GB2538804A

GB2538804A - Improved method for inversion modelling

Info

Publication number: GB2538804A
Application number: GB1509332.1A
Authority: GB
Inventors: Warner Mike; Guasch Lluis
Original assignee: SUB SALT SOLUTIONS Ltd
Current assignee: SUB SALT SOLUTIONS Ltd
Priority date: 2015-05-29
Filing date: 2015-05-29
Publication date: 2016-11-30
Also published as: WO2016193180A1; GB201509332D0

Abstract

The method comprises generating a geophysical representation of a portion of the volume of the Earth from a seismic measurement of at least one physical parameter. The method comprises: providing 100 an observed data set, comprising data values derived from seismic measured values of said portion of the volume of the Earth; generating 104, using a subsurface model comprising a plurality of model coefficients, a modelled data set comprising a plurality of modelled data values; updating the subsurface model by: generating 106 an objective function which measures the mismatch/similarity between the observed and predicted data set; determining 108, using a first proportion of the total number of data values of the observed data set, the gradient of the objective function; determining 110, using a reduced objective function utilising a second proportion of the total number of data values, the step length for a subsurface model update, wherein the second proportion comprises 40% or less of the total number of data values of the first proportion of the observed data set; and updating 112 the subsurface model using the determined gradient and step length; and providing an updated subsurface model for subsurface exploration.

Description

Improved Method for Inversion Modelling The present invention relates to a method of, and apparatus for, inversion modelling. More particularly, the present invention relates to an improved methodology for inverse problems which enables a more accurate determination of step length. Additionally, the present invention relates to an improved method of, and apparatus for, inversion modelling which requires fewer computational resources than known methods.

There is significant interest in surveying sections of the Earth to detect natural mineral resources or other sites of geological interest. One approach to detection of such natural features is seismic surveying. Seismic surveys are the principal means by which the petroleum industry can explore the subsurface of the Earth for oil and gas reserves. Typically, seismic survey data is acquired and analysed with regard to identifying locations suitable for direct investigation of the sub-surface by drilling. Seismic surveying also has applications within the mining industry and within other industrial sectors that have an interest in details of the subsurface of the Earth.

In a seismic survey, one or more natural or artificial seismic sources are arranged to generate vibrational energy which is directed into the subsurface of the Earth. Reflected, refracted and other signals returned from subsurface features are then detected and analysed. These signals can be used to map the subsurface of the Earth.

A schematic illustration of an experimental set up 10 for an undersea seismic survey is shown in Figure 1. However, this example is intended to be non-limiting and an equivalent experiment can be carried out on land. The present invention is applicable to subsurface exploration in any suitable environment, for example land or marine measurements of a portion of the subsurface of the Earth. The present invention may be applicable to identification of numerous subsurface resources, and is intended to include oil exploration and gas prospecting.

The skilled person would be readily aware of the suitable environments in which data could be gathered for analysis and exploration purposes as set out in the present disclosure.

In this example, the experimental set up 10 comprises a source 12. In this example, the source 12 is located on a ship 14, although this need not be the case and the source may be located on land, or within the sub-surface, or on any other suitable vessel or vehicle.

The source 12 generates acoustic and/or elastic waves having sufficient vibrational energy to penetrate the subsurface of the Earth and generate sufficient return signals to aid useful detection. The source 12 may comprise, for example, an explosive device, or alternatively an air gun or other mechanical device capable of creating sufficient vibrational disturbance.

Commonly, for many seismic survey experiments a single source is used which is shot from multiple locations. Naturally occurring sources may also be employed.

A plurality of detectors 16 is provided. The detectors 16 may comprise any suitable vibrational detection apparatus. Commonly, two types of device are used. Geophones which detect particle motion, and hydrophones which detect pressure variations. Commonly, a large number of detectors 16 are laid out in lines for 2D data acquisition. Alternatively, the detectors 16 can be arranged in sets of lines or in a grid for 3D data acquisition. Detectors 16 may also be located within the subsurface, for example down boreholes. The detectors 16 are connected to trace acquisition apparatus such as a computer or other electronic storage device. In this example, the acquisition apparatus is located on a further ship 18. However, this need not be the case and other arrangements are possible.

In use, elastic waves 20 generated by the source 12 propagate into the subsurface 22 of the Earth. The subsurface 22, in general, comprises one or more layers or strata 24, 26, 28 formed from rock or other materials. The elastic waves 20 are transmitted and refracted through the layers and/or reflected off the interfaces between them and/or scattered from other heterogeneities in the sub-surface and a plurality of return signals 30 is detected by the detectors 16.

In general, the returning signals 30 comprise elastic waves having different polarisations. Primary or pressure waves (known as P-waves) are approximately longitudinally polarised and comprise alternating rarefactions and compressions in the medium in which the wave is travelling. In other words, in an isotropic environment, the oscillations of a P-wave are parallel to the direction of propagation. P-waves typically have the highest velocity and so are typically the first to be recorded. P-waves travel at a velocity Vp in a particular medium. Vp may vary with position, with direction of travel, with frequency, and with other parameters, and is, effectively, the speed of sound in a medium. It is this quantity Vp which is most commonly of particular interest in seismic inversion.

Shear or secondary waves (known as S-waves) may also be generated. S-waves have an approximately transverse polarisation. In other words, in an isotropic environment, the polarisation is perpendicular to the direction of propagation. S-waves are in general, more slowly moving than P-waves in materials such as rock. Whilst S-wave analysis is possible and falls within the scope of the present invention, the following description will focus on the analysis of P-waves.

A seismic survey is typically composed of a large number of individual source excitation events. The Earth's response to these events is recorded at each receiver location, as a w seismic trace for each source-receiver pair. For a two dimensional survey, the tens of thousands of individual traces may be taken. For the three dimensional case, this number may run into the millions.

A seismic trace comprises a sequence of measurements in time made by one or more of the multiplicity of detectors 16, of the returning reflected, refracted and/or scattered acoustic and/or elastic waves 30 originating from the source 12. In general, a partial reflection of the acoustic wave 20 occurs at a boundary or interface between two dissimilar materials, or when the elastic properties of a material changes. Traces are usually sampled in time at discrete intervals of the order of milliseconds.

Seismic surveys at the surface or seabed can be used to extract rock properties and construct reflectivity images of the subsurface. Such surveys can, with the correct interpretation, provide an accurate picture of the subsurface structure of the portion of the Earth being surveyed. This may include subsurface features associated with mineral resources such as hydrocarbons (for example, oil and natural gas). Features of interest in prospecting include: faults, folds, anticlines, unconformities, salt domes, reefs.

Key to this process of modelling and imaging a portion of earth is the seismic velocity V. In a portion of the volume of the Earth, Vp may be estimated in various ways.

Full-wavefield inversion (FWD is one known method for analysing seismic data. FWI is able to produce models of physical properties such as Vp in a subsurface region that have high fidelity and that are well resolved spatially. FVVI seeks to extract the properties of subsurface rocks from a given seismic dataset recorded at the surface or seabed. A detailed velocity estimate can be produced using an accurate model with variations on the scale of a seismic wavelength.

The FWI technique involves generating a two or three dimensional model to represent the measured portion of the Earth and attempting to modify the properties and parameters of the Earth model to generate predicted data that matches the experimentally obtained seismic trace data.

The predicted data is calculated from the subsurface model typically using the full two-way wave equation. The final model has potentially far higher resolution and accuracy however the method can fail due to the sensitivity of the predicted waveforms to the model. FWI is an iterative process requiring a starting model. A sufficiently accurate starting model for FWI may be provided by travel-time tomography.

FWI can extract many physical properties (Vp and Vs velocities, attenuation, density, anisotropy) of the modelled portion of the Earth. However, Vp, the P-wave velocity, is a particularly important parameter which the subsequent construction of the other parameters depends heavily upon. Nevertheless, other parameters may be used with the present invention, either alone or in combination. The nature and number of parameters used in a model of a portion of the Earth will be readily apparent to the skilled person.

An example of a basic starting model is shown in Figure 2. The model shows a simple estimation of the subsurface of a portion of the Earth. The source of acoustic waves is shown as a star and a plurality of receivers shown as circles. Both the source and the receivers are located at or above the seabed. For clarity, the basic model shows a gradually increasing Vp with increasing depth without any detailed structure.

A modelled seismic gather is shown in Figure 3 for one shot and one line of receivers. The modelled seismic traces in the gather have been generated using the basic model shown in Figure 2. This is done by applying the isotropic acoustic wave equation to the model of Figure 2 and then modelling the reflected and refracted signals as they would be detected.

The modelled seismic shot gather is made up of individual traces at surface receiver positions showing pressure recorded as a function of time.

