneous source inversion method, the stacking suggested by this method reduces the amount of information in the data to be inverted and therefore the inversion is less well constrained than it would have been with the original data.

0013Mora [9] proposes inverting data that is the sum of widely spaced sources. Thus, this publication suggests improving the efficiency of inversion using non-encoded simultaneous source simulation. Summing widely spaced sources has the advantage of preserving much more information than the coherent sum proposed by Berkhout. However, summation of widely spaced sources implies that the aperture (model region inverted) that must be used in the inversion must be increased to accommodate all the widely spaced sources. Since the compute time is proportional to the area of this aperture, Mora's method does not produce as much efficiency gain as could be achieved if the summed sources were near each other.

0014Ober [10] suggests speeding up seismic migration, a special case of non-iterative inversion, by using simultaneous encoded sources. After testing various coding methods, Ober found that the resulting migrated images had significantly reduced signal-to-noise ratio due to the fact that broad band encoding functions are necessarily only approximately orthogonal. Thus, when summing more than 16 shots, the quality of the inversion was not satisfactory. Since non-iterative inversion is not very costly to begin with, and since high signal-to-noise ratio inversion is desired, this technique is not widely practiced in the geophysical industry.

0015Ikelle [11] suggests a method for fast forward simulation by simultaneously simulating point sources that are activated (in the simulation) at varying time intervals. A method is also discussed for decoding these time-shifted simultaneous-source simulated data back into the separate simulations that would have been obtained from the individual point sources. These decoded data could then be used as part of any conventional inversion procedure. A problem with Ikelle's method is that the proposed decoding method will produce separated data having noise levels proportional to the difference between data from adjacent sources. This noise will become significant for subsurface models that are not laterally constant, for example from models containing dipping reflectors. Furthermore, this noise will grow in proportion to the number of simultaneous sources. Due to these difficulties Ikelle's simultaneous source approach may result in unacceptable levels of noise if used in inverting a subsurface that is not laterally constant.

0016<patcit id="pcit0002" dnum="EP1094338A"><text>EP1094338</text></patcit> relates to a method of estimating elastic and compositional parameters by the inversion of multiple seismic and echo-acoustic data full or partial stacks. <patcit id="pcit0003" dnum="US6021094A"><text>US6021094</text></patcit> provides a method of migrating seismic records that retains the information in the seismic records and allows migration with significant reduction in computing cost. <patcit id="pcit0004" dnum="US5924049A"><text>US5924049</text></patcit> relates to a method for acquiring and processing seismic survey data from two or more seismic sources activated simultaneously or nearly simultaneously or for a single source moved to and fired at different locations.

0017What is needed is a more efficient method of iteratively inverting data, without significant reduction in the accuracy of the resulting inversion.

<u>SUMMARY OF THE INVENTION</u>

0018A physical properties model gives one or more subsurface properties as a function of location in a region. Seismic wave velocity is one such physical property, but so are (for example) p-wave velocity, shear wave velocity, several anisotropy parameters, attenuation (q) parameters, porosity, permeability, and resistivity. Referring to the flow chart of <figref idref="f0008">Fig. 10</figref>, in one embodiment the invention is a computer-implemented method for inversion of measured geophysical data to determine a physical properties model for a subsurface region, comprising: <ol id="ol0002"><li>(a) obtaining a group of two or more encoded gathers of the measured geophysical data, wherein each gather is associated with a single generalized source or, using source-receiver reciprocity, with a single receiver, and wherein each gather is encoded with a different encoding signature selected from a set non-equivalent encoding signatures;</li><li>(b) summing (<b>4</b>) the encoded gathers in the group by summing all data records in each gather that correspond to a single receiver (or source if reciprocity is used), and repeating for each different receiver, resulting in a simultaneous encoded gather;</li><li>(c) assuming a physical properties model <b>5</b> of the subsurface region, said model providing values of at least one physical property at locations throughout the subsurface region;</li><li>(d) calculating an update <b>6</b> to the assumed physical properties model that is more consistent with the simultaneous encoded gather from step (b), said calculation involving one or more encoded simultaneous source forward (or reverse) simulation operations that use the assumed physical properties model and encoded source signatures using the same encoding functions used to encode corresponding gathers of measured data, wherein an entire simultaneous encoded gather is simulated in a single simulation operation;</li><li>(e) repeating step (d) at least one more iteration, using the updated physical properties model from the previous iteration of step (d) as the assumed model to produce a further updated physical properties model 7 of the subsurface region that is more consistent with a corresponding simultaneous encoded gather of measured data, using the same encoding signatures for source signatures in the simulation as were used in forming the corresponding simultaneous encoded gather of measured data; and</li><li>(f) downloading the further updated physical properties model or saving it to computer storage.</li></ol>

0019It may be desirable in order to maintain inversion quality or for other reasons to perform the simultaneous encoded-source simulations in step (b) in more than one group. In such case, steps (a)-(b) are repeated for each additional group, and inverted physical properties models from each group are accumulated before performing the model update in step (d). If the encoded gathers are not obtained already encoded from the geophysical survey as described below, then gathers of geophysical data <b>1</b> are encoded by applying encoding signatures <b>3</b> selected from a set of non-equivalent encoding signatures <b>2</b>.

0020In another embodiment, the present invention is a computer-implemented method for inversion of measured geophysical data to determine a physical properties model for a subsurface region, comprising: <ol id="ol0003"><li>(a) obtaining a group of two or more encoded gathers of the measured geophysical data, wherein each gather is associated with a single generalized source or, using source-receiver reciprocity, with a single receiver, and wherein each gather is encoded with a different encoding signature selected from a set non-equivalent encoding signatures;</li><li>(b) summing the encoded gathers in the group by summing all data records in each gather that correspond to a single receiver (or source if reciprocity is used), and repeating for each different receiver, resulting in a simultaneous encoded gather;</li><li>(c) assuming a physical properties model of the subsurface region, said model providing values of at least one physical property at locations throughout the subsurface region;</li><li>(d) simulating a synthetic simultaneous encoded gather corresponding to the simultaneous encoded gather of measured data, using the assumed physical properties model, wherein the simulation uses encoded source signatures using the same encoding functions used to encode the simultaneous encoded gather of measured data, wherein an entire simultaneous encoded gather is simulated in a single simulation operation;</li><li>(e) computing a cost function measuring degree of misfit between the simultaneous encoded gather of measured data and the simulated simultaneous encoded gather;</li><li>(f) repeating steps (a), (b), (d) and (e) for at least one more cycle, accumulating costs from step (e);</li><li>(g) updating the physical properties model by optimizing the accumulated costs;</li><li>(h) iterating steps (a)-(g) at least one more time using the updated physical properties model from the previous iteration as the assumed physical properties model in step (c), wherein a different set non-equivalent encoding signatures may be used for each iteration, resulting in a further updated physical properties model; and</li><li>(i) downloading the further updated physical properties model or saving it to computer storage.</li></ol>

0021In another embodiment, the invention is a computer-implemented method for inversion of measured geophysical data to determine a physical properties model for a subsurface region, comprising: <ol id="ol0004"><li>(a) obtaining a group of two or more encoded gathers of the measured geophysical data, wherein each gather is associated with a single generalized source or, using source-receiver reciprocity, with a single receiver, and wherein each gather is encoded with a different encoding signature selected from a set non-equivalent encoding signatures;</li><li>(b) summing the encoded gathers in the group by summing all data records in each gather that correspond to a single receiver (or source if reciprocity is used), and repeating for each different receiver, resulting in a simultaneous encoded gather;</li><li>(c) assuming a physical properties model of the subsurface region, said model providing values of at least one physical property at locations throughout the subsurface region;</li><li>(d) selecting an iterative series solution to a scattering equation describing wave scattering in said subsurface region;</li><li>(e) beginning with the first <i>n</i> terms of said series, where <i>n</i> ≥ 1, said first <i>n</i> terms corresponding to the assumed physical properties model of the subsurface region;</li><li>(f) computing the next term in the series, said calculation involving one or more encoded simultaneous source forward (or reverse) simulation operations that use the assumed physical properties model and encoded source signatures using the same encoding functions used to encode corresponding gathers of measured data, wherein an entire simultaneous encoded gather is simulated in a single simulation operation and the simulated encoded gather and measured encoded gather are combined in a manner consistent with the iterative series selected in step (d);</li><li>(g) updating the model by adding the next term in the series calculated in step (f) to the assumed model;</li><li>(h) repeating steps (f) and (g) for at least one time to add at least one more term to the series to further update the physical properties model; and</li><li>(i) downloading the further updated physical properties model or saving it to computer storage.</li></ol>

