Patent Description:
Marine seismology companies invest heavily in the development of marine seismic surveying equipment and seismic data processing techniques in order to obtain accurate, high-resolution images of subterranean formations located beneath a body of water. Such images are used, for example, to determine the structural features of the subterranean formations, to discover oil and natural gas reservoirs, and to monitor oil and natural gas reservoirs during production. A typical marine seismic survey is performed with one or more survey vessels that tow a seismic source and many streamers through the body of water. The survey vessel contains seismic acquisition equipment, such as navigation control, seismic source control, seismic receiver control, and recording equipment. A seismic source control controls activation of the one or more seismic sources at selected times or locations. A seismic source may be an impulsive source comprised of an array of air guns that are activated to produce impulses of acoustic energy. Alternatively, a seismic source may be a marine vibrator that emits acoustic energy over a longer time period. The acoustic energy generated by a seismic source spreads out in all directions. A portion of the acoustic energy travels down through the water and into a subterranean formation to propagate as sound waves within the subterranean formation. At each interface between different types of liquid, rock and sediment, a portion of the sound wave is refracted, a portion is transmitted, and another portion is reflected into the body of water to propagate as a reflected wavefield toward the water surface. The streamers are elongated spaced apart cable-like structures towed behind a survey vessel in the direction the survey vessel is traveling and are typically arranged substantially parallel to one another. Each streamer contains many seismic receivers or sensors that detect pressure and/or particle motion wavefields of the sound waves. The streamers collectively form a seismic data acquisition surface that records wavefields as seismic data in the recording equipment. Alternatively, a seismic data acquisition surface may be created by deploying the receivers at the bottom of the body of water and directly on or near the surface of the subterranean formation. The recorded pressure and/or particle motion wavefields are processed to generate and display images of the subterranean formation, enabling geoscientist to identify potential oil and natural gas reservoirs and to monitor oil and natural gas reservoirs under production.

International Publication No. <CIT> describes a method for processing seismic data from a survey of a subterranean formation using least squares migration or wave equation migration velocity analysis over a cost function to output an updated reflectivity model or an updated velocity model. In <CIT>, Wang describes a true amplitude pre-stack depth migration based on an initial velocity model of a subterranean formation to obtain a reflectivity model, and then Born modeling using the reflectivity model is used to generate synthetic data. Imaged-based reflection FWI is applied to a cost function of differences between seismic data recorded in a survey and the synthetic data to update the initial velocity model. In "<NPL>, the authors describe a joint migration inversion, which is an FWI process, that does not use local velocity and density in finite-difference-type modeling. Migration is executed in terms of elastic reflectivity operators and propagation operators. In "<NPL>, Berkhout describes extending the theory of FWI for velocity estimation to recursive FWI. In "<NPL>, the authors describe a method to produce long wavelength updates in gradient-based FWI. <CIT> describes techniques for executing FWI using a travel-time cost function in which time shifts are weighted using cross-correlation coefficients of respective time-shifted recorded data and synthetic data generated based on a current velocity model.

Wave equation-based seismic imaging is a two-step process for generating images and/or reflectivity models of a subterranean formation from seismic data recorded in a marine survey. At step one, an acoustic wave equation is used to forward propagate a source wavefield and backward propagate reflection events recorded in the seismic data. At step two, an imaging condition is applied to the propagated wavefields to obtain an image that reveals the detailed structural properties or attributes of the subterranean formation. The acoustic wave equation employed at step one models propagation of acoustic waves in a subterranean formation and is traditionally expressed in terms of a seismic velocity model. The seismic velocity model is a map of the seismic velocities associated with layers of the subterranean formation.

Least-squares reverse time migration ("LSRTM") is an iterative seismic imaging process performed in the data space domain to update and improve an image or a reflectivity model of the subterranean formation at each iteration The iterative process minimizes a difference between the reflection events recorded at the receiver locations during the survey and reflection events that are simulated during forward propagation of the source wavefield and is finished when the resulting image or reflectivity model minimizes the difference. However, the velocity models typically used in iterative LSRTM do not represent all the impedance contrasts of the subterranean formations that simulate the reflection events. Thus, a first-order approximation Born theory is used to generate these reflections. The corresponding wave equation is an approximation and does not generate all the reflection events in the recorded seismic data. In addition, at each iteration, two different wave equations are solved during forward and background propagation.