In general, the parameters of the model are estimated at a plurality of points set out in a grid or volume, but they may be estimated from any suitable parameterisafion. The model is used to generate a modelled representation of the seismic data set. The modelled data set is then compared to the real-world experimentally obtained seismic data set. Then, through use of a convergent numerical iteration, the parameters of the model are modified until the modelled data set generated from the Earth model matches the actual observed seismic data to a sufficient degree of accuracy or until sufficient convergence is obtained. Examples of this technique are illustrated in "An overview of full-waveform inversion in exploration geophysics", J. Virieux and S. Operto, Geophysics Vol. 74 No. 6 and US-A-7,725,266.

FWI typically operates on the principle of iteratively updating the starting model to minimise or maximise an objective function through repeated steepest-descent direction calculation, or an analogous technique. An objective function represents some measure of the mismatch or some measure of similarity between the recorded data and the predicted data. A measure of mismatch, obtained for example by subtracting two traces, should be minimised; whereas a measure of similarity, obtained for example by cross-correlating two traces, should be maximised.

Due to the non-linearity in the relationship between the model and the data, the objective function used in FWI will oscillate with changes in the model. This makes it necessary to have a sufficiently accurate starting model for global minimum convergence. The objective function can be formulated in the frequency domain, the time domain or other suitable domain. The choice of domain allows the use of pre-conditioning on either the data or the model update direction that could improve convergence or the linearity of the inverse problem.

Frequency domain inversion is equivalent to time domain inversion if all the frequencies are inverted simultaneously. However, the global minimum broadens at lower frequencies reducing how accurate the starting model needs to be for localised inversion to be successful.

There are numerous ways to quantify the difference (also known as the residual) between the datasets. The residual expresses the misfit between the two datasets as a single number. This parameter is known as the objective function, although it often takes other names such as the objective function, the cost function or the functional. The objective function f is a real, positive, scalar quantity, and it is a function of the model m.

An example of a generic objective function f is shown in equation 1). 1)

Where d(m) is the data where d(m) is the data modelled using a model m and do is the observed data.

FWI is a local iterative inversion scheme. A starting model mo that is assumed to be sufficiently close to the true, ideal model is prepared. The process then makes a series of step-wise improvements to this model which successively reduces the objective function towards zero. Therefore, across an iterative step of the calculation, the objective function needs to be considered for a starting model mo and a new model m = mo + Om.

A typical, generalised FWI-type method involves the following steps: 1 Start from model mo; 2 Evaluate the gradient of the objective function for the current model; 3 Find the step length a; 4 Subtract a times the gradient from the current model to obtain a new model; and Iterate from step 2 using the new model until the objective function is minimised.

Therefore, the step length a controls the scaling of an update to be applied to the model. This is, clearly, a critical aspect of the inversion process. Consider a method starting from a starting model mo that generates data do and residuals brio. A new model mk,fri for the k+1th iteration can be defined as: 2) Ink+1 = + athnk Where ornk+1 is the normalised update for the kth iteration of the model Mk. The step length a can be calculated in a number of different ways. Two possible approaches are as follows: 1. Fit a parabola shape using three values of the functional f. The three values are computed using: the current model mk and two new models mk + aouessi Oink and mk + aouesszamk-This approach requires two additional forward iterations (one for each guess of a) to evaluate the functional values for each new model. Figure 4 illustrates an example of this.

In Figure 4, the value of the functional f is plotted as a function of a. The thicker line shows the actual distribution of f as a function of a, and the thinner line the estimated parabola. As shown, the parabola is estimated based on the two obtained a values, agu",s, and aguess2. The optimal step length is the value of a where f is at a minimum. As shown, this method can result in errors when only a few points are used to generate an estimated parabola. As shown, the estimated optimal step-length is shifted from the actual (real) optimal step-length.

2. An alternative approach is to assume a linear relation between the data residuals and the model parameters. This approach relies upon the Born approximation, which assumes that the perturbation to a wavefield produced by changing a model is linearly related to the change in the model. This is equivalent to considering only first-order scattering by the perturbation. In other words, by applying the Born approximation, it can be assumed that changes in the model perturbation are linearly related to changes in the residuals.

However, both of the above methods have disadvantages. A parabolic fit may not be sufficiently accurate and useful when only using three values of a as known in existing arrangements. Further, given the computational cost of the method, it is not feasible in known methods to use additional guess values.

The second approach also has disadvantages. In conventional FVVI, the residuals are 6d. However, the Born approximation may not hold in all cases, leading to an inaccurate step length determination. For example, in Adaptive Waveform Inversion (AWI), model-dependent Wiener filters w(m) are defined which enable the modelled data to be fitted to the measured data. The method of AWI is described in United Kingdom Patent GB2509223B.

In AWI, the residuals take the form set out in equation 3): 3) where w is the Wiener filter operable to tum predicted data into observed data. In this method, the residuals and the model parameters are not linearly related, so the assumptions made in the FWI case are no longer valid.

As a result, the Born approximation does not apply. Furthermore, typically, the shape of the functional in the update direction (the thicker line in Figure 4) is a combination of concave and convex curves for this method. Therefore, attempts to fit a parabola as set out in the first method above can result in inaccurate estimates for the step length.

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However, whilst of particular application to the AWI methodology, the problem of inaccuracies in step length calculation is of relevance to all types of FWI, and to inversion problems generally.

One approach to addressing this problem is to sample the functional curve more frequently -for example, to use a number of guess values of a greater than two to increase the number of sampling points along the curve. This will then provide more information on the true shape of the curve so that the minimum can be identified more readily. However, a problem with this approach is that, for each additional sample point, a full forward model iteration would be w required. This would quickly be prohibitively expensive in terms of computational resources required. For example, the cost of a single iteration in FWI, AWI or related methods is at least three (and potentially up to six) forward runs of the model. Therefore, if it was desired to sample twenty points of the functional, the method would require around six times the computing resources. Such additional computational expense is likely to be unacceptable in most environments.

Therefore, a technical problem exists in the art that no accurate and computationally efficient method for calculating the step length of an iteration is known in the art. The present invention seeks to address these issues.

According to a first aspect of the present invention, there is provided a method of subsurface exploration, the method comprising generating a geophysical representation of a portion of the volume of the Earth from a seismic measurement of at least one physical parameter, the method comprising the steps of: providing an observed data set, the observed data set comprising data values derived from seismic measured values of said portion of the volume of the Earth; generating, using a subsurface model of a portion of the Earth comprising a plurality of model coefficients, a modelled data set comprising a plurality of modelled data values; updating the subsurface model by: generating an objective function operable to measure the mismatch or similarity between the observed data set and the predicted data set; determining, using a first proportion of the total number of data values of the observed data set, the gradient of the objective function; determining, using a reduced objective function utilising a second proportion of the total number of data values of the observed data set, the step length for a subsurface model update, wherein the second proportion comprises 40% or less of the total number of data values of the first proportion of the observed data set; and updating the subsurface model of a portion of the Earth using the determined gradient and step length; and providing an updated subsurface model of a portion of the Earth for subsurface exploration.

In one embodiment, the second proportion is 30% or less of the total number of data values of the first proportion of the observed data set. In one embodiment, the second proportion is 15% or less of the total number of data values of the first proportion of the observed data set.

In one embodiment, the second proportion is 10% or less of the total number of data values of the first proportion of the observed data set. In one embodiment, the second proportion is 5% or less of the total number of data values of the first proportion of the observed data set. In one embodiment, the first proportion is 100% of the total number of data values in the observed data set.

In one embodiment, the observed data set comprises a plurality of shots, each shot comprising a plurality of seismic traces grouped by source or receiver location and comprising data values derived from seismic measured values of said portion of the volume of the Earth; and the modelled data set comprises a corresponding plurality of shots comprising a plurality of modelled traces grouped by source or receiver location and comprising modelled data values.

In one embodiment, the first proportion of the observed data set comprises a first number of shots and the second proportion of the observed data set comprises a second number of shots, the second number being 40% or less than the first number. In one embodiment, the second number is 30% or less of the first number. In one embodiment, the second number is 15% or less of the first number. In one embodiment, the second number is 10% or less of the first number. In one embodiment, the second number is 5% or less of the first number.

In one embodiment, step f) comprises determining the step length for a subsurface model update by obtaining a plurality of trial values of the step length and identifying the minimum step length therefrom.

In one embodiment, the number of trial values of the step length is 5 or more. In one embodiment, the number of trial values of the step length is 10 or more. In one embodiment, the number of trial values of the step length is 50 or more.

In one embodiment, step d) further comprises: generating at least one convolutional filter, which when convolved with the observed data, increases the similarity between the predicted data and the convolved observed data; generating at least one convolutional reference filter, which when convolved with the observed data, leaves the observed data unaltered to some level of precision; and generating an objective function operable to measure the similarity or mismatch between the filter coefficients for the or each convolutional filter and the filter coefficients for the or each convolutional reference filter.