0022In another embodiment, the invention is a computer-implemented method for inversion of measured geophysical data to determine a physical properties model for a subsurface region, comprising: <ol id="ol0005"><li>(a) obtaining measured geophysical data from a geophysical survey the subsurface region;</li><li>(b) assuming an initial physical properties model and inverting it by iterative inversion involving simultaneous simulation of survey data representing a plurality of survey sources (or receivers if source-receiver reciprocity is used) wherein source signatures in the simulation are encoded, resulting in a simulated simultaneous encoded gather of geophysical data, the inversion process involving updating the physical properties model to reduce misfit between the simulated simultaneous encoded gather and a corresponding simultaneous encoded gather formed by summing gathers of measured survey data encoded with the same encoding functions used in the simulation; and</li><li>(c) downloading the updated physical properties model or saving it to computer storage.</li></ol>

<u>BRIEF DESCRIPTION OF THE DRAWINGS</u>

0023The present invention and its advantages will be better understood by referring to the following detailed description and the attached drawings in which: <ul id="ul0002" list-style="none"><li><figref idref="f0001">Fig. 1</figref> is a flow chart showing steps in a method for preparing data for simultaneous encoded-source inversion;</li><li><figref idref="f0002">Fig. 2</figref> is a flow chart showing steps in one embodiment of the present inventive method for simultaneous source computation of the data inversion cost function;</li><li><figref idref="f0003">Fig. 3</figref> is a base velocity model for an example demonstrating the computation of the full wavefield cost function;</li><li><figref idref="f0004">Fig. 4</figref> is a data display showing the first 3 of 256 sequential source data records simulated in the Example from the base model of <figref idref="f0003">Fig. 3</figref>;</li><li><figref idref="f0004">Fig. 5</figref> shows a single simultaneous encoded-source gather produced from the 256 sequential source data records of which the first three are shown in <figref idref="f0004">Fig. 4</figref>;</li><li><figref idref="f0005">Fig. 6</figref> illustrates one of the perturbations of the base model in <figref idref="f0003">Fig. 3</figref> that is used in the Example to demonstrate computation of the full wave inversion cost function using simultaneous sources;</li><li><figref idref="f0006">Fig. 7</figref> shows the cost function computed for the present invention's simultaneous source data shown in <figref idref="f0004">Fig. 5</figref>;</li><li><figref idref="f0006">Fig. 8</figref> shows the cost function computed for the sequential source data shown in <figref idref="f0004">Fig. 4</figref>, i.e., by traditional inversion;</li><li><figref idref="f0007">Fig. 9</figref> shows the cost function for a prior-art "super-shot" gather, formed by simply summing the sequential source data shown in <figref idref="f0004">Fig. 4</figref>; and</li><li><figref idref="f0008">Fig. 10</figref> is a flow chart showing basic steps in one embodiment of the present inventive method.</li></ul>

0024The invention will be described in connection with its preferred embodiments. However, to the extent that the following detailed description is specific to a particular embodiment or a particular use of the invention, this is intended to be illustrative only, and is not to be construed as limiting the scope of the invention. On the contrary, it is intended to cover all alternatives, modifications and equivalents that may be included within the scope of the invention, as defined by the appended claims.

<u>DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS</u>

0025The present invention is a method for reducing the computational time needed to iteratively invert geophysical data by use of simultaneous encoded-source simulation.

0026Geophysical inversion attempts to find a model of subsurface elastic properties that optimally explains observed geophysical data. The example of seismic data is used throughout to illustrate the inventive method, but the method may be advantageously applied to any method of geophysical prospecting involving at least one source, activated at multiple locations, and at least one receiver. The data inversion is most accurately performed using iterative methods. Unfortunately iterative inversion is often prohibitively expensive computationally. The majority of compute time in iterative inversion is spent computing forward and/or reverse simulations of the geophysical data (here forward means forward in time and reverse means backward in time). The high cost of these simulations is partly due to the fact that each geophysical source in the input data must be computed in a separate computer run of the simulation software. Thus, the cost of simulation is proportional to the number of sources in the geophysical data (typically on the order of 1,000 to 10,000 sources for a geophysical survey). In this invention, the source signatures for a group of sources are encoded and these encoded sources are simulated in a single run of the software, resulting in a computational speedup proportional to the number of sources computed simultaneously.

0027As discussed above in the Background section, simultaneous source methods have been proposed in several publications for reducing the cost of various processes for inversion of geophysical data [3,6,7,8,9]. In a more limited number of cases, simultaneous <i>encoded</i>-source techniques are disclosed for certain purposes [10,11]. These methods have all been shown to provide increased efficiency, but always at significant cost in reduced quality, usually in the form of lower signal-to-noise ratio when large numbers of simultaneous sources are employed. The present invention mitigates this inversion quality reduction by showing that simultaneous encoded-source simulation can be advantageously used in connection with <i>iterative</i> inversion. Iteration has the surprising effect of reducing the undesirable noise resulting from the use of simultaneous encoded sources. This is considered unexpected in light of the common belief that inversion requires input data of the highest possible quality. In essence, the simultaneous encoded-source technique produces simulated data that appear to be significantly degraded relative to single source simulation (due to the data encoding and summation which has the appearance of randomizing the data), and uses this apparently degraded data to produce an inversion that has, as will be shown below, virtually the same quality as the result that would have been obtained by the prohibitively expensive process of inverting the data from the individual sources. (Each source position in a survey is considered a different "source" for purposes of inversion.)

0028The reason that these apparently degraded data can be used to perform a high quality iterative inversion is that by encoding the data before summation of sources the information content of the data is only slightly degraded. Since there is only insignificant information loss, these visually degraded data constrain an iterative inversion just as well as conventional sequential source data. Since simultaneous sources are used in the simulation steps of the inversion, the compute time is significantly reduced, relative to conventional sequential source inversion.

0029Two iterative inversion methods commonly employed in geophysics are cost function optimization and series methods. The present invention can be applied to both of these methods. Simultaneous encoded-source cost function optimization is discussed first.

<i>Iterative Cost Function Optimization</i>

0030Cost function optimization is performed by minimizing the value, with respect to a subsurface model <i>M,</i> of a cost function <i>S</i>(<i>M</i>) (sometimes referred to as an <i>objective function</i>), which is a measure of misfit between the observed (measured) geophysical data and corresponding data calculated by simulation of the assumed model. A simple cost function S often used in geophysical inversion is: <maths id="math0001" num="(1)"><math display="block"><mrow><mi>S</mi><mfenced><mi>M</mi></mfenced><mo>=</mo><mstyle displaystyle="true"><mrow><munderover><mrow><mo>∑</mo></mrow><mrow><mi>g</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>N</mi><mi>R</mi></msub></mrow></munderover></mrow></mstyle><mstyle displaystyle="true"><mrow><munderover><mrow><mo>∑</mo></mrow><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>N</mi><mi>r</mi></msub></mrow></munderover></mrow></mstyle><mstyle displaystyle="true"><mrow><munderover><mrow><mo>∑</mo></mrow><mrow><mi>t</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>N</mi><mi>t</mi></msub></mrow></munderover></mrow></mstyle><msup><mrow><mfenced open="|" close="|"><msub><mi>ψ</mi><mi mathvariant="italic">calc</mi></msub><mfenced separators=",,,,"><mi>M</mi><mi>g</mi><mi>r</mi><mi>t</mi><msub><mi>w</mi><mi>g</mi></msub></mfenced><mo>−</mo><msub><mi>ψ</mi><mi mathvariant="italic">obs</mi></msub><mfenced separators=",,,"><mi>g</mi><mi>r</mi><mi>t</mi><msub><mi>w</mi><mi>g</mi></msub></mfenced></mfenced></mrow><mi>N</mi></msup></mrow></math><img file="EP2067112B1_D0001.tif" /></maths> where <dl id="dl0001" compact="compact"><dt><i>N</i></dt><dd>= norm for cost function (typically the least squares or L2-Norm is used in which case <i>N</i> = 2),</dd><dt><i>M</i></dt><dd>= subsurface model,</dd><dt><i>g</i></dt><dd>= gather index (for point source data this would correspond to the individual sources),</dd><dt><i>N</i>g</dt><dd>= number of gathers,</dd><dt><i>r</i></dt><dd>= receiver index within gather,</dd><dt><i>N</i>_r</dt><dd>= number of receivers in a gather,</dd><dt><i>t</i></dt><dd>= time sample index within a data record,</dd><dt><i>N</i>_t</dt><dd>= number of time samples,</dd><dt><i>ψ_calc</i></dt><dd>= calculated geophysical data from the model <i>M,</i></dd><dt><i>ψ_obs</i></dt><dd>= measured geophysical data, and</dd><dt><i>w</i>_g</dt><dd>= source signature for gather g, i.e. source signal without earth filtering effects.</dd></dl>