Full-waveform inversion ("FWI") is a similar iterative process to that of LSRTM, except that instead of updating a reflectivity model of a subterranean formation, FWI also improves resolution of a velocity model of the subterranean formation. Conventional FWI does not require reflection events and refraction events are enough to improve the velocity model, when refraction events are available. However, maximum penetration depth from refraction events is limited to a maximum source-receiver offset of the marine survey. For example, in typical deep-water marine surveys performed with a maximum offset of about <NUM>, the maximum depth update of the velocity model is severely constrained. By using reflection events in FWI, the depth limitation is removed and it is possible to correctly update the velocity model to a maximum depth where reflection events are generated at the boundaries of the subterranean formations. In addition, the reflectivity model may be updated at each iteration once the velocity model is improved.

As in reflection-based FWI, a smooth velocity model is usually used and most of the reflection events cannot be simulated from such a model. Thus, a density model is used in some approaches. However, building accurate density models of a subterranean formation is challenging and expensive because the process requires interpretation and well integration, which in some cases is not possible. Where wells are available, density models may also be inaccurate away from actual well locations. Other reflection-based FWI approaches use the reflectivity model (or image) and the first-order Born theory to generate the reflection events. In order to generate the full-wavefield, it is necessary to solve two different wave equations at each modeling realization, in addition to the inaccuracy due to the limitation of generating multiple scattering.

Processes and systems described herein are directed to using a novel parameterization of an acoustic wave equation to build accurate high-resolution velocity and reflectivity models. The acoustic wave equation enables accurate and efficient simulation of transmitted and reflected components of acoustic waves propagating within the subterranean formation. In particular, the acoustic wave equation may be used with FWI to build accurate, high-resolution velocity and reflectivity models of the subterranean formation and may be used with LSRTM to build a reflectivity model of the subterranean formation. The velocity and reflectivity models reveal subsurface properties of features and layers of a subterranean formation in terms of structure and lithology. Oil and natural gas reservoirs are typically found in layers of sandstone, clastic rocks, and carbonates, such as limestones. These layers have associated seismic velocities and are embedded in particular structural features that are revealed by the reflectivity model or image, which are used to distinguish the layers from other layers in an image of a subterranean formation. For example, shales have seismic velocities in a range of about <NUM> - <NUM>/s, oil has seismic velocities in a range of about <NUM> - <NUM>/s, sandstones have seismic velocities in a range of about <NUM> - <NUM>/s, and granite and basalt have seismic velocities in a range of about <NUM> - <NUM>/s. (See e.g., <FIG> in <NPL>) Geoscientists in the oil and gas industry carefully examine images and/or reflectivity models of a subterranean formation and may use velocity models of the subterranean formation to identify rock interfaces or layers that potentially contain oil and natural gas reservoirs. Without accurate seismic images or reflectivity models and associated velocity models of subterranean formations, geoscientists would have to resort to randomly drilling test wells in the hopes of finding a reservoir of oil and natural gas.

The novel acoustic wave equation described herein provides advantages over traditional acoustic wave equations used in velocity model building and seismic imaging: (<NUM>) The acoustic wave equation does not require construction of a density model and/or high velocity contrasts of the subterranean formation to simulate reflection events used the iterative velocity model building, such as FWI, and imaging, such as LSRTM. As a result, reflection events may be used to update the velocity and reflectivity models at depths beyond the penetration depth of transmitted waves in FWI. (<NUM>) The acoustic wave equation enables generation of a reflectivity model with a smooth velocity model in LSRTM. (<NUM>) Use of the acoustic wave equation to determine velocity and reflectivity models in FWI and LSRTM is computationally more efficient than traditional FWI and LSRTM, which use a first-order Born approximation to perturbation theory.