In one embodiment, step d) further comprises: generating at least one convolutional filter, which when convolved with the predicted data, increases the similarity between the observed data and the predicted data; generating at least one convolutional reference filter, which when convolved with the predicted data, leaves the predicted data unaltered to some level of precision; and generating an objective function operable to measure the similarity or w mismatch between the filter coefficients for the or each convolutional filter and the filter coefficients for the or each convolutional reference filter.

In one embodiment, prior to step d), the method comprises: generating at least one non-trivial convolutional filter, the or each filter comprising three or more non-zero filter coefficients; generating a convolved observed data set by convolving the or each convolutional filter with said observed seismic data set; generating one or more filter objective functions operable to measure the similarity and/or mismatch between said convolved observed dataset and said predicted dataset; maximising and/or minimising at least one of said filter objective functions by modifying at least one filter coefficient of the or each convolutional filter; and generating one or more pre-determined reference filters comprising at least three reference coefficients.

In one embodiment, said at least one convolutional filter is operable to transform at least a portion of said observed data set to render the observed data set and predicted data set approximations of one another.

In one embodiment, one or more of the convolutional filters comprise Wiener filters.

In one embodiment, a plurality of observed data sets and/or a plurality of modelled data sets is provided.

In one embodiment, the objective function and/or the reduced objective function comprise different functions.

In one embodiment, the objective function and/or the reduced objective function comprises a norm misfit objective function.

In one embodiment, the objective function and/or the reduced objective function comprises an Lrnorm misfit objective function.

In one embodiment, the objective function and/or the reduced objective function comprises a least-squares misfit objective function.

In one embodiment, at least a portion of the observed data set is numerically propagated to a subsurface region of the model that is spatially removed from the location at which they were originally recorded.

In one embodiment, further comprising, subsequent to step g), repeating steps b) to f) for the updated subsurface model of a portion of the Earth until a convergence criterion is met.

In one embodiment, step h) further comprises: utilising said updated model for sub-surface exploration.

In one embodiment, said at least one physical parameter comprises pressure, particle velocity or displacement.

In one embodiment, the observed data set and the predicted data set comprise values of a plurality of physical parameters.

According to a second aspect of the present invention, there is provided a computer program product executable by a programmed or programmable processing apparatus, comprising one or more software portions for performing the steps of the first aspect.

According to a third aspect of the present invention, there is provided a computer usable storage medium having a computer program product according to the second aspect stored 20 thereon Embodiments of the present invention will now be described in detail with reference to the accompanying drawings, in which: Figure 1 is a schematic illustration of a typical seismic survey experiment in which seismic traces are obtained from an undersea portion of the Earth; Figure 2 is a schematic illustration of a basic starting model for full waveform inversion modelling; Figure 3 is a schematic illustration of modelled seismic trace data generated from the basic starting model of Figure 2 for an individual seismic shot; Figure 4 shows a graph of an objective function as a function of the step length a for a typical functional. Guess values of a are shown to enable the illustrated parabolic fit; Figure 5 shows a seismic trace data shot for observed and modelled data sets as a function of time; Figure 6 shows the misfit of the observed and modelled data sets in Figure 5 as a function of time for conventional FWI; Figure 7 shows a method according to a first embodiment of the present invention; Figure 8 shows a method according to a second embodiment of the present invention; Figure 9 shows a graph of an objective function as a function of step length a for a full observed data set of 91 shots and for a reduced observed data set of 3 shots; Figure 10 shows a trial model for testing the accuracy of FWI; Figure 11 shows the convergence achieved for conventional FWI relative to said trial model; Figure 12 shows the convergence achieved for AWI relative to said trial model; and Figure 13 shows a schematic representation of the variation of the objective function with V, and V, with different step sizes The present invention relates to an improved methodology for inverse problems which enables a more accurate determination of step length. Therefore, more accurate and reliable convergence can be obtained using the present invention when compared to known approaches. More particularly, the present invention provides a robust method to calculate the step-length in Full Waveform Inversion (FWD-like methods where the residuals to be minimised do not have a linear relation with the model parameters.

The present invention is further operable to improve the convergence rate of conventional FWI by computing more accurate step-lengths. This, therefore, assists in eliminating the linear assumption commonly adopted by the scientific community that leads to step-length calculations relying on a parabolic fit of relatively a small number of data points The present invention provides a method whereby only a small subset of the original data or datasets is used to perform an explicit line search of the functional values in the update (preconditioned gradient) direction. When the number of points computed in the line search times the number of shots used to evaluate the functional value is equal to the total number of shots used in the inversion (or in any particular iteration) then the cost of both the conventional and our new method is exactly the same.

The following embodiments illustrate the application of the present invention in practice. The first embodiment outlines the general approach of the present invention. The second embodiment relates to a specific implementation of the present invention utilising AWI methods using Wiener filters.

The following embodiments are described with regard to time domain analysis. However, the described methods are equally applicable to other domains, for example, frequency domain analysis. This may be achieved by performing a Fourier analysis on the time domain data and extracting particular frequencies for analysis. These aspects are to be considered to form part of the present disclosure and the skilled person would be readily aware of implementations of this.

A method according to the present invention will now be described with reference to Figure 7. Figure 7 shows a flow diagram of a first embodiment of the present invention.

Step 100: Obtain observed data set Initially, it is necessary to obtain a set of experimentally gathered data in order to initiate subsurface exploration. This may be gathered by an experimental arrangement such as the set up shown and described with reference to Figure 1.

The gathered seismic data may be optionally pre-processed in various ways including by propagating numerically to regions of the surface or subsurface where experimental data have not been acquired directly. The skilled person would readily be able to design and undertake such pre-processing as might be necessary or desirable. With or without such pre-processing, the resultant seismic dataset representing experimentally-gathered data is known as an "observed data set".

As shown in Figure 1, a large number of receivers or detectors 16 are positioned at well known positions on the surface of the portion of the Earth to be explored. The detectors 16 may be arranged in a two dimensional (such as a line) or a three dimensional (such as a grid or plurality of lines) arrangement. The physical location of the detectors 16 is known from, for example, location tracking devices such as GPS devices. Additionally, the location of the source 12 is also well known by similar location tracking means.

The observed data set generally multiple source 12 emission traces known in the art as "shots". A typical observed data set may comprise hundreds of individual traces (or shots). A shot is a group of observed data traces belonging to the same source or receiver. For the same source, a shot would be a group of observed data trace taken at different receiver locations for the same source. Conversely, for a shot identified by individual receiver, a shot would be a group of observed data trace taken at the same receiver location for different source locations. Multiple shots comprise the "observed data set".

Figure 5 shows an example of an observed data shot which forms part of the observed data set do Figure 5 also shows a predicted seismic data trace forming part of a predicted data set dm. The predicted data set will be described in the next steps. Both traces are shown as a function of time (on the vertical axis).

The data comprises pressure as a function of receiver position (on the x-axis) with respect to time (on the y-axis). This is because, in general, a detector such as a hydrophone measures the scalar pressure at its location. However, other arrangements may be used.

The seismic trace data comprises a plurality of observed data points. In one example, each measured discrete data point has a minimum of seven associated location values -three spatial dimensions (x, y and z) for receiver (or detector) position (r), three spatial dimensions (x, y, z) for source location (s), and one temporal dimension measuring the time of observation relative to the time of source initiation, together with pressure magnitude data. The seven coordinates for each discrete data point define its location in space and time.

The seismic trace data also comprises one or more measurement parameters which denote the physical property being measured. In this embodiment, a single measurement parameter, pressure is measured. The observed data set is defined as do and, in this embodiment, is in the time domain. For clarity, the following discussion considers a single source-receiver pair and so r, s are not needed.

However, it is possible to measure other parameters using appropriate technology; for example, to measure the velocity or displacements of particles in three spatial dimensions in addition to the pressure. The present invention is applicable to the measurement of these additional variables.

The actual gathering of the seismic data set is described here for clarity. However, this is not to be taken as limiting and the gathering of the data may or may not form part of the present invention. The present invention simply requires a real-world observed data set upon which analysis can be performed to facilitate subsurface exploration of a portion of the Earth. The method now proceeds to step 102.

Step 102: Provide starting model At step 102, an initial starting model of the specified subsurface portion of the Earth is provided. The model may be provided in either a two dimensional or a three dimensional form. Whilst the illustrated examples are of a two-dimensional form, the skilled person would be readily aware that the present invention is applicable to three dimensional approaches.

The form of the model is not material to the present invention, and may take any suitable form. However, a general example will be described.

A commonly generated model generally consists of values of the coefficient Vp and, possibly, other physical values or coefficients, typically defined over a discrete grid representing the subsurface. Such starting models are routinely generated and represent the general trends of the major features within the subsurface region to be modelled and could be readily generated by the skilled person.