0031The gathers in Equation 1 can be any type of gather that can be simulated in one run of a forward modeling program. For seismic data, the gathers correspond to a seismic shot, although the shots can be more general than point sources [<b>6</b>]. For point sources, the gather index g corresponds to the location of individual point sources. For plane wave sources, g would correspond to different plane wave propagation directions. This generalized source data, <i>ψ_obs</i>, can either be acquired in the field or can be synthesized from data acquired using point sources. The calculated data <i>ψ_calc</i> on the other hand can usually be computed directly by using a generalized source function when forward modeling (e.g. for seismic data, forward modeling typically means solution of the anisotropic visco-elastic wave propagation equation or some approximation thereof). For many types of forward modeling, including finite difference modeling, the computation time needed for a generalized source is roughly equal to the computation time needed for a point source. The model <i>M</i> is a model of one or more physical properties of the subsurface region. Seismic wave velocity is one such physical property, but so are (for example) p-wave velocity, shear wave velocity, several anisotropy parameters, attenuation (q) parameters, porosity, and permeability. The model M might represent a single physical property or it might contain many different parameters depending upon the level of sophistication of the inversion. Typically, a subsurface region is subdivided into discrete cells, each cell being characterized by a single value of each parameter.

0032Equation 1 can be simplified to: <maths id="math0002" num="(2)"><math display="block"><mrow><mi>S</mi><mfenced><mi>M</mi></mfenced><mo>=</mo><mstyle displaystyle="true"><mrow><munderover><mrow><mo>∑</mo></mrow><mrow><mi>g</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>N</mi><mi>g</mi></msub></mrow></munderover></mrow></mstyle><msup><mrow><mfenced open="|" close="|"><mi>δ</mi><mfenced separators=",,"><mi>M</mi><mi>g</mi><msub><mi>w</mi><mi>g</mi></msub></mfenced></mfenced></mrow><mi>N</mi></msup></mrow></math><img file="EP2067112B1_D0002.tif" /></maths> where the sum over receivers and time samples is now implied and, <maths id="math0003" num="(3)"><math display="block"><mrow><mi>δ</mi><mfenced separators=",,"><mi>M</mi><mi>g</mi><msub><mi>w</mi><mi>g</mi></msub></mfenced><mo>=</mo><msub><mi>ψ</mi><mi mathvariant="italic">calc</mi></msub><mfenced separators=",,"><mi>M</mi><mi>g</mi><msub><mi>w</mi><mi>g</mi></msub></mfenced><mo>−</mo><msub><mi>ψ</mi><mi mathvariant="italic">obs</mi></msub><mfenced separators=","><mi>g</mi><msub><mi>w</mi><mi>g</mi></msub></mfenced></mrow></math><img file="EP2067112B1_D0003.tif" /></maths> One major problem with iterative inversion is that computing <i>ψ_calc</i> takes a large amount of computer time, and therefore computation of the cost function, <i>S</i>, is very time consuming. Furthermore, in a typical inversion project this cost function must be computed for many different models <i>M.</i>

0033The computation time for <i>ψ_calc</i> is proportional to the number of gathers (for point source data this equals the number of sources), <i>N_g,</i> which is on the order of 10,000 to 100,000 for a typical seismic survey. The present invention greatly reduces the time needed for geophysical inversion by showing that <i>S</i>(<i>M</i>) can be well approximated by computing <i>ψ_calc</i> for many encoded generalized sources which are activated simultaneously. This reduces the time needed to compute <i>ψ_calc</i> by a factor equal to the number of simultaneous sources. A more detailed version of the preceding description of the technical problem being addressed follows.

0034The cost function in Equation 2 is replaced with the following: <maths id="math0004" num="(5)"><math display="block"><mrow><msub><mi>S</mi><mi mathvariant="italic">sim</mi></msub><mfenced><mi>M</mi></mfenced><mo>=</mo><mstyle displaystyle="true"><mrow><munderover><mrow><mo>∑</mo></mrow><mrow><mi>G</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>N</mi><mi>G</mi></msub></mrow></munderover></mrow></mstyle><msup><mfenced open="|" close="|"><mstyle displaystyle="true"><mrow><munderover><mrow><mo>∑</mo></mrow><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow><mspace width="1em" /></munderover></mrow></mstyle><mi>δ</mi><mfenced><mi>M</mi><mo>,</mo><mi>g</mi><mo>,</mo><msub><mi>c</mi><mi>g</mi></msub><mo>⊗</mo><msub><mi>w</mi><mi>g</mi></msub></mfenced></mfenced><mi>N</mi></msup></mrow></math><img file="EP2067112B1_D0004.tif" /></maths> where a summation over receivers and time samples is implied as in Equation 2 and: <maths id="math0005"><math display="inline"><mrow><mstyle displaystyle="true"><mrow><munderover><mrow><mo>∑</mo></mrow><mrow><mi>g</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>N</mi><mi>g</mi></msub></mrow></munderover></mrow></mstyle><mo>=</mo><mstyle displaystyle="true"><mrow><munderover><mrow><mo>∑</mo></mrow><mrow><mi>G</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>N</mi><mi>G</mi></msub></mrow></munderover></mrow></mstyle><mstyle displaystyle="true"><mrow><munderover><mrow><mo>∑</mo></mrow><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow><mspace width="1em" /></munderover></mrow></mstyle></mrow></math><img file="EP2067112B1_D0005.tif" /></maths>  defines a sum over gathers by sub groups of gathers, <i>S_sim</i>  = cost function for simultaneous source data, <i>G</i>  = the groups of simultaneous generalized sources, and <i>N_G</i>  = the number of groups, <i>c_g</i>  = functions of time that are convolved (⊗) with each gather's source signature to encode the gathers, these encoding functions are chosen to be different, i.e. non-equivalent, for each gather index g (e.g. different realizations of random phase functions).

0035The outer summation in Equation 5 is over groups of simultaneous generalized sources corresponding to the gather type (e.g. point sources for common shot gathers). The inner summation, over g, is over the gathers that are grouped for simultaneous computation. For some simulation methods, such as finite difference modeling, the computation of the model for summed sources (the inner sum over <i>g∈G</i>) can be performed in the same amount of time as the computation for a single source. Thus, Equation 5 can be computed in a time that is <i>N_g</i>/<i>N_G</i> times faster than Equation 2. In the limiting case, all gathers are computed simultaneously (i.e. <i>G</i> contains all <i>N_g</i> sources and <i>N_G</i> = <i>1</i>) and one achieves a factor of <i>N_g</i> speedup.

0036This speedup comes at the cost that <i>S_sim</i>(<i>M</i>) in Equation 5 is not in general as suitable a cost function for inversion as is <i>S</i>(<i>M</i>) defined in Equation 2. Two requirements for a high quality cost function are: <ol id="ol0006"><li>1. It has a global minimum when the model M is close to the true subsurface model,</li><li>2. It has few local minima and they are located far from the true subsurface model.</li></ol> It is easy to see that in the noise-free case the global minimum of both <i>S</i>(<i>M</i>) and <i>S_sim</i>(<i>M</i>) will occur when <i>M</i> is equal to the true subsurface model and that their value at the global minimum is zero. Experience has shown that the global minimum of <i>S_sim</i>(<i>M</i>) is also close to the actual subsurface model in the case where the data are noisy. Thus, <i>S_sim</i>(<i>M</i>) satisfies requirement number 1 above. Next it will be shown how <i>S_sim</i>(<i>M</i>) can be made to satisfy the second enumerated requirement.

0037One cannot in general develop a cost function for data inversion that has no local minima. So it would be unreasonable to expect <i>S_sim</i>(<i>M</i>) to have no local minima as desired by requirement 2 above. However, it is at least desirable that <i>S_sim</i>(<i>M</i>) has a local minima structure not much worse than <i>S(M).</i> According to the present invention, this can be accomplished by a proper choice of encoding signatures.