<FIG> show a side-elevation view and a top view, respectively, of an example marine seismic data acquisition system comprising an exploration seismology survey vessel <NUM> and a source <NUM>. A seismic data acquisition system is not limited to one source as shown in <FIG>. In practice, the number of sources can range from as few as a single source towed by a survey vessel to multiple sources towed by different survey vessels. The body of water can be, for example, an ocean, a sea, or any portion thereof. In this example, the survey vessel <NUM> tows six streamers <NUM>-<NUM> below the free surface of a body of water. Each streamer is attached at one end to the survey vessel <NUM> via a streamer-data-transmission cable. A data acquisition surface is not limited to six streamers as shown in <FIG>. In practice, the number of streamers used to form a data acquisition surface can range from as few as one streamer to as many as <NUM> or more streamers. The source <NUM> may be an impulsive source, such as an array of airguns, or the source <NUM> may be a vibrational source, such one or more marine vibrators. Additional survey vessels (not shown) may be used to tow additional sources.

<FIG> includes an xz-plane <NUM>, and <FIG> includes an xy-plane <NUM>, of a Cartesian coordinate system having three orthogonal, spatial coordinate axes labeled x, y and z. The coordinate system specifies orientations and coordinate locations within the body of water. The x-axis specifies the position of a point in a direction parallel to the length of the streamers or the direction of the survey vessel and is referred to as the "in-line" direction. The y-axis specifies the position of a point in a direction perpendicular to the x-axis and substantially parallel to the free surface <NUM> and is referred to as the "cross-line" direction. The z-axis, also referred to as the "depth" axis, specifies the position of a point in a direction perpendicular to the xy-plane (i.e., perpendicular to the free surface <NUM>) with the positive z-direction pointing downward away from the free surface <NUM>.

The streamers may be towed to form a planar horizontal seismic data acquisition surface with respect to the free surface <NUM>. However, in practice, the streamers may be smooth varying due to active sea currents and weather conditions. A seismic data acquisition surface is not limited to the parallel streamers shown in <FIG> and <FIG>. In other implementations, the streamers may be towed with progressively larger streamer separation in the crossline direction toward longer distances from the survey vessel <NUM> in a process called "streamer fanning. " Streamer fanning spreads the streamers farther apart with increasing distance from the survey vessel in the inline direction. Streamer fanning may improve coverage at far source/receiver offsets without compromising seismic data resolution or seismic data quality and may also increase acquisition efficiency by reducing seismic data infill. In still other implementations, the streamers may be towed with a downward slant with increasing distance from the survey vessel.

The streamers <NUM>-<NUM> are typically long cables containing power and data-transmission lines coupled to receivers (represented by shaded rectangles) such as receiver <NUM> that are spaced-apart along the length of each streamer. The data transmission lines couple receivers to seismic data acquisition equipment, computers, and data-storage devices located onboard the survey vessel <NUM>. Streamer depth below the free surface <NUM> can be estimated at various locations along the streamers using depth-measuring devices attached to the streamers. For example, the depth-measuring devices can measure hydrostatic pressure or utilize acoustic distance measurements. The depth-measuring devices can be integrated with depth controllers, such as paravanes or water kites that control and maintain the depth and position of the streamers as the streamers are towed through the body of water. The depth-measuring devices are typically placed at intervals (e.g., about <NUM>-meter intervals in some implementations) along each streamer. Note that in other implementations buoys may be attached to the streamers and used to maintain the orientation and depth of the streamers below the free surface <NUM>.