Predicted seismic data may be generated based on an analysis of the acoustic isotropic two-way wave equation as set out below in equation 4): 4) ) p V. i -I V p = 5 \P where the acoustic pressure p and driving source s vary in both space and time, and the acoustic velocity c and density p vary in space. This equation applies to small-amplitude pressure waves propagating within an inhomogeneous, isotropic, non-attenuating, non-dispersive, stationary, fluid medium. It is relatively straightforward to add elastic effects, attenuation and anisotropy to the wave equation. Introduction of these parameters changes the detailed equations and numerical complexity, but not the general approach.

The wave equation 4) represents a linear relationship between a wave field p and the source s that generates the wave field. After discretisation (with, for example, finite differences) we can therefore write equation 4) as a matrix equation 5): 5) Ap = s where p and s are column vectors that represent the source and wavefield at discrete points in space and time, and A is a matrix that represents the discrete numerical implementation of the operator set out in equation 6): 6) 102

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72 (I V) Although the wave equation represents a linear relationship between p and s, it also represents a non-linear relationship between a model m and wavefield p. Thus equation 4) can be rewritten as equation 7): 7) G(m)= p where m is a column vector that contains the model parameters. Commonly these will be the values of c (and p if density is an independent parameter) at every point in the model, but they may be any set of parameters that is sufficient to describe the model, for example slowness 1/c, acoustic modulus c2 p, or impedance cp.

In equation 7), G is not a matrix. Instead it is a non-linear Green's function that describes how to calculate a wavefield p given a model m.

Once the model has been generated, the method then proceeds to step 104 Step 104: Generate predicted data set At step 104, a modelled data set is generated. The predicted data is required to correspond to the same source-receiver location data positions as the actual measured trace data so that the modelled and observed data can be compared. In other words, the predicted data set corresponds discrete point to discrete point to the observed dataset. The predicted data set is generated for the same measurement parameter(s) at the same frequency or frequencies.

From the above analysis, predicted seismic data can be generated for one or more physical parameters in the time domain. If done in the frequency domain it could be done for one or more selected frequencies. This forms the modelled data set d(m). Figure 5 shows an predicted seismic data trace forming part of a predicted data set d(m).

The method now proceeds to step 106 Step 106: Construct objective function At step 106, an objective (or misfit) function is configured. In one example, the objective function is configured to measure the dis-similarity between the modelled and observed data sets (or analogues thereof). Alternatively, an objective function may be configured that measures similarity; in this case, step 108 will operable to maximise rather than minimise the objective function.

The objective function f is a real, positive, scalar quantity, and it is a function of the model m.

In practice, a factor of a half is often included in the definition of the objective function f because it makes later results simpler. An example of a generic objective function f is shown in equation 8): 8) f(m) = -11c1(Tri) -Where d(m) is the data modelled using model m and do is the observed data.

There are numerous ways to quantify the difference ad (also known as the residual) between the datasets. However, amongst the most common is a least-squares formulation where the sum of the squares of the differences between the two datasets is minimised over all sources and receivers and over all times. In other words, in many configurations, a model is sought that minimises the L2-nonn of the data residuals.

The L2-norm expresses the misfit between the two datasets as a single number. This parameter is known as the objective function, although it often takes other names such as the misfit function, the cost function or the functional. The objective function f is a real, positive, scalar quantity, and it is a function of the model m.

In practice, a factor of a half is often included in the definition of the objective function f because it makes later results simpler. An exemplary objective function f(m) using the L2-norm is shown below in equation 9).

f (n) = Ild (n) -(10112 = Ilsc1112 = s On) -c10112 where n, nr and nt are the number of sources, receivers and time samples in the observed data set.

Figure 6 shows an exemplary objective function f corresponding to the data of Figure 5. In general, the value of the objective function, or misfit, is oscillatory as a function of time as shown.

The skilled person would readily understand how to design such objective functions and how to minimise or maximise them. The method then proceeds to step 108.

Step 108: Calculate gradient Commonly, localised gradient-based methods are used with FWI. These methods iteratively update an existing model in a direction that derives from the objective function's direction of steepest descent. The central purpose of FVVI is to find a model of the subsurface that minimises the difference between an observed seismic dataset and a predicted seismic dataset generated by the model for the same real-world spatial data points as the observed seismic dataset.

FWI or AWI are local iterative inversion schemes. Improvements to the starting model are made which successively reduces the objective function towards zero. Therefore, across an iterative step of the calculation, the objective function needs to be considered for a starting model mo and a new model m1 at iteration 1. This can be generalised to a model mk at iteration k and a new model mk+, at iteration k+1, where: 10) ink+, = m.k + aomk And where Sink is the normalised update for the km iteration of the model mk which minimises the objective function, and is the a step length The first stage is to determine the normalised update by calculating the gradient. This may be done by any suitable method. However, a generic example of gradient of the objective function is obtained in accordance with known conventional FWI.

An expression can be derived which expresses the update to the model Omk: 1 1) ri T1 1 Where 7f is the gradient of the objective function f with respect to the model parameters, and H is the Hessian matrix of second differentials, both evaluated at mk.

If the model has M parameters, then the gradient is a column vector of length M and the Hessian is an Mx M symmetric matrix.

If the number of model parameters M is large, then calculating the Hessian is computationally demanding, and inverting the Hessian exactly is normally computationally intractable. Consequently the method that is typically used is to replace the inverse of the Hessian in equation 11) by the scalar step length a. Equation 11) can then be expressed as: 12) Based on equation 12), conventional FWI can use the method of steepest descent. To calculate the gradient and determine the step length, a Jacobian matrix is used and an expression can be derived for the gradient: 13) Vlx),1= P 0A.1, (Aod) i)m So to find the gradient, the forward wavefield p is calculated, the numerical operator A is differentiated with respect to the model parameters and the final term of equation 13) is calculated, which represents a back-propagated residual wavefield.

These terms are then multiplied together for all times and all sources, and summed together to give a value corresponding to each parameter within the model, typically to give one value of the gradient at each grid point within the model.

The final term in equation 13) can be written to arrive at equation 14): 14) Equation 14) simply describes a wavefield p that is generated by a (virtual) source 5d, and that is propagated by the operator AT which is the adjoint of the operator in the original wave equation. So the term that we need to compute in equation 11) is just the solution of a modified wave equation with the data residuals used as a source.

The gradient for each of the seismic trace groups (or shots) forming the observed data set can then be obtained. The gradient is determined for the total number (or a large subset thereof) of shots in the observed data set. This may, typically, be in the region of 100 shots for a single parameter analysis. However, if more than one parameter is used (e.g. Vp and Vs are calculated), then the number of shots increases. For a 3D computation, the number of shots may be around 10000. Once the update is obtained using the gradient method, this can be normalised to provide On,.

The method then proceeds to step 110.

Step 110: Determine step length At step 110, the step length a is determined. Step 108 enables determination of the gradient for each shot, summed across each shot. The gradient identifies in which direction the model should be changed. However, it does not specify by how much. To determine this, the step length a is computed.

This is normally done by perturbing the model by a small amount in the opposite direction to the gradient, calculating the data residuals for this new model, and using this to optimise the step length by assuming a linear relationship between changes in the residuals and changes in the model.

Therefore, the aim is to find a new model ma = mo + cram that generates residuals Sma, where a minimises: 15) Two common approaches are as follows to determine the step length a: Starting from equation 10), consider the previously-discussed model mk+1 for the k+1th iteration.

1. Fit a parabola shape using three values of the functional f. The three values are computed using: the current model mk and two new models m -k aguessl ärnk and mk + aguess2C5Mk.

This approach requires two additional forward iterations (one for each guess of a) to evaluate the functional values for each new model. Figure 4 illustrates an example of a curve fit using these values.

In Figure 4, the value of the functional f is plotted as a function of a. The thicker line shows the actual distribution of f as a function of a, and the thinner line the estimated parabola. As shown, the parabola is estimated based on the two obtained a values, -uuessl and n -guess2. The optimal step length is the value of a where f is at a minimum.

2. An alternative, and more efficient, approach is to assume a linear relation between the data residuals and the model parameters. This approach relies upon the Born approximation, which assumes that the perturbation to a wavefield produced by changing a model is linearly related to the change in the model. This is equivalent to considering only first-order scattering by the perturbation. In other words, by applying the Born approximatbn, it can be assumed that changes in the model perturbation are linearly related to changes in the residuals. Thus, the residuals can be written as a function of a and minimised to find the optimum step length.