0038When the cost function uses an L2-Norm, choosing the encoding signatures to be random phase functions gives a simultaneous source cost function that has a local minima structure similar to the sequential source cost function. This can be seen by developing a relationship between <i>S_sim</i>(<i>M</i>) to <i>S</i>(<i>M</i>) as follows. First, Equation 5 is specialized to the L2-Norm case: <maths id="math0006" num="(6)"><math display="block"><mrow><msub><mi>S</mi><mi mathvariant="italic">sim</mi></msub><mfenced><mi>M</mi></mfenced><mo>=</mo><mstyle displaystyle="true"><mrow><munderover><mrow><mo>∑</mo></mrow><mrow><mi>G</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>N</mi><mi>G</mi></msub></mrow></munderover></mrow></mstyle><msup><mfenced open="|" close="|"><mstyle displaystyle="true"><mrow><munderover><mrow><mo>∑</mo></mrow><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow><mspace width="1em" /></munderover></mrow></mstyle><mi>δ</mi><mfenced><mi>M</mi><mo>,</mo><mi>g</mi><mo>,</mo><msub><mi>c</mi><mi>g</mi></msub><mo>⊗</mo><msub><mi>w</mi><mi>g</mi></msub></mfenced></mfenced><mn>2</mn></msup></mrow></math><img file="EP2067112B1_D0006.tif" /></maths> The square within the sum over groups can be expanded as follows: <maths id="math0007" num="(7)"><math display="block"><mrow><msub><mi>S</mi><mi mathvariant="italic">sim</mi></msub><mfenced><mi>M</mi></mfenced><mo>=</mo><mstyle displaystyle="true"><mrow><munderover><mrow><mo>∑</mo></mrow><mrow><mi>G</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>N</mi><mi>G</mi></msub></mrow></munderover></mrow></mstyle><mfenced><msup><mfenced open="|" close="|"><mstyle displaystyle="true"><mrow><munderover><mrow><mo>∑</mo></mrow><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow><mspace width="1em" /></munderover></mrow></mstyle><mi>δ</mi><mfenced><mi>M</mi><mo>,</mo><mi>g</mi><mo>,</mo><msub><mi>c</mi><mi>g</mi></msub><mo>⊗</mo><msub><mi>w</mi><mi>g</mi></msub></mfenced></mfenced><mn>2</mn></msup><mo>+</mo><mstyle displaystyle="true"><mrow><munderover><mrow><mo>∑</mo></mrow><mrow><mtable columnalign="left" width="auto"><mtr><mtd><mi>g</mi><mo>,</mo><mi>g</mi><mo>′</mo><mo>∈</mo><mi>G</mi></mtd></mtr><mtr><mtd><mi>g</mi><mo>≠</mo><mi>g</mi><mo>′</mo></mtd></mtr></mtable></mrow><mspace width="1em" /></munderover></mrow></mstyle><mrow><mo>|</mo><mi>δ</mi><mfenced><mi>M</mi><mo>,</mo><mi>g</mi><mo>,</mo><msub><mi>c</mi><mi>g</mi></msub><mo>⊗</mo><msub><mi>w</mi><mi>g</mi></msub></mfenced><mo>|</mo><mfenced open="|" close="|"><mi>δ</mi><mfenced><mi>M</mi><mo>,</mo><mi>g</mi><mo>′</mo><mo>,</mo><msub><mi>c</mi><mi>g</mi></msub><mo>,</mo><mo>⊗</mo><msub><mi>w</mi><mrow><mi>g</mi><mo>′</mo></mrow></msub></mfenced></mfenced></mrow></mfenced></mrow></math><img file="EP2067112B1_D0007.tif" /></maths> By choosing the <i>c_g</i> so that they have constant amplitude spectra, the first term in Equation 7 is simply <i>S</i>(<i>M</i>), yielding: <maths id="math0008" num="(8)"><math display="block"><mrow><msub><mi>S</mi><mi mathvariant="italic">sim</mi></msub><mfenced><mi>M</mi></mfenced><mo>=</mo><mi>S</mi><mfenced><mi>M</mi></mfenced><mo>+</mo><mstyle displaystyle="true"><mrow><munderover><mrow><mo>∑</mo></mrow><mrow><mi>G</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>N</mi><mi>G</mi></msub></mrow></munderover></mrow></mstyle><mstyle displaystyle="true"><mrow><munderover><mrow><mo>∑</mo></mrow><mrow><mtable><mtr><mtd><mi>g</mi><mo>∈</mo><mi>G</mi></mtd></mtr><mtr><mtd><mtable><mtr><mtd><mi>g</mi><mo>′</mo><mo>∈</mo><mi>G</mi></mtd></mtr><mtr><mtd><mi>g</mi><mo>′</mo><mo>≠</mo><mi>g</mi></mtd></mtr></mtable></mtd></mtr></mtable></mrow><mspace width="1em" /></munderover></mrow></mstyle><mfenced open="|" close="|"><mi>δ</mi><mfenced><mi>M</mi><mo>,</mo><mi>g</mi><mo>,</mo><msub><mi>c</mi><mi>g</mi></msub><mo>⊗</mo><msub><mi>w</mi><mi>g</mi></msub></mfenced></mfenced><mfenced open="|" close="|"><mi>δ</mi><mfenced><mi>M</mi><mo>,</mo><mi>g</mi><mo>′</mo><mo>,</mo><msub><mi>c</mi><mi>g</mi></msub><mo>,</mo><mo>⊗</mo><msub><mi>w</mi><mrow><mi>g</mi><mo>′</mo></mrow></msub></mfenced></mfenced></mrow></math><img file="EP2067112B1_D0008.tif" /></maths> Equation 8 reveals that <i>S</i>_sim(<i>M</i>) is equal <i>S</i>(<i>M</i>) plus some cross terms. Note that due to the implied sum over time samples, the cross terms |<i>δ(M,g,c_g</i>⊗<i>w_g</i>)||δ<i>(M,g', c_g'⊗w_g'</i>)| are really cross correlations of the residual from two different gathers. This cross correlation noise can be spread out over the model by choosing the encoding functions <i>c_g</i> such that <i>c_g</i> and <i>c_g'</i> are different realizations of random phase functions. Other types of encoding signatures may also work. Thus, with this choice of the <i>c_g, S</i>_sim(<i>M</i>) is approximately equal to <i>S</i>(<i>M</i>). Therefore, the local minima structure of <i>S</i>_sim(<i>M</i>) is approximately equal to <i>S</i>(<i>M</i>).

0039In practice, the present invention can be implemented according to the flow charts shown in <figref idref="f0001">Figs. 1</figref> and <figref idref="f0002">2</figref>. The flow chart in <figref idref="f0001">Fig. 1</figref> may be followed to encode and sum the geophysical survey data to be inverted to form simultaneous gather data. In step <b>20,</b> the input data <b>10</b> are separated into groups of gathers that will be encoded and summed to form simultaneous encoded gathers. In step <b>40</b>, each gather in one of the gather groups from step <b>20</b> are encoded. This encoding is performed by selecting a gather from the gather group and selecting an encoding signature from the set of non-equivalent encoding signatures <b>30</b>. All the traces from the gather are then temporally convolved with that selected encoding signature. Each gather in the gather group is encoded in the same manner, choosing a different encoding signature from <b>30</b> for each gather. After all gathers have been encoded in <b>40</b>, all the gathers are summed in <b>50</b>. The gathers are summed by summing all traces corresponding to the same receiver from each gather. This forms a simultaneous encoded-source gather which is then saved in step <b>60</b> to the output set of simulated simultaneous encoded gathers <b>70.</b> At step <b>80</b>, steps <b>40-60</b> are typically repeated until all gather groups from step <b>20</b> have been encoded. When all gather groups have been encoded, this process is ended and the file containing the simultaneous encoded gathers <b>70</b> will contain one simultaneous encoded gather for each gather group formed in step <b>20</b>. How many gathers to put in a single group is a matter of judgment. The considerations involved include quality of the cost function vs. speedup in inversion time. One can run tests like those in the examples section below, and ensure that the grouping yields a high quality cost function. In some instances, it may be preferable to sum all the gathers into one simultaneous gather, i.e., use a single group.

0040<figref idref="f0001">Figure 1</figref> describes how simultaneous encoded gathers are obtained in some embodiments of the invention. In other embodiments, the geophysical data are acquired from simultaneous encoded sources, eliminating the need for the process in <figref idref="f0001">Fig. 1</figref>. It may be noted that acquiring simultaneous encoded-source data in the field could significantly reduce the cost of acquiring the geophysical data and also could increase the signal-to-noise ratio relative to ambient noise. Thus the present invention may be advantageously applied to (using a seismic vibrator survey as the example) a single vibrator truck moved sequentially to multiple locations, or to a survey in which two or more vibrator trucks are operating simultaneously with different encoded sweeps in close enough proximity that the survey receivers record combined responses of all vibrators. In the latter case only, the data could be encoded in the field.