In <FIG>, curve <NUM>, the formation surface, represents a top surface of the subterranean formation <NUM> located at the bottom of the body of water. The subterranean formation <NUM> may have subterranean layers of sediment and rock. Curves <NUM>, <NUM>, and <NUM> represent interfaces between subterranean layers of different compositions. A shaded region <NUM>, bounded at the top by a curve <NUM> and at the bottom by a curve <NUM>, represents a subterranean hydrocarbon deposit, such as oil and natural gas, the depth and positional coordinates of which may be determined, at least in part, by the processes and systems described herein. As the survey vessel <NUM> moves over the subterranean formation <NUM>, the source <NUM> is activated (i.e., fired or shot) to produce an acoustic signal. <FIG> shows an acoustic signal expanding outward from the source <NUM> as a pressure wavefield <NUM> represented by semicircles of increasing radius centered at the source <NUM>. The outwardly expanding wavefronts from the source may be spherical but are shown in vertical plane cross section in <FIG>. The outward and downward expanding portion of the pressure wavefield <NUM> and any portion of the pressure wavefield <NUM> reflected from the free-surface <NUM> are called the "source wavefield. " The source wavefield eventually reaches the formation surface <NUM> of the subterranean formation <NUM>, at which point the source wavefield may be partially reflected from the formation surface <NUM> and partially refracted downward into the subterranean formation <NUM>, becoming elastic waves within the subterranean formation <NUM>. In other words, in the body of water, the acoustic signal primarily comprises compressional pressure waves, or P-waves, while in the subterranean formation <NUM>, the waves include both P-waves and transverse waves, or S-waves. Within the subterranean formation <NUM>, at each interface between different types of materials or at discontinuities in density or in one or more of various other physical characteristics or parameters, downward propagating waves may be partially reflected and partially refracted. As a result, each point of the formation surface <NUM> and each point of the interfaces <NUM>, <NUM>, and <NUM> may be a reflector or reflection point that becomes a potential secondary point source from which acoustic and elastic wave energy, respectively, may emanate upward toward the receivers <NUM> in response to the acoustic signal generated by the source <NUM> and downward-propagating elastic waves generated from the pressure impulse. As shown in <FIG>, waves of significant amplitude may be generally reflected from points on or close to the formation surface <NUM>, such as reflection point <NUM>, and from reflection points on or very close to interfaces in the subterranean formation <NUM>, such as reflection points <NUM> and <NUM>. The upward expanding waves reflected from the subterranean formation <NUM> are collectively the "reflected wavefield.

The waves that compose the reflected wavefield may be generally reflected at different times within a range of times following the initial source wavefield. A point on the formation surface <NUM>, such as the reflection point <NUM>, may receive a pressure disturbance from the source wavefield more quickly than a point within the subterranean formation <NUM>, such as reflection points <NUM> and <NUM>. Similarly, a reflection point on the formation surface <NUM> directly beneath the source <NUM> may receive the pressure disturbance sooner than a more distant-lying reflection point on the formation surface <NUM>. Thus, the times at which waves are reflected from various reflection points within the subterranean formation <NUM> may be related to the distance, in three-dimensional space, of the reflection points from the activated source <NUM>.

Acoustic and elastic waves may travel at different velocities within different materials as well as within the same material under different pressures. Therefore, the travel times of the source wavefield and reflected wavefield are functions of distance from the source <NUM> as well as the materials and physical characteristics of the materials through which the wavefields travel. In addition, expanding wavefronts of the wavefields may be altered as the wavefronts cross interfaces and as the velocity of sound varies in the media traversed by the wavefront. The superposition of waves reflected from within the subterranean formation <NUM> in response to the source wavefield may be a generally complicated wavefield that includes information about the shapes, sizes, and material characteristics of the subterranean formation <NUM>, including information about the shapes, sizes, and locations of the various reflecting features within the subterranean formation <NUM> of interest to geoscientists.

Each receiver <NUM> may be a multi-component sensor including particle motion sensors and a pressure sensor. A pressure sensor detects variations in water pressure over time. The term "particle motion sensor" refers to a sensor that detects particle displacement, particle velocity, or particle acceleration over time. Each pressure sensor and particle motion sensor may include an analog-to-digital converter that converts time-dependent analog signals into discrete time series that consist of consecutively measured values called "amplitudes" separated in time by a sample rate. The time series data generated by a pressure or particle motion sensor is called a "trace," which may consist of thousands of samples collected at a typical sample rate of about <NUM> to <NUM> samples per millisecond. A trace is a recording of acoustic energy, such as the acoustic energy in a subterranean formation response to the source wavefield that passes from the source <NUM> and into the subterranean formation where a portion of the acoustic energy is reflected and/or refracted, and ultimately detected by a sensor. In general, each trace is an ordered set of discrete spatial and time-dependent pressure or particle motion sensor amplitudes denoted by: <MAT> where.