Consider mkt, in equation 10) which will have residuals 5mk. Thus, the residuals can now be written as a function of a as set out in equation 16) below: 16) 6d(a) = cicik + a(d(irt9) -clk) Where mu = mk +g5m for a small predetermined value of g. g is a small perturbation to the model. In other words, to calculate the step length, a forward calculation is made with a perturbed model, and the residuals from both the original and the perturbed models combined Thus, it is desired to find a model that minimises equation 7) with respect to a: 17) -116(lk a(4171.9)-dk)112 = kiclk + [6. dk + aq] where q=d(mg)-dk to simplify the expressions. This is done by taking the derivative of equation 17) with respect to a: o[ac 1 k+aqp+ [aak-FoN1 = IT (sdk aq) = (IF (sdk aq' q) 18) This is then made equal to zero and a is isolated to give equation 19): 19) (IT (sdk agTo = 0 From which expression 20) can be derived: 20) (frit This approach is advantageous since it only requires one step of forward modelling to determine a Irrespective of the method used, the present invention is operable to determine the step length more accurately than for known methods.

In the present method, each forward model run to determine the step length is carried using a reduced objective function utilising only a sub-set of the full observed data set. In embodiments, a small sub-set of the full observed data set is used. For example, 40% or less qTadk of the data of the observed data set used for calculation of the gradient in the previous step may be used in the determination of step length in step 110. In embodiments, 15% or less of the data or shots) of the data of the observed data set used for calculation of the gradient in the previous step may be used in the determination of step length.

In embodiments, the sub-set of the observed data set used for determination of step length can be reduced below 10% to, for example, 5% or even lower. The inventors have found that useful results can be obtained using only a very small sub-set of the observed data set used in the objective function for determining the gradient in step 108. For example, 2-4% of the to observed data set used in the calculation of the gradient may be used to determine in the or each calculation of the step length.

In embodiments, the sub-set of the observed data set used to determine step length may be differentiated in terms of numbers of "shots". As described above, a shot is a group of observed data traces belonging to the same source or receiver. For the same source, a shot would be a group of observed data traces taken at different receiver locations for the same source. Conversely, for a shot identified by individual receiver, a shot would be a group of observed data trace taken at the same receiver location for different source locations.

Further, for the avoidance of doubt, whilst the reduced data set is identified with respect to the observed data set used for calculation of the gradient, this is not to be taken as limiting. As described above, the size of the data set is, at least in part, determined by the size of the observed data set. This is because, as described above, a predicted data set is generated based on the individual data points (i.e. source and receiver locations) in the observed data set. The differences between the two are then minimised. Therefore, the number of data points and number of sampled parameters in the observed data set defines the overall size of the data set for processing in the objective function.

It is noted that different types of data set may be used. For example, composite shot data sets may be used with the method of the present invention.

The form and nature of the data set is not material to the present invention, provided that the data set used to determine the step length is reduced in size when compared to the data set used to determine the gradient in step 108. Indeed, the two data sets need not overlap.. For example, it is to be considered to be within the scope of the present invention for a sub-set of the original observed data set to be used to calculate the gradient in step 108. This may be done either through appropriate selection of data or through processing such as composition of data shots or other pre-calculation processing. A different sub-set of the original data may then be used for calculation of the step length, provided the absolute size of the data set used is smaller than the data set used for determination of the gradient.

The inventors have found that surprisingly accurate results can be obtained using this method. Figure 9 shows an example of step length determination for an objective function using the AWI method. An objective function computed using all shots (i.e. the full observed data set) is shown. In this example, the objective function was computed using an observed to data set comprising 91 shots. A reduced objective function computed using only 3 shots is also shown. Note that the reduced objective function is identical to the objective function save for the use of a reduced number of shots in its calculation.

As shown, the two graphs vary in form, as may be expected. However, importantly, the reduced objective function calculated using only three shots has the same global minimum in a as the full objective function calculated using 91 shots. This is the key parameter to determine the optimum step length a, and so this example shows that the reduced objective function calculation can provide an accurate determination of a from a reduced sub-set of shots. However, the calculation of the reduced objective function is significantly less expensive computationally, and so the calculation can be completed with a reduced load on computational resources.

It has been determined above that the computation of the reduced objective function can provide accurate information on the minimum value of a, albeit whilst requiring significantly fewer computational resources. Therefore, a further advantage of this approach is that a number of different computations of a can be carried out for the same or similar computational resource cost as a single conventional objective function determination of step length. Therefore, a can be sampled at a greater number of points along the objective function curve to obtain a more accurate value of the step length. This is particularly important for methods such as AWI, where the Born approximation may not hold under all circumstances. However, it also has significant implications for FWI-type methods where a local minimum in a may be found using conventional approaches, leading to convergence of the model to an inaccurate local minimum.

In embodiments, the number of sampled points is five or more. However, 50 to 100 samples could be used to sample the curve of the objective functional at multiple points to provide a highly accurate estimate for the step length. Such a large number of samples would be computationally unfeasible using conventional methods.

Step 112: Update model At step 112, the model is updated using the gradient obtained in step 108 and the step length obtained in step 110. The model update derives from the gradient i.e. the partial derivative with respect to a point perturbation of the model m at each position. Ultimately gradients from separate shots will be summed when forming the final model update.

As for the computational structure of conventional FWI method, this is the result of two wavefields: an incident wavefield emitted by a source at the source location, and a back-propagated wavefield which is emitted by a (multi-point) source located the receiver positions.

The method now proceeds to step 114 Step 114: Convergence criteria met? At step 114 it is determined whether convergence criteria have been met. For example, when the method is deemed to have reached convergence when the difference between the data sets reaches a threshold percentage or other value. If the criteria as set out above have been met, then the method proceeds to step 116 and finishes with the resultant Earth model generated. If the criteria have not been met, then the method proceeds back to repeat steps 104 to 112 as discussed above.

Step 116: Finish When, at step 116, it has been determined that the convergence criteria has been met, the method finishes and the modelled subsurface portion of the Earth is deemed to be sufficiently accurate to be used for subsurface exploration. This may involve the direct interpretation of the recovered model, and/or involve the process of depth-migration to generate a subsurface reflectivity image to be used for the identification of subsurface features such as cavities or channels which may contain natural resources such as hydrocarbons. Examples of such hydrocarbons are oil and natural gas.

Once these features have been identified in the subsurface model and/or reflectivity image, then measures can be taken to investigate these resources. For example, survey vessels or vehicles may be dispatched to drill pilot holes to determine whether natural resources are indeed present in these areas.

A second embodiment of the invention is illustrated in Figure 8. The second embodiment focuses on a specific application of the present invention to an AWI-type method using Wiener filters.

Step 200: Obtain observed data set Step 200 corresponds substantially to method step 100 of the previous embodiment. Therefore, this will not be described again here. Only the steps which are new to this embodiment of the method of the present invention will be described. The method now proceeds to step 202.

Step 202: Provide starting model At step 202, an initial starting model of the specified subsurface portion of the Earth is provided. The model may be provided in either a two dimensional or a three dimensional form. Whilst the illustrated examples are of two-dimensional form, the skilled person would be readily aware that the present invention is applicable to three dimensional approaches.

The model is generated in this step in accordance with step 102 above.

The generated model consists of values of the coefficient V, and, possibly, other physical values or coefficients over a discrete grid representing the subsurface. Such starting models are routinely generated and represent the general trends of the major features within the subsurface region to be modelled and could be readily generated by the skilled person.

The method then proceeds to step 204.

Step 204: Generate predicted data set In order to model accurately the subsurface region under investigation, it is necessary to generate predicted data which corresponds to the same source-receiver location data position as the actual measured trace data so that the modelled and observed data can be compared. In other words, the predicted data set corresponds discrete point to discrete point to the observed dataset. The predicted data set is generated for the same measurement parameter(s) at the same frequency or frequencies.

The predicted data set is generated using the full two-way wave equation. In one approach, the modelled trace data may be generated using the time domain full two-way wave equation as set out above.

From the above analysis, predicted seismic data can be generated for one or more physical parameters and at one or more selected frequencies. This forms the modelled data set d d(r,$). The method now proceeds to step 206.

Step 206: Scale data At step 206, the data is scaled so that the predicted d (r,$) and observed do(r,$) data sets comprise matching root mean square amplitudes. This may be accomplished by any known approach. Either or both of the predicted d(r,$) and observed dd(r,$) data sets may be scaled as appropriate.

Additionally, this step may be optional if it is possible to generate predicted data which intrinsically matches the observed data.

The method proceeds to step 208.

Step 208: Design Wiener filter A Wiener filter is a convolutional filter of finite length that is operable to convert an input wavelet into a desired output wavelet in a least-squares manner. In other words, if b is the input wavelet, and c is the desired output wavelet, then the Wiener filter w minimises expression 21). 21) w

Where expression 21) represents, inter alia, the convolution of input wavelet b with filter w. However, any convolution can be expressed as a matrix operation, and so b * w = c can be expressed as equation 22): 22) Bw If w is a column vector of length M, and c is a column vector of length M +N -1, then B is a rectangular matrix with M columns and M + N -1 rows that contains the elements of b arranged as: Based on equation 23), BTB represents a matrix containing the autocorrelafion of b in each column with the zero lag on the main diagonal, and I3Tc is the cross-correlation of b and c.