0041<figref idref="f0002">Figure 2</figref> is a flowchart showing basic steps in the present inventive method for computing the data inversion cost function for the simultaneous encoded-source data. The simultaneous encoded gathers <b>120</b> are preferably either the data formed at <b>70</b> of <figref idref="f0001">Figure 1</figref> or are simultaneous encoded gathers that were acquired in the field. In step <b>130,</b> a simultaneous encoded gather from <b>120</b> is forward modeled using the appropriate signatures from the set of encoding signatures <b>110</b> that were used to form the simultaneous encoded gathers <b>120.</b> In step <b>140,</b> the cost function for this simultaneous encoded gather is computed. If the cost function is the L2 norm cost function, then step <b>140</b> would constitute summing, over all receivers and all time samples, the square of the difference between the simultaneous encoded gather from <b>120</b> and the forward modeled simultaneous encoded gather from <b>130.</b> The cost value computed in <b>140</b> is then accumulated into the total cost in step <b>150.</b> Steps <b>130-150</b> are typically repeated for another simultaneous encoded gather from <b>120,</b> and that cycle is repeated until all desired simultaneous encoded gathers from <b>120</b> have been processed <b>(160).</b>

0042There are many techniques for inverting data. Most of these techniques require computation of a cost function, and the cost functions computed by this invention provide a much more efficient method of performing this computation. Many types of encoding functions <i>c_g</i> can be used to ensure that <i>S</i>_sim(<i>M</i>) is approximately equal to <i>S</i>(<i>M</i>) including but not limited to: <ul id="ul0003" list-style="bullet"><li>Linear, random, chirp and modified chirp frequency dependent phase encoding as presented in Romero et al. [12];</li><li>The frequency independent phase encoding as presented in Jing et al. [13];</li><li>Random time shift encoding.</li></ul> Some of these encoding techniques will work better than others depending upon the application, and some can be combined. In particular, good results have been obtained using frequency dependent random phase encoding and also by combining frequency independent encoding of nearby sources with frequency dependent random phase encoding for more widely separated sources. An indication of the relative merits of different encodings can be obtained by running test inversions with each set of encoding functions to determine which converges faster.

0043It should be noted that the simultaneous encoded-source technique can be used for many types of inversion cost function. In particular it could be used for cost functions based on other norms than L2 discussed above. It could also be used on more sophisticated cost functions than the one presented in Equation 2, including regularized cost functions. Finally, the simultaneous encoded-source method could be used with any type of global or local cost function inversion method including Monte Carlo, simulated annealing, genetic algorithm, evolution algorithm, gradient line search, conjugate gradients and Newton's method.

<i>Iterative Series Inversion</i>

0044Besides cost function optimization, geophysical inversion can also be implemented using iterative series methods. A common method for doing this is to iterate the Lippmann-Schwinger equation [3]. The Lippmann-Schwinger equation describes scattering of waves in a medium represented by a physical properties model of interest as a perturbation of a simpler model. The equation is the basis for a series expansion that is used to determine scattering of waves from the model of interest, with the advantage that the series only requires calculations to be performed in the simpler model. This series can also be inverted to form an iterative series that allows the determination of the model of interest, from the measured data and again only requiring calculations to be performed in the simpler model. The Lippmann-Schwinger equation is a general formalism that can be applied to all types of geophysical data and models, including seismic waves. This method begins with the two equations: <maths id="math0009" num="(9)"><math display="block"><mrow><mi mathvariant="bold">LG</mi><mo>=</mo><mo>−</mo><mi mathvariant="bold">I</mi></mrow></math><img file="EP2067112B1_D0009.tif" /></maths><maths id="math0010" num="(10)"><math display="block"><mrow><msub><mi mathvariant="bold">L</mi><mn>0</mn></msub><msub><mi mathvariant="bold">G</mi><mn>0</mn></msub><mo>=</mo><mo>−</mo><mi mathvariant="bold">I</mi></mrow></math><img file="EP2067112B1_D0010.tif" /></maths> where <b>L, L</b>₀ are the actual and reference differential operators, <b>G</b> and <b>G</b>₀ are the actual and reference Green's operators respectively and <b>I</b> is the unit operator. Note that <b>G</b> is the measured point source data, and <b>G₀</b> is the simulated point source data from the initial model. The Lippmann-Schwinger equation for scattering theory is: <maths id="math0011" num="(11)"><math display="block"><mrow><mi mathvariant="bold">G</mi><mo>=</mo><msub><mi mathvariant="bold">G</mi><mn>0</mn></msub><mo>+</mo><msub><mi mathvariant="bold">G</mi><mn>0</mn></msub><mi mathvariant="bold">VG</mi></mrow></math><img file="EP2067112B1_D0011.tif" /></maths> where <b>V</b> = <b>L - L</b>₀ from which the difference between the true and initial models can be extracted.

0045Equation 11 is solved iteratively for <b>V</b> by first expanding it in a series (assuming G = Go for the first approximation of G and so forth) to get: <maths id="math0012" num="(12)"><math display="block"><mrow><mi mathvariant="bold">G</mi><mo>=</mo><msub><mi mathvariant="bold">G</mi><mn>0</mn></msub><mo>+</mo><msub><mi mathvariant="bold">G</mi><mn>0</mn></msub><msub><mi mathvariant="bold">VG</mi><mn>0</mn></msub><mo>+</mo><msub><mi mathvariant="bold">G</mi><mn>0</mn></msub><msub><mi mathvariant="bold">VG</mi><mn>0</mn></msub><msub><mi mathvariant="bold">VG</mi><mn>0</mn></msub><mo>+</mo><mo>⋯</mo></mrow></math><img file="EP2067112B1_D0012.tif" /></maths> Then V is expanded as a series: <maths id="math0013" num="(13)"><math display="block"><mrow><mi mathvariant="bold">V</mi><mo>=</mo><msup><mi mathvariant="bold">V</mi><mrow><mfenced><mn>1</mn></mfenced></mrow></msup><mo>+</mo><msup><mi mathvariant="bold">V</mi><mrow><mfenced><mn>2</mn></mfenced></mrow></msup><mo>+</mo><msup><mi mathvariant="bold">V</mi><mrow><mfenced><mn>3</mn></mfenced></mrow></msup><mo>+</mo><mo>⋯</mo></mrow></math><img file="EP2067112B1_D0013.tif" /></maths> where <b>V</b>⁽ⁿ⁾ is the portion of <b>V</b> that is n^th order in the residual of the data (here the residual of the data is <b>G - G</b>₀ measured at the surface). Substituting Equation 13 into Equation 12 and collecting terms of the same order yields the following set of equations for the first 3 orders: <maths id="math0014" num="(14)"><math display="block"><mrow><mi mathvariant="bold">G</mi><mo>−</mo><msub><mi mathvariant="bold">G</mi><mn>0</mn></msub><mo>=</mo><msub><mi mathvariant="bold">G</mi><mn>0</mn></msub><msup><mi mathvariant="bold">V</mi><mfenced><mn>1</mn></mfenced></msup><msub><mi mathvariant="bold">G</mi><mn>0</mn></msub></mrow></math><img file="EP2067112B1_D0014.tif" /></maths><maths id="math0015" num="(15)"><math display="block"><mrow><mn>0</mn><mo>=</mo><msub><mi mathvariant="bold">G</mi><mn>0</mn></msub><msup><mi mathvariant="bold">V</mi><mrow><mfenced><mn>2</mn></mfenced></mrow></msup><msub><mi mathvariant="bold">G</mi><mn>0</mn></msub><mo>+</mo><msub><mi mathvariant="bold">G</mi><mn>0</mn></msub><msup><mi mathvariant="bold">V</mi><mrow><mfenced><mn>1</mn></mfenced></mrow></msup><msub><mi mathvariant="bold">G</mi><mn>0</mn></msub><msup><mi mathvariant="bold">V</mi><mrow><mfenced><mn>1</mn></mfenced></mrow></msup><msub><mi mathvariant="bold">G</mi><mn>0</mn></msub></mrow></math><img file="EP2067112B1_D0015.tif" /></maths><maths id="math0016" num="(16)"><math display="block"><mrow><mn>0</mn><mo>=</mo><msub><mi mathvariant="bold">G</mi><mn>0</mn></msub><msup><mi mathvariant="bold">V</mi><mfenced><mn>3</mn></mfenced></msup><msub><mi mathvariant="bold">G</mi><mn>0</mn></msub><mo>+</mo><msub><mi mathvariant="bold">G</mi><mn>0</mn></msub><msup><mi mathvariant="bold">V</mi><mfenced><mn>1</mn></mfenced></msup><msub><mi mathvariant="bold">G</mi><mn>0</mn></msub><msup><mi mathvariant="bold">V</mi><mfenced><mn>2</mn></mfenced></msup><msub><mi mathvariant="bold">G</mi><mn>0</mn></msub><mo>+</mo><msub><mi mathvariant="bold">G</mi><mn>0</mn></msub><msup><mi mathvariant="bold">V</mi><mfenced><mn>2</mn></mfenced></msup><msub><mi mathvariant="bold">G</mi><mn>0</mn></msub><msup><mi mathvariant="bold">V</mi><mfenced><mn>1</mn></mfenced></msup><msub><mi mathvariant="bold">G</mi><mn>0</mn></msub><mo>+</mo><msub><mi mathvariant="bold">G</mi><mn>0</mn></msub><msup><mi mathvariant="bold">V</mi><mfenced><mn>1</mn></mfenced></msup><mo>+</mo><msub><mi mathvariant="bold">G</mi><mn>0</mn></msub><msup><mi mathvariant="bold">V</mi><mfenced><mn>1</mn></mfenced></msup><mo>+</mo><msub><mi mathvariant="bold">G</mi><mn>0</mn></msub><msup><mi mathvariant="bold">V</mi><mfenced><mn>1</mn></mfenced></msup><msub><mi mathvariant="bold">G</mi><mn>0</mn></msub></mrow></math><img file="EP2067112B1_D0016.tif" /></maths> and similarly for higher orders in <b>V.</b> These equations may be solved iteratively by first solving Equation 14 for <b>V</b>⁽¹⁾ by inverting Go on both sides of <b>V</b>⁽¹⁾ to yield: <maths id="math0017" num="(17)"><math display="block"><mrow><msup><mi mathvariant="bold">V</mi><mrow><mfenced><mn>1</mn></mfenced></mrow></msup><mo>=</mo><msubsup><mi mathvariant="bold">G</mi><mn>0</mn><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mfenced><mi mathvariant="bold">G</mi><mo>−</mo><msub><mi mathvariant="bold">G</mi><mn>0</mn></msub></mfenced><msubsup><mi mathvariant="bold">G</mi><mn>0</mn><mrow><mo>−</mo><mn>1</mn></mrow></msubsup></mrow></math><img file="EP2067112B1_D0017.tif" /></maths>