The coordinate location <MAT> of each receiver may be determined from global position information obtained from one or more global positioning devices located along the streamers, survey vessel, and buoys and the known geometry and arrangement of the streamers and receivers. The coordinate location <MAT> of the source <NUM> may also be obtained from one or more global positioning devices located at each source and the know geometry and arrangement of source elements of the source <NUM>. The source and receiver coordinates define an acquisition geometry for recording seismic data. In the following discussion the source coordinate location is suppressed. Each trace also includes a trace header not represented in Equation (<NUM>) that identifies the specific receiver that generated the trace, receiver and source GPS spatial coordinates, and may include the time sample rate and the number of time samples.

<FIG> shows a side-elevation magnified view <NUM> of the receiver <NUM>. In this example, the magnified view <NUM> reveals that the receiver <NUM> is a multi-component sensor comprising a pressure sensor <NUM> and a particle motion sensor <NUM>. The pressure sensor may be, for example, a hydrophone. Each pressure sensor is a non-directional sensor that measures changes in hydrostatic pressure over time to produce a trace of pressure data denoted by <MAT>. The particle motion sensors may be responsive to water motion. The particle motion sensors are directional sensors that detect particle motion (i.e., displacement, velocity, or acceleration) in a particular direction and may be responsive to such directional displacement of the particles, velocity of the particles, or acceleration of the particles. A particle motion sensor that measures particle displacement produces a trace of particle displacement data denoted by <MAT>, where the vector <MAT> represents the direction along which particle displacement is measured. A particle motion sensor that measures particle velocity (i.e., particle velocity sensor) generates a trace of particle velocity data denoted by <MAT>. A particle motion sensor that measures particle acceleration (i.e., accelerometer) generates a trace of particle acceleration data denoted by <MAT>. The data generated by one type of particle motion sensor may be converted to another type. For example, particle displacement data may be differentiated to obtain particle velocity data, and particle acceleration data may be integrated to obtain particle velocity data.

The term "particle motion data" refers to particle displacement data, particle velocity wavefield data, or particle acceleration data. The term "seismic data" refers to pressure wavefield data and/or particle motion data. Pressure wavefield data may also be called the "pressure wavefield. " Particle displacement data represents a particle displacement wavefield, particle velocity wavefield data represents a particle velocity wavefield, and particle acceleration data represents a particle acceleration wavefield. The particle displacement, velocity, and acceleration wavefield data are correspondingly called particle displacement, velocity, and acceleration wavefields.

The particle motion sensors are typically oriented so that the particle motion is measured in the vertical direction (i.e., <MAT>) in which case <MAT> is called vertical displacement wavefield, <MAT> is called vertical velocity wavefield, and <MAT> is called vertical acceleration wavefield. Alternatively, each receiver <NUM> may include two additional particle motion sensors that measure particle motion in two other directions, <MAT> and <MAT>, that are orthogonal to <MAT> (i.e., <MAT>, where "·" is the scalar product) and orthogonal to one another (i.e., <MAT>). In other words, each receiver <NUM> may include a pressure sensor and three particle motion sensors that measure particle motion in three orthogonal directions. For example, in addition to having a particle motion sensor that measures particle velocity in the z-direction to give <MAT>, each receiver may include a particle motion sensor that measures the wavefield in the in-line direction in order to obtain the in-line velocity wavefield, <MAT>, and a particle motion sensor that measures the wavefield in the cross-line direction in order to obtain the cross-line velocity wavefield, <MAT>. In certain implementations, the receivers may be only pressure sensors, and in other implementations, the receivers may be only particle motion sensors. The three orthogonal velocity data sets form a velocity vector <MAT>.

The streamers <NUM>-<NUM> and the survey vessel <NUM> may include sensing electronics and data-processing facilities that allow seismic data generated by each receiver to be correlated with the location of the source <NUM>, absolute positions on the free surface <NUM>, and absolute three-dimensional positions with respect to an arbitrary three-dimensional coordinate system. The seismic data may be stored at the receiver and/or may be sent along the streamers in data transmission cables to the survey vessel <NUM>, where the seismic data may be stored on data-storage devices located onboard the survey vessel <NUM> and/or transmitted onshore to a seismic data-processing facility.