The solution to the least squares problem in expression 21) is now: 24) Therefore, the approach is to find the auto-correlation BTB of the input trace b, the cross-correlation I3Tc of the input trace with the desired output trace c, and deconvolve the latter using the former. Because of the Toeplitz structure of B, fast algorithms (such as the Levinson algorithm) to find the filter coefficients if B is square which can be ensured by suitably padding b and c with zeros. In practice, equation 25) would be solved: 25) NV = (BIB ± X1) I This approach can be applied to FWI. At each receiver, for each source, a Wiener filter w can be designed that operates on the observed data set dobs to convert the observed data set deps into the predicted data set dpied. In this embodiment, the filter w operates to convert the observed data into the predicted data. The third embodiment discloses an alternative approach.

Equation 26) and 27) denote the approach of this embodiment.

26) dobs B'B) do,",:d. 27)

Here B is the matrix representation of convolution by dobs, and BT is the matrix representation of cross-correlation with dob,. The method proceeds to step 210.

Step 210: Generate reference filter Once the convolutional filter is designed in step 208, an analogous convolutional filter is designed such that, when applied to an input dataset, this reference filter will generate an output dataset that provides an equivalent approximation to all or parts of the input dataset In other words, the reference filter does not modify the input data in essence.

In one embodiment, the reference filter comprises a sequence of zeros and a value of one at the zero lag time. This is, in essence, similar to an impulse function centred upon time t=0.

The method then proceeds to step 212.

Step 212: Construct objective function At step 212, an objective function is configured. The objective function (or objective function) is configured to measure the dis-similarity between the actual filter coefficients and reference filter coefficients. Alternatively, an objective function may be configured that measures similarity; in this case, step 214 (described later) will operable to maximise rather than minimise the objective function.

An example of an objective function configured to measure the dis-similarity between a simple one-dimensional convolutional filter in time and a reference function that consist of only one non-zero value at zero lag, would be to weight the convolution filter coefficients by the modulus of their temporal lag. The objective function would then wnsist of some norm of these weighted coefficients divided by the same norm of the unweighted coefficients. If the 1_2 norm is used here, then this objective function will provide the least-squares solution, but other norms are possible and potentially desirable.

The norm of the weighted coefficients must be normalised by the norm of the unweighted coefficients in this example otherwise the objective function could be simply minimised by driving the predicted data to large values, and hence driving the filter coefficients to small values.

In this example, the coefficients generated for each source receiver pair r,s in step 208 are weighted as a function of the modulus of the temporal lag. In other words, the coefficients are weighted based on the data position in time for a time domain analysis.

However, it is to be understood that other types of weighting could be used. For example, more complicated functions of the temporal lag are possible such as weighting with a Gaussian function of lag centred on zero lag. In general two types of weighting are desirable; those that increase monotonically away from zero lag, such as the modulus, and those that decrease monotonically away from zero lag, such as a Gaussian weighting. The former type of weighting will lead to objective functions that should be minimised and the latter type will lead to objective functions that should be maximised. Combinations of these two types are also possible. The skilled person would readily understand how to design such objective functions and how to minimise or maximise them.

A model is then sought that makes the Wiener filters as close as possible to being just a spike at zero time lag. In other words, the predicted and modelled data matches apart from a scale factor.

Therefore, a model m is desired that minimises or maximises: 28) where w is a column vector containing all the Wiener filter coefficients, and T is the weighting function, for example the modulus of temporal lag. The filters vary with source and receiver position, and with the model. The Wiener filters are obtained by solving equation (27). The method proceeds to step 214.

Step 214: Minimise objective function (determine gradient) The gradient method is used to determine the minimum of the objective function. In this embodiment, the gradient is straightforward to derive from equation 28): 29) 13 (13' 13) This is the usual expression for FWI, but now the adjoint source that must be used in the back propagation is no longer simply the residual data but is instead the entity represented -,') by: w'w) .1 As set out above, in contrast to conventional FWI, the AWI approach seeks to minimise the difference between the convolutional and reference filters, rather than between the data sets themselves. Therefore, because the convolutional filters are not periodic, cycle skipping does not occur.

The gradient can be obtained based on equation 29) as follows. A set of Wiener filters is found in step 208, one filter per data trace. The filter coefficients are normalised by their inner product, trace-by-trace. The normalised coefficients are weighted by a function of temporal lag. The resultant sequence is deconvolved by the auto-correlation of the observed data, and convolved with the observed data. This forms an adjoint source for each source-receiver pair. These adjoint sources are then back-propagated and combined in the normal way with the forward wavefield to produce the gradient.

The gradient calculation is performed using a "full" objective function. In other words, the gradient calculation is performed using the entire observed data set, pre-processed derivative thereof, or pre-defined sub-set.

Step 216: Minimise objective function (determine step length) At step 216, the step length a is determined. The gradient identifies in which direction the model should be changed. However, it does not specify by how much. To determine this, the step length a is computed. Aspects of the method are as described in step 110. Subject matter in common will not be repeated here.

The use of the present invention is particularly advantageous in the case of AWI. This is because the residuals and the model parameters are not linearly related, and so the second method of step 110, i.e. the Born approximation method for calculating step length, is not valid. Additionally, the shape of the objective function in the update direction is a combination of concave and convex curves. Therefore, the method of fitting a parabola is often inaccurate.

Consequently, the present invention provides a robust method for determining step length in such cases.

Consider, as for step 110, model update mkiq which will have residuals 5mk. In AWI, the residuals amk have the form 30) Tw aInk = liwTwil Where w is the Wiener filter that turns the predicted data into the observed data.

In the present method, each forward model run to determine the step length is carried using a reduced objective function utilising only a sub-set of the full observed data set. In embodiments, a small sub-set of the full observed data set is used. For example, less than 40% of the data of the observed data set used for calculation of the gradient in the previous step may be used in the determination of step length. In embodiments, less than 15% of the data or shots) of the data of the observed data set used for calculation of the gradient in the previous step may be used in the determination of step length.

In embodiments, the sub-set of the observed data set used for determination of step length can be reduced below 10%, or even further. The inventors have found that useful results can be obtained using only a very small sub-set of the observed data set used in the objective function for determining the gradient in step 108. For example, 2-4% of the observed data set used in the calculation of the gradient may be used to determine in the or each calculation of the step length.

In embodiments, the sub-set of the observed data set used to determine step length may be differentiated in terms of numbers of "shots". As described above, a shot is a group of observed data traces belonging to the same source or receiver. For the same source, a shot would be a group of observed data trace taken at different receiver locations for the same source. Conversely, for a shot identified by individual receiver, a shot would be a group of observed data trace taken at the same receiver location for different source locations.

Further, for the avoidance of doubt, whilst the reduced data set is identified with respect to the observed data set used for calculation of the gradient, this is not to be taken as limiting.

As described above, the size of the data set is, at least in part, determined by the size of the observed data set. This is because, as described above, a predicted data set is generated based on the individual data points (i.e. source and receiver locations) in the observed data set. The differences between the two are then minimised. Therefore, the number of data points and number of sampled parameters in the observed data set defines the overall size of the data set for processing in the objective function.

The form and nature of the data set is not material to the present invention, provided that the data set used to determine the step length is reduced in size when compared to the data set used to determine the gradient in step 212. Indeed, the two data sets need not overlap entirely (although in practice this will be necessary). For example, it is to be considered to be within the scope of the present invention for a sub-set of the original observed data set to be used to calculate the gradient in step 212. This may be done either through appropriate selection of data or through processing such as composition of data shots or other pre-calculation processing. A different sub-set of the original data may then be used for calculation of the step length, provided the absolute size of the data set used is smaller than the data set used for determination of the gradient.

The inventors have found that surprisingly accurate results can be obtained using this method. As described for the previous embodiment, Figure 9 shows an example of step length determination for an objective function using the AWI method. An objective function computed using all shots (i.e. the full observed data set) is shown. In this example, the objective function was computed using an observed data set comprising 91 shots. A reduced objective function computed using only 3 shots is also shown. Note that the reduced objective function is identical to the objective function save for the use of a reduced number of shots in its calculation.

w It has been determined above that the computation of the reduced objective function can provide accurate information on the minimum value of a, albeit whilst requiring significantly fewer computational resources. Therefore, a further advantage of this approach is that a number of different computations of a can be carried out for the same or similar computational resource cost as a single conventional objective function determination of step length. Therefore, a can be sampled at a greater number of points along the objective function curve to obtain a more accurate value of the step length. This is particularly important for methods such as AWI, where the Born approximation may not hold under all circumstances. However, it also has significant implications for FWI-type methods where a local mimimum in a may be found using conventional approaches, leading to convergence of the model to an inaccurate local minimum.