0046<b>V</b>⁽¹⁾ from Equation 17 is then substituted into Equation 15 and this equation is solved for <b>V</b>⁽²⁾ to yield: <maths id="math0018" num="(18)"><math display="block"><mrow><msup><mi mathvariant="bold">V</mi><mrow><mfenced><mn>2</mn></mfenced></mrow></msup><mo>=</mo><mo>−</mo><msubsup><mi mathvariant="bold">G</mi><mn>0</mn><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><msub><mi mathvariant="bold">G</mi><mn>0</mn></msub><msup><mi mathvariant="bold">V</mi><mrow><mfenced><mn>1</mn></mfenced></mrow></msup><msub><mi mathvariant="bold">G</mi><mn>0</mn></msub><msup><mi mathvariant="bold">V</mi><mrow><mfenced><mn>1</mn></mfenced></mrow></msup><msub><mi mathvariant="bold">G</mi><mn>0</mn></msub><msubsup><mi mathvariant="bold">G</mi><mn>0</mn><mrow><mo>−</mo><mn>1</mn></mrow></msubsup></mrow></math><img file="EP2067112B1_D0018.tif" /></maths> and so forth for higher orders of V.

0047Equation 17 involves a sum over sources and frequency which can be written out explicitly as: <maths id="math0019" num="(17)"><math display="block"><mrow><msup><mi mathvariant="bold">V</mi><mfenced><mn>1</mn></mfenced></msup><mo>=</mo><mstyle displaystyle="true"><mrow><munderover><mrow><mo>∑</mo></mrow><mi>ω</mi><mspace width="1em" /></munderover></mrow></mstyle><mstyle displaystyle="true"><mrow><munderover><mrow><mo>∑</mo></mrow><mi>s</mi><mspace width="1em" /></munderover></mrow></mstyle><msubsup><mi mathvariant="bold">G</mi><mn>0</mn><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mfenced><msub><mi mathvariant="bold">G</mi><mi>s</mi></msub><mo>−</mo><msub><mi mathvariant="bold">G</mi><mrow><mn>0</mn><mi>s</mi></mrow></msub></mfenced><msubsup><mi mathvariant="bold">G</mi><mrow><mn>0</mn><mi>s</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msubsup></mrow></math><img file="EP2067112B1_D0019.tif" /></maths> where <b>G</b><i>_s</i> is the measured data for source s, <b>G</b>_0<i>s</i> is the simulated data through the reference model for source s and <b>G</b>₀<i>_s</i>^<i>-</i>1 can be interpreted as the downward extrapolated source signature from source <i>s</i>. Equation 17 when implemented in the frequency domain can be interpreted as follows: (1) Downward extrapolate through the reference model the source signature for each source (the <b>G</b>_0<i>s</i>^-1 term), (2) For each source, downward extrapolate the receivers of the residual data through the reference model (the <b>G</b>₀^-1(<b>G</b><i>_s</i><b><i>-</i>G</b>_0<i>s</i>) term), (3) multiply these two fields then sum over all sources and frequencies. The downward extrapolations in this recipe can be carried out using geophysical simulation software, for example using finite differences.

0048The simultaneous encoded-source technique can be applied to Equation 17 as follows: <maths id="math0020" num="(18)"><math display="block"><mrow><msup><mrow><mover><mi mathvariant="bold">V</mi><mrow><mo>˜</mo></mrow></mover></mrow><mfenced><mn>1</mn></mfenced></msup><mo>=</mo><mstyle displaystyle="true"><mrow><munderover><mrow><mo>∑</mo></mrow><mi>ω</mi><mspace width="1em" /></munderover></mrow></mstyle><msubsup><mi mathvariant="bold">G</mi><mn>0</mn><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mfenced open="[" close="]"><mstyle displaystyle="true"><mrow><munderover><mrow><mo>∑</mo></mrow><mi>s</mi><mspace width="1em" /></munderover></mrow></mstyle><msup><mi>e</mi><mrow><msub><mi mathvariant="italic">iφ</mi><mi>s</mi></msub><mfenced><mi>ω</mi></mfenced></mrow></msup><msub><mi mathvariant="bold">G</mi><mi>s</mi></msub><mo>−</mo><mstyle displaystyle="true"><mrow><munderover><mrow><mo>∑</mo></mrow><mi>s</mi><mspace width="1em" /></munderover></mrow></mstyle><msup><mi>e</mi><mrow><msub><mi mathvariant="italic">iφ</mi><mi>s</mi></msub><mfenced><mi>ω</mi></mfenced></mrow></msup><msub><mi mathvariant="bold">G</mi><mrow><mn>0</mn><mi>s</mi></mrow></msub></mfenced><mstyle displaystyle="true"><mrow><munderover><mrow><mo>∑</mo></mrow><mrow><mi>s</mi><mo>′</mo></mrow><mspace width="1em" /></munderover></mrow></mstyle><msup><mfenced><msup><mi>e</mi><mrow><msub><mi mathvariant="italic">iφ</mi><mrow><mi>s</mi><mo>′</mo></mrow></msub><mfenced><mi>ω</mi></mfenced></mrow></msup><msub><mi mathvariant="bold">G</mi><mrow><mn>0</mn><mo>′</mo><mi>s</mi></mrow></msub></mfenced><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math><img file="EP2067112B1_D0020.tif" /></maths> where a choice has been made to encode by applying the phase function <i>φ_s</i>(<i>ω</i>) which depends on the source and may depend on the frequency <i>ω</i>. As was the case for Equation 17, Equation 18 can be implemented by: (1) Encoding and summing the measured data (the first summation in brackets), (2) Forward simulating the data that would be acquired from simultaneous encoded sources using the same encoding as in step 1 (the second term in the brackets, (3) Subtract the result for step 2 from the result from step 1, (4) Downward extrapolate the data computed in step 3 (the first <b>G</b>₀^-1 term applied to the bracketed term), (5) Downward extrapolate the simultaneous encoded sources encoded with the same encoding as in step 1, (6) Multiply these two fields and sum over all frequencies. Note that in this recipe the simulations are all performed only once for the entire set of simultaneous encoded sources, as opposed to once for each source as was the case for Equation 17. Thus, Equation 18 requires much less compute time than Equation 17.