The method then proceeds to step 218.

Step 218: Update model At step 218, the model is updated using the gradient obtained in step 214 and the step length in step 216. The model update derives from the gradient of equation 11) i.e. the partial derivative with respect to a point perturbation of the model m at each position. Ultimately gradients from separate shots will summed when forming the final model update.

As with the computational structure of conventional FWI method, this is the product of two wavefields: an incident wavefield emitted by a source at source location back-propagated wavefield which is emitted by a (multi-point) source located the receiver positions.

As noted above, gradient methods can be enhanced using an approximate form for the Hessian and conjugate directions for the model update. Further a step-length is then calculated to scale the search direction and give the final model update.

For any useful convolutional filter, and for any useful measure of difference or similarity, as t0 the convolutional filter moves towards the reference filter, the modelled data set will move towards the observed data set. Thus, the starting model will move towards the true model.

With an appropriate choice of convolutional filter and measure of difference/similarity, this scheme is unaffected or less affected by cycle skipping than is conventional FWI. It can therefore be applied more readily to observed data that lack low frequencies and/or in the absence of a good starting model.

The method now proceeds to step 220 Step 220: Convergence criteria met? At step 220 it is determined whether convergence criteria have been met. For example, when the method is deemed to have reached convergence when the difference between the data sets reaches a threshold percentage. If the criteria as set out above have been met, then the method proceeds to step 222 and finishes with the resultant Earth model generated. If the criteria have not been met, then the method proceeds back to repeat steps 204 to 218 as discussed above.

Step 222: Finish When, at step 222, it has been determined that the convergence criteria has been met, the method finishes and the modelled subsurface portion of the Earth is deemed to be sufficiently accurate to be used for subsurface exploration. This may involve the direct interpretation of the recovered model, and/or involve the process of depth-migration to generate a subsurface reflectivity image to be used for the identification of subsurface features such as cavities or channels which may contain natural resources such as hydrocarbons. Examples of such hydrocarbons are oil and natural gas.

The effectiveness of the AVVI method (with or without the additional modification described in the present invention) is illustrated with reference to Figures 10 to 12. Figure 10 shows a relatively extreme subsurface model including a strong anomaly to resolve.

Figure 11 shows a model derived from conventional FWI. As shown, the model fails to converge correctly due to cycle skipping. Consequently, the minimised model bears little resemblance to the actual model as set out in Figure 10.

In contrast, Figure 12 shows the model recovered using the method of the present invention. As shown, the model is fully recovered subject only to the limitations of geometry and bandwidth. Clearly, the present invention enables accurate modelling of complex subsurface features where conventional FWI fails.

When utilising the step length determination of the present invention, accurate convergence of the model can be achieved with computational efficiency in this method, further improving the basic AWI process.

Variations of the above embodiments will be apparent to the skilled person. The precise configuration of hardware and software components may differ and still fall within the scope of the present invention.

The method outlined above for performing FWI is commonly extended and modified in a variety of ways to improve both its efficiency and effectiveness. For example, various improved approximations to the Hessian matrix H and its inverse can be made including use of Newton, Gauss-Newton, quasi-Newton and Limited Memory Broyden-Fletcher-GoldfarbShanno (L-BFGS) methods.

Various modifications upon simple steepest-descent are known in the art; for example conjugate gradient methods. The predicted data, observed data, data residuals, forward wavefield, backward-propagated wavefield, gradient and Hessian may all be pre-processed in various ways.

Further, additional constraints may be placed upon model updates and/or recovered model. Sources and other sub-sets of the data may be selected, modified and/or combined in various ways, and these used in the inversion in preference to using the original physical sources. Many of these enhancements are discussed and described in "An overview of full-waveform inversion in exploration geophysics", J. Virieux and S. Operto, Geophysics Vol. 74, WCC1-WCC26, doi: 10.1190/1.3238367.

Further, whilst the example here has used the scalar parameter of pressure as its focus (i.e. P-waves), it is also possible (with appropriate equipment such as geophones) to measure the particle motion in the receiver location and so determine S-wave parameters. Data so-generated could then be utilised and processed in the present invention.

Whilst the present invention has been described in relation to single parameter inversions, the present invention can also be used in multi-parameter inversions. Indeed, the use of a reduced data set for step length calculations could provide improvements where multi-parameter initial data sets are very large.

In multi-parameter inversion, instead of searching the minimum along a one-dimensional curve, the target is the global minimum in a surface with as many dimensions as different inverted parameters. Typically, the number of parameters will be two or three. For example, Vp, and V, may be inverted in the case of elastic inversion. Alternatively, Vp and E could be inverted for anisotropic inversion.

Figure 13 shows a schematic representation of an objective function for a multi-parameter inversion. It is represented as a 3D surface as a function of different Vp, and V, step sizes. A limitation when applying the method to multi-parameter inversion is that the density of points mapped along each parameter step size is decimated by a factor equal to the number of different parameters to invert. For example, if an acoustic inversion maps the reduced functional with 49 points, an elastic equivalent could generate a 2D map with a resolution of 7x7 points as shown in Figure 13.

Additionally, the present invention may be an alternative to subspace methods (or other techniques that aim at balancing the different parameter updates at each iteration).

Further modifications that are intended to fall within the scope of the present invention as discussed below.

It is possible to sub-sample the total number of observed data shots. For example, only a proportion of the total number of shots or total data in the observed data set may be used in the calculation of the gradient, then a corresponding reduced proportion of that data used to determine the step length. For example, only a proportion of the receivers of the full observed data set may be used to determine the gradient.

Alternatively or additionally, the proportion of data selected for either the gradient calculation (i.e. the first proportion of the original data set) or the proportion of the data used for the gradient calculation to be used in the step length calculation may be determined in a number of organised ways.

For example, the proportion of data to be used may be done by selecting a fraction of the receivers in the original observed data set. Altemafively, only a fraction of the sources in the observed data set may be selected. Alternatively, different post-processing techniques could be used to remove less significant shots or data elements from the observed data set to reduce the data load.

If the number of processes is smaller than the number of points we want to use to map the curve, then probably start coarser and rene in the interesting regions in the 'second pass' 25 when the nodes When calculating different guess values for the step length, the step length calculated in the previous iteration of the model update could be used to inform the next iteration of the model update Further, the method is applicable to composite shots without modification. Problems arising from cross-talk in composite shot methods are unlikely to be problematic because only the objective function is evaluated. Cross-talk problems arise from the fact that forward-and back-propagated energy from different shots correlate. The use of composite shots is likely to be restricted to acquisition geometries where different shots have receivers in common.

With regard to embodiments in which AWI is implemented, whilst the above embodiments are illustrated with regard to one dimensional Wiener filters, multidimensional filters could also be used. In the simple one-dimensional scheme above, the filter for each source-receiver pair is designed only using data from that source-receiver pair. The scheme is also implemented in time -the two sequences that are being matched represent data that varies in time, and the Wiener filter has its coefficients arranged by temporal lag.

However, it is possible to implement an analogous scheme in multiple dimensions in space, or in a mixture of time and one or more space dimensions, or in any of many transformed to spaces, for example in intercept-slowness (tau-p) space, in frequency-wavenumber (f-k) space, or in parabolic Radon space.

Considering a single source and a single receiver, then, using a source estimate, the predicted data can be calculated from that source not only at the receiver also in a three dimensional volume around it. For that receiver, a one-dimensional observed dataset and a four-dimensional predicted dataset is now available. That is, three space dimensions plus time. Now a convolutional filter can be designed that takes all or some 1D, 2D or 3D subset of this 4D predicted dataset as input and generates the 1D observed dataset as output. This can be done in step 208.

If this convolutional filter is a multi-dimensional Wiener convolutional filter, then the corresponding reference filter will be unity at the coefficient that corresponds to zero lag in time and zero lag in all of the space dimensions -that is at the position of the receiver -and it will be zero everywhere else. In this context, the method remains unchanged but now a 1D, 2D, 3D or 4D convolutional filter can be designed, and this can be combined with an appropriate function of all the temporal and spatial lags involved. The associated objective functional can then be minimised.

In practice, it is unlikely to be necessary to use all four dimensions together because the computational effort would be large. However, different subsets of dimensions on different iterations may be required.

Further, it is possible to generate data at or around the receiver, for multiple source locations, and for different source initiation times. It takes four dimensions to describe the source time and location. Therefore, up to eight dimensions of predicted data are available. A filter can be designed using any combination of these as input though in a practical scheme we would be unlikely to want to use all of them simultaneously. There is also redundancy between the two time dimensions unless at least one space dimension is also involved in designing the filter.

The dimensionality of the output can also be increased. Thus, rather than design a filter that seeks to match the observed data in just one dimension, the observed data could be matched in two or more dimensions, for example with the observed data predicted in time over a two-dimensional array of surface receivers, giving a 3D output from the convolutional filter.