0049Separating the summations over s and s' into portions where <i>s = s'</i> and <i>s ≠ s'</i> in Equation 18 gives: <maths id="math0021" num="(19)"><math display="block"><mrow><msup><mrow><mover><mi mathvariant="bold">V</mi><mrow><mo>˜</mo></mrow></mover></mrow><mfenced><mn>1</mn></mfenced></msup><mo>=</mo><mstyle displaystyle="true"><mrow><munderover><mrow><mo>∑</mo></mrow><mi>ω</mi><mspace width="1em" /></munderover></mrow></mstyle><mstyle displaystyle="true"><mrow><munderover><mrow><mo>∑</mo></mrow><mi>s</mi><mspace width="1em" /></munderover></mrow></mstyle><msubsup><mi mathvariant="bold">G</mi><mn>0</mn><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mfenced><msub><mi mathvariant="bold">G</mi><mi>s</mi></msub><mo>−</mo><msub><mi mathvariant="bold">G</mi><mrow><mn>0</mn><mi>s</mi></mrow></msub></mfenced><msubsup><mi mathvariant="bold">G</mi><mrow><mn>0</mn><mi>s</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo>+</mo><mstyle displaystyle="true"><mrow><munderover><mrow><mo>∑</mo></mrow><mi>ω</mi><mspace width="1em" /></munderover></mrow></mstyle><msubsup><mi mathvariant="bold">G</mi><mn>0</mn><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mstyle displaystyle="true"><mrow><munderover><mrow><mo>∑</mo></mrow><mi>s</mi><mspace width="1em" /></munderover></mrow></mstyle><mstyle displaystyle="true"><mrow><munderover><mrow><mo>∑</mo></mrow><mrow><mi>s</mi><mo>′</mo><mo>≠</mo><mi>s</mi></mrow><mspace width="1em" /></munderover></mrow></mstyle><msup><mi>e</mi><mrow><msub><mi mathvariant="italic">iφ</mi><mi>s</mi></msub><mfenced><mi>ω</mi></mfenced></mrow></msup><msup><mi>e</mi><mrow><msub><mi mathvariant="italic">iφ</mi><mrow><mi>s</mi><mo>′</mo></mrow></msub><mfenced><mi>ω</mi></mfenced></mrow></msup><mfenced><msub><mi mathvariant="bold">G</mi><mi>s</mi></msub><mo>−</mo><msub><mi mathvariant="bold">G</mi><mrow><mn>0</mn><mi>s</mi></mrow></msub></mfenced><msubsup><mi mathvariant="bold">G</mi><mrow><mn>0</mn><mi>s</mi><mo>′</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msubsup></mrow></math><img file="EP2067112B1_D0021.tif" /></maths> The first term in Equation 19 may be recognized as Equation 17 and therefore: <maths id="math0022" num="(20)"><math display="block"><mrow><msup><mrow><mover><mi mathvariant="bold">V</mi><mrow><mo>˜</mo></mrow></mover></mrow><mfenced><mn>1</mn></mfenced></msup><mo>=</mo><msup><mi mathvariant="bold">V</mi><mfenced><mn>1</mn></mfenced></msup><mo>+</mo><mi mathvariant="italic">crossterms</mi></mrow></math><img file="EP2067112B1_D0022.tif" /></maths>

0050The cross terms in Equation 19 will be small if <i>φ_s</i> ≠ <i>φ_s'</i> when <i>s ≠ s'.</i> Thus, as was the case for cost function optimization, the simultaneous encoded-source method speeds up computation of the first term of the series and gives a result that is similar to the much more expensive sequential source method. The same simultaneous encoded-source technique can be applied to higher order terms in the series such as the second and third-order terms in Equations 15 and 16.

<i>Further considerations</i>

0051The present inventive method can also be used in conjunction with various types of generalized source techniques, such as those suggested by Berkhout [6]. In this case, rather than encoding different point source gather signatures, one would encode the signatures for different synthesized plane waves.

0052A primary advantage of the present invention is that it allows a larger number of gathers to be computed simultaneously. Furthermore, this efficiency is gained without sacrificing the quality of the cost function. The invention is less subject to noise artifacts than other simultaneous source techniques because the inversion's being iterative implies that the noise artifacts will be greatly suppressed as the global minimum of the cost function is approached.

0053Some variations on the embodiments described above include: <ul id="ul0004" list-style="bullet"><li>The <i>c_g</i> encoding functions can be changed for each computation of the cost function. In at least some instances, using different random phase encodings for each computation of the cost function further reduces the effect of the cross terms in Equation 8.</li><li>In some cases (e.g., when the source sampling is denser than the receiver sampling) it may be advantageous to use reciprocity to treat the actual receivers as computational sources, and encode the receivers instead of the sources.</li><li>This invention is not limited to single-component point receivers. For example, the receivers could be receiver arrays or they could be multicomponent receivers.</li><li>The method may be improved by optimizing the encoding to yield the highest quality cost function. For example the encoding functions could be optimized to reduce the number of local minima in the cost function. The encoding functions could be optimized either by manual inspection of tests performed using different encoding functions or using an automated optimization procedure.</li><li>Acquisition of simultaneous encoded-source data could result in significant geophysical data acquisition cost savings.</li><li>For marine seismic data surveys, it would be very efficient to acquire encoded source data from simultaneously operating marine vibrators that operate continuously while in motion.</li><li>Other definitions for the cost function may be used, including the use of a different norm (e.g. L1 norm (absolute value) instead of L2 norm), and additional terms to regularize and stabilize the inversion (e.g. terms that would penalize models that aren't smooth or models that are not sparse).</li></ul>

0054While the invention includes many embodiments, a typical embodiment might include the following features: <ol id="ol0007"><li>1. The input gathers are common point source gathers.</li><li>2. The encoding signatures <b>30</b> and <b>110</b> are changed between iterations.</li><li>3. The encoding signatures <b>30</b> and <b>110</b> are chosen to be random phase signatures from Romero et. al. [12]. Such a signature can be made simply by making a sequence that consists of time samples which are a uniform pseudo-random sequence.</li><li>4. In step <b>40</b>, the gathers are encoded by convolving each trace in the gather with that gather's encoding signature.</li><li>5. In step <b>130</b>, the forward modeling is performed with a finite difference modeling code in the space-time domain.</li><li>6. In step <b>140</b>, the cost function is computed using an L2 norm.</li></ol>

<i>Examples</i>

0055<figref idref="f0003 f0004 f0005 f0006">Figures 3-8</figref> represent a synthetic example of computing the cost function using the present invention and comparison with the conventional sequential source method. The geophysical properties model in this simple example is just a model of the acoustic wave velocity. <figref idref="f0003">Figure 3</figref> is the base velocity model (the model that will be inverted for) for this example. The shading indicates the velocity at each depth. The background of this model is a linear gradient starting at 1500 m/s at the top of the model and having a gradient of 0.5 sec^-1. Thirty-two 64 m thick horizontal layers (<b>210</b>) having a plus or minus 100 m/s velocity are added to the background gradient. The darker horizontal bands in <figref idref="f0003">Figure 3</figref> represent layers where 100 m/s is added to the linear gradient background, and the alternating lighter horizontal bands represent layers where 100 m/s is subtracted from the linear gradient background. Finally a rectangular anomaly (<b>220</b>) that is 256 m tall and 256 m wide and having a velocity perturbation of 500 m/s is added to the horizontally layered model.

0056Conventional sequential point source data (corresponding to item <b>10</b> in <figref idref="f0001">Figure 1</figref>) were simulated from the model in <figref idref="f0003">Figure 3</figref>. 256 common point source gathers were computed, and <figref idref="f0004">Fig. 4</figref> shows the first three of these gathers. These gathers have a six second trace length and are sampled at 0.8 msec. The source signature (corresponding to <i>w_s</i> in Equation 2) is a 20 Hz Ricker wavelet. The distance between sources is 16 m and the distance between receivers is 4 m. The sources and receivers cover the entire surface of the model, and the receivers are stationary.