Equivalent schemes may also be generated that match the observed data to the predicted data. In this case, we would use observed data from multiple receivers and/or multiple sources as input, generating predicted data over the same multi-dimensional volume or some subset of it. These multi-dimensional filters could be extended by applying them to virtual sources and receivers located anywhere within the sub-surface including at the source position.

Further, whilst the above embodiments have been illustrated in the time domain, frequency domain analysis could also be used. The different frequency components can be extracted through the use of a Fourier transform. Fourier transforms and associated coefficients are well known in the art. Since only a few frequency components may be required, a discrete Fourier transform may be computed from the inputted seismic trace data. Alternatively, a conventional fast Fourier transform (FFT) approach may be used. As a further alternative, partial or full Fourier transforms may be used. The Fourier transform may be taken with respect to complex or real valued frequencies.

One or more observed data sets at one or more frequencies may be extracted. For example, a plurality of observed data sets may be provided having frequency components of, for example, 4.5 Hz, 5 Hz and 5.5 Hz. These 3 frequencies can be inverted individually with the output of one becoming the input of the next.

As can be understood, further variations to this method are possible. For example, the process could be applied using the reverse filter,that using a filter that takes the observed data as input and that generates an approximation to the predicted data as output. The data could be pre-processed in any desired manner the observed and/or field data prior to the formation of the Wiener filter coefficients. The auto and/or cross correlation functions could be pre-processed in order to form the Wiener filter. The Wiener filter coefficients and/or the weighted Wiener coefficients could be processed before forming the quantity to be minimised or maximised. Alternatively, post-processing could be performed on the quantity to be minimised or maximised.

The scheme could be applied to data, to models, and/or to filter coefficients that have been transformed into some other domain or domains including but not limited to the single or multi-dimensional Fourier domain, the Laplace domain, the Radon domain, the wavelet domain, the curvelet domain.

The observed and/or predicted data could be transformed into a sequence of complex numbers, for example by forming an imaginary part using the Hilbert transform, such that the Wiener coefficients are themselves complex numbers, and optionally combine this with the minimisation of the imaginary portion of the Wiener coefficients.

Additional constraints could be added to the Wiener coefficients including but not limited to constraints generated by the observed data and/or the predicted data and/or the model and/or the other Wiener coefficients and/or the accuracy of the convolutional filter.

Wiener filters could be used that vary continuously or discontinuously in time, in space, with receiver position, with source position, with source-receiver azimuth, with source-receiver offset, with source-receiver midpoint, and/or with other characteristics of the data, the model, the model updates, the source, the receiver, the filer coefficients. In other words, the Wiener filter need not be one dimensional and may encompass multiple dimensions.

In addition, whilst embodiments have illustrated the use of Wiener filters, the present invention is not limited to such filters. The skilled person would readily be aware of other filter types suitable for use with the present invention. Non-limiting examples may include: Kalman filters; least mean squares filters; normalised least mean squares filters; and recursive least squares filters.

Further, whilst the present invention has been described with reference to a model which involves solving the acoustic wave equation, the present invention is equally applicable to the models which involve solving the visco-acoustic, elastic, visco-elastic or poro-elastic wave equation.

Embodiments of the present invention have been described with particular reference to the examples illustrated. While specific examples are shown in the drawings and are herein described in detail, it should be understood, however, that the drawings and detailed description are not intended to limit the invention to the particular form disclosed. It will be appreciated that variations and modifications may be made to the examples described within the scope of the present invention.

Claims

CLAIMS1 A method of subsurface exploration, the method comprising generating a geophysical representation of a portion of the volume of the Earth from a seismic measurement of at least one physical parameter, the method comprising the steps of: a) providing an observed data set, the observed data set comprising data values derived from seismic measured values of said portion of the volume of the Earth; b) generating, using a subsurface model of a portion of the Earth comprising a plurality of model coefficients, a modelled data set comprising a plurality of modelled data values; c) updating the subsurface model by: d) generating an objective function operable to measure the mismatch or similarity between the observed data set and the predicted data set; e) determining, using a first proportion of the total number of data values of the observed data set, the gradient of the objective function; f) determining, using a reduced objective function utilising a second proportion of the total number of data values of the observed data set, the step length for a subsurface model update, wherein the second proportion comprises 40% or less of the total number of data values of the first proportion of the observed data set; and g) updating the subsurface model of a portion of the Earth using the determined gradient and step length; and h) providing an updated subsurface model of a portion of the Earth for subsurface exploration.
2. A method according to claim 1, wherein the second proportion is 15% or less of the total number of data values of the first proportion of the observed data set.
3. A method according to claim 2, wherein the second proportion is 10% or less of the total number of data values of the first proportion of the observed data set.
4. A method according to claim 3, wherein the second proportion is 5% or less of the total number of data values of the first proportion of the observed data set.
5. A method according to any one of the preceding claims, wherein the first proportion is 100% of the total number of data values in the observed data set.
6. A method according to claim 1, wherein the observed data set comprises a plurality of shots, each shot comprising a plurality of seismic traces grouped by source or receiver location and comprising data values derived from seismic measured values of said portion of the volume of the Earth; and the modelled data set comprises a corresponding plurality of shots comprising a plurality of modelled traces grouped by source or receiver location and comprising modelled data values.
7. A method according to claim 6, wherein the first proportion of the observed data set comprises a first number of shots and the second proportion of the observed data set comprises a second number of shots, the second number being 40% or less than the first number.
8. A method according to claim 7, wherein the second number is 15% or less of the first number.
9. A method according to claim 8, wherein the second number is 10% or less of the first number.
10. A method according to claim 9, wherein the second number is 5% or less of the first number.
11. A method according to any one of the preceding claims, wherein step f) comprises determining the step length for a subsurface model update by obtaining a plurality of trial values of the step length and identifying the step length therefrom.
12. A method according to claim 11, wherein the number of trial values of the step length is 5 or more.
13. A method according to claim 12, wherein the number of trial values of the step length is 10 or more.
14. A method according to claim 13, wherein the number of trial values of the step length is 50 or more.
A method according to any one of the preceding claims, wherein step d) further comprises: i) generating at least one convolutional filter, which when convolved with the observed data, increases the similarity between the predicted data and the convolved observed data; j) generating at least one convolutional reference filter, which when convolved with the observed data, leaves the observed data unaltered to some level of precision; and k) generating an objective function operable to measure the similarity or mismatch between the filter coefficients for the or each convolutional filter and the filter coefficients for the or each convolutional reference filter.
16. A method according to any one of the preceding claims, wherein step d) further comprises: I) generating at least one convolutional filter, which when convolved with the predicted data, increases the similarity between the observed data and the predicted data; m) generating at least one convolutional reference filter, which when convolved with the predicted data, leaves the predicted data unaltered to some level of precision; and n) generating an objective function operable to measure the similarity or mismatch between the filter coefficients for the or each convolutional filter and the filter coefficients for the or each convolutional reference filter.
17. A method according to claim 15 or 16, wherein one or more of the convolutional filters comprise Wiener filters
18. A method according to any one of the preceding claims, wherein a plurality of observed data sets and/or a plurality of modelled data sets is provided.
19. A method according to any one of the preceding claims, wherein the objective function and/or the reduced objective function comprise different functions.
20. A method according to any one of the preceding claims, wherein the objective function and/or the reduced objective function comprises a norm misfit objective function.
21. A method according to claim 20, wherein the objective function and/or the reduced objective function comprises an 1_1-norm misfit objective function.
22. A method according to claim 21, wherein the objective function and/or the reduced objective function comprises a least-squares misfit objective function.
23. A method according to any one of the preceding claims, wherein at least a portion of the observed data set is numerically propagated to a subsurface region of the model that is spatially removed from the location at which they were originally recorded.
24. A method according to any one of the preceding claims, further comprising, subsequent to step g), repeating steps b) to f) for the updated subsurface model of a portion of the Earth until a convergence criterion is met.
25. A method according to any one of the preceding claims, wherein step h) further comprises: utilising said updated model for sub-surface exploration.
26. A method according to any one of the preceding claims, wherein said at least one physical parameter comprises pressure, particle velocity or displacement.
27. A method according to any one of the preceding claims, wherein the observed data set and the predicted data set comprise values of a plurality of physical parameters.
28. A computer program product executable by a programmed or programmable processing apparatus, comprising one or more software portions for performing the steps of any one of claims 1 to 27.
29. A computer usable storage medium having a computer program product according to claim 28 stored thereon.
30. A method substantially as shown in and/or described with reference to any one or more of Figures 4,7-9 and 13 of the accompanying drawings.
31. A computer program product substantially as described with reference to any one or more of Figures 4, 7-9 and 13 of the accompanying drawings.