0057The flow outlined in <figref idref="f0001">Fig. 1</figref> is used to generate simultaneous encoded-source data from the sequential source data shown in <figref idref="f0004">Fig. 4</figref>. In step <b>20</b> of <figref idref="f0001">Fig. 1</figref>, all. 256 simulated sequential gathers were formed into one group. These gathers were then encoded by convolving the traces from each point source gather with a 2048 sample (1.6384 sec long) uniform pseudo-random sequence. A different random sequence was used for each point source gather. These gathers were then summed to produce the single simultaneous encoded-source gather shown in <figref idref="f0004">Fig. 5</figref>. It should be noted that this process has converted 256 sequential source gathers to a single simultaneous encoded-source gather.

0058To compute a cost function, the base model is perturbed and seismic data are simulated from this perturbed model. For this example the model was perturbed by changing the depth of the rectangular anomaly. The depth of the anomaly was perturbed over a range of -400 to +400 m relative to its depth in the base model. One perturbation of that model is shown in <figref idref="f0005">Fig. 6</figref>, with the anomaly indicated at <b>310</b>.

0059For each perturbation of the base model a single gather of simultaneous encoded-source data was simulated to yield a gather of traces similar to the base data shown in <figref idref="f0004">Fig. 5</figref>. The encoding signatures used to simulate these perturbed gathers were exactly the same as those used to encode the base data in <figref idref="f0004">Fig. 5</figref>. The cost function from Equation 6 was computed for each perturbed model by subtracting the perturbed data from the base data and computing the L2 norm of the result. <figref idref="f0006">Figure 7</figref> is a graph of this simultaneous encoded-source cost function. This cost function may be compared to the conventional sequential source cost function for the same model perturbations shown in <figref idref="f0006">Fig. 8</figref> (computed using the data in <figref idref="f0004">Fig. 4</figref> as the base data and then simulating sequential source data from the perturbed models). <figref idref="f0006">Figure 8</figref> corresponds to the cost function in Equation 2 with <i>N</i> = 2. The horizontal axis in <figref idref="f0006">Figs. 7 and 8</figref> is the perturbation of the depth of the rectangular anomaly relative to its depth in the base model. Thus, a perturbation of zero corresponds to the base model. It is important to note that for this example the simultaneous encoded-source cost function was computed 256 times faster than the sequential source cost function.

0060Two things are immediately noticeable upon inspection of <figref idref="f0006">Figs. 7 and 8</figref>. One is that both of these cost functions have their global minimum (<b>410</b> for the simultaneous source data and <b>510</b> for the sequential source data) at zero perturbation as should be the case for an accurate inversion. The second thing to note is that both cost functions have the same number of local minima (<b>420</b> for the simultaneous source data and <b>520</b> for the sequential source data), and that these local minima are located at about the same perturbation values. While local minima are not desirable in a cost function, the local minima structure of the simultaneous encoded-source cost function is similar to the sequential source cost function. Thus, the simultaneous encoded-source cost function (<figref idref="f0006">Fig. 7</figref>) is just as good as the sequential source cost function (<figref idref="f0006">Fig. 8</figref>) for seismic inversion.

0061The factor of 256 computational time reduction of the simultaneous encoded-source cost function, along with similar quality of the two cost functions for seismic inversion, leads to the conclusion that for this example the simultaneous encoded-source cost function is strongly preferred. The perturbed models represent the various model guesses that might be used in a real exercise in order to determine which gives the closest fit, as measured by the cost function, to the measured data.

0062Finally, to demonstrate the importance of encoding the gathers before summing, <figref idref="f0007">Fig. 9</figref> shows the cost function that would result from using Mora's [9] suggestion of inverting super shot gathers. This cost function was computed in a manner similar to that <figref idref="f0006">Figure 7</figref> except that the source gathers were not encoded before summing. This sum violates Mora's suggestion that the sources should be widely spaced (these sources are at a 16 m spacing). However, this is a fair comparison with the simultaneous encoded-source method suggested in this patent, because the computational speedup for the cost function of <figref idref="f0007">Fig. 9</figref> is equal to that for <figref idref="f0006">Fig. 7</figref>, while Mora's widely spaced source method would result in much less speedup. Note that the global minimum for the super shot gather data is at zero perturbation (<b>610</b>), which is good. On the other hand, the cost function shown in <figref idref="f0007">Fig. 9</figref> has many more local minima (<b>620</b>) than either the cost functions in <figref idref="f0006">Fig. 7 or Fig. 8</figref>. Thus, while this cost function achieves the same computational speedup as the simultaneous encoded-source method of this patent, it is of much lower quality for inversion.

0063The foregoing application is directed to particular embodiments of the present invention for the purpose of illustrating it. It will be apparent, however, to one skilled in the art, that many modifications and variations to the embodiments described herein are possible. All such modifications and variations are intended to be within the scope of the present invention, as defined in the appended claims. Persons skilled in the art will readily recognize that in preferred embodiments of the invention, at least some of the steps in the present inventive method are performed on a computer, i.e. the invention is computer implemented. In such cases, the resulting updated physical properties model may either be downloaded or saved to computer storage.

<i>References</i>

0064<ol id="ol0008"><li>1. <nplcit id="ncit0001" npl-type="s"><text>Tarantola, A., "Inversion of seismic reflection data in the acoustic approximation," Geophysics 49, 1259-1266 (1984</text></nplcit>).</li><li>2. <nplcit id="ncit0002" npl-type="s"><text>Sirgue, L., and Pratt G. "Efficient waveform inversion and imaging: A strategy for selecting temporal frequencies," Geophysics 69, 231-248 (2004</text></nplcit>).</li><li>3. <nplcit id="ncit0003" npl-type="s"><text>Weglein, A. B., Araujo, F. V., Carvalho, P. M., Stolt, R. H., Matson, K. H., Coates, R. T., Corrigan, D., Foster, D. J., Shaw, S. A., and Zhang, H., "Inverse scattering series and seismic exploration," Inverse Problems 19, R27-R83 (2003</text></nplcit>).</li><li>4. <nplcit id="ncit0004" npl-type="s"><text>Fallat, M. R., Dosso, S. E., "Geoacoustic inversion via local, global, and hybrid algorithms," Journal of the Acoustical Society of America 105, 3219-3230 (1999</text></nplcit>).</li><li>5. <nplcit id="ncit0005" npl-type="s"><text>Van Manen, D. J., Robertsson, J.O.A., Curtis, A., "Making wave by time reversal," SEG International Exposition and 75th Annual Meeting Expanded Abstracts, 1763-1766 (2005</text></nplcit>).</li><li>6. <nplcit id="ncit0006" npl-type="s"><text>Berkhout, A. J., "Areal shot record technology," Journal of Seismic Exploration 1, 251-264 (1992</text></nplcit>).</li><li>7. <nplcit id="ncit0007" npl-type="s"><text>Zhang, Y., Sun, J., Notfors, C., Gray, S. H., Cherris, L., Young, J., "Delayed-shot 3D depth migration," Geophysics 70, E21-E28 (2005</text></nplcit>).</li><li>8. Van Riel, P., and Hendrik, W. J. D., "Method of estimating elastic and compositional parameters from seismic and echo-acoustic data," <patcit id="pcit0005" dnum="US6876928B"><text>U.S. Patent No. 6,876,928 (2005</text></patcit>).</li><li>9. <nplcit id="ncit0008" npl-type="s"><text>Mora, P., "Nonlinear two-dimensional elastic inversion of multi-offset seismic data," Geophysics 52, 1211-1228 (1987</text></nplcit>).</li><li>10. Ober, C. C., Romero, L. A., Ghiglia, D. C., "Method of Migrating Seismic Records," <patcit id="pcit0006" dnum="US6021094A"><text>U.S. Patent No. 6,021,094 (2000</text></patcit>).</li><li>11. Ikelle, L. T., "Multi-shooting approach to seismic modeling and acquisition," <patcit id="pcit0007" dnum="US6327537B"><text>U.S. Patent No. 6,327,537 (2001</text></patcit>).</li><li>12. <nplcit id="ncit0009" npl-type="s"><text>Romero, L. A., Ghiglia, D. C., Ober, C. C., Morton, S. A., "Phase encoding of shot records in prestack migration," Geophysics 65, 426-436 (2000</text></nplcit>).</li><li>13. <nplcit id="ncit0010" npl-type="s"><text>Jing X., Finn, C. J., Dickens, T. A., Willen, D. E., "Encoding multiple shot gathers in prestack migration," SEG International Exposition and 70th Annual Meeting Expanded Abstracts, 786-789 (2000</text></nplcit>).</li></ol>