As explained above, the reflected wavefield typically arrives first at the receivers located closest to the sources. The distance from the sources to a receiver is called the "source-receiver offset," or simply "offset. " A larger offset generally results in a longer arrival time delay. Traces are sorted according to different source and receiver locations and are collected to form "gathers" that can be further processed using various seismic data processing techniques to obtain information about the structure of the subterranean formation. The traces may be sorted into different domains such as, for example, a common-shot domain, common-receiver domain, common-receiver-station domain, and common-midpoint domain. A collection of traces sorted into the common-shot domain is called a common-shot gather. A collection of traces sorted into common-receiver domain is called a common-receiver gather.

The portion of the acoustic signal reflected into the body of water from the subterranean formation and that travels directly to the receivers is called a primary reflected wavefield or simply a "primary. " Other portions of the acoustic signal reflected into the body of water may be reflected many times between the free surface and interfaces within the subterranean formation before reaching the receivers. These multiple reflected wavefields are simply called "multiples. " Still other portions of the acoustic signal may create head waves and diving waves within the subterranean formation before being reflected into the body of water. Head waves are created when a portion of the acoustic signal traveling downward through a low-velocity layer reaches a higher velocity layer at the critical angle. Head waves travel in the higher velocity layer parallel to an interface between the layers before being reflected upward toward the formation surface. Diving waves are created when a portion of the acoustic signal travels within a progressively compacted layer, creating a velocity gradient in which velocities increase with depth. Diving waves are continuously refracted along curved ray paths that turn upward toward the surface. The deepest point along the curved ray path is called the "turning point.

<FIG> shows a side elevation view of different example ray paths acoustic energy travels between the source <NUM> and a receiver <NUM> of the streamer <NUM>. Differently patterned directional arrows <NUM>-<NUM> represent ray paths of different portions of an acoustic signal generated by the source <NUM>. Ray paths <NUM>-<NUM> represent different portions of acoustic energy that interact with the subterranean formation <NUM>. Ray path <NUM> represents a portion of the acoustic energy that travels directly to the receiver <NUM>. Ray paths <NUM> and <NUM> represent paths of acoustic energy that strike the interface <NUM> and surface <NUM> at critical angles <NUM> and <NUM>, respectively, creating head waves that travel along ray paths <NUM> and <NUM> adjacent to the interface <NUM> and surface <NUM>. Ray paths <NUM> and <NUM> represent the paths of head waves that travel within the higher velocity layer overlain by a low velocity layer. Ray paths <NUM> and <NUM> represent upward reflections of the acoustic energy of the head waves to the receiver <NUM>. Ray paths <NUM> and <NUM> represent different portions of the acoustic energy traveling along ray paths <NUM> and <NUM>, respectively, that are reflected upward from the interface <NUM> and the surface <NUM> to the receiver <NUM>. Ray path <NUM> reaches the interface <NUM> of a layer <NUM> with progressive compaction, creating a vertical velocity gradient in which velocities increase with increasing depth. Curved ray path <NUM> represents a continuously refracted path of a diving wave that reaches a turning point <NUM> within the layer <NUM> and is turned upward to the receiver <NUM> along ray path <NUM>. Ray path <NUM> represents a portion of the acoustic energy traveling along ray path <NUM> that is reflected upward from the interface <NUM> to the receiver <NUM>. Deeper penetrating acoustic energy (not shown) tends to be reflected back toward the surface <NUM> but may reach the surface <NUM> too far away to be recorded by the receivers.

<FIG> shows an example common-shot gather <NUM> of example traces of reflected wavefields measured by the receivers located along the streamer <NUM> shown in <FIG>. Vertical axis <NUM> represents time. Horizontal axis <NUM> represents channels or trace numbers with trace "<NUM>" representing a trace of seismic data generated by a receiver located closer to the source <NUM> than trace "<NUM>" representing a trace of seismic data generated by a receiver located farther away from the source <NUM>. Wavelets represent reflection events from an interface or a surface. The distance along a trace from time zero to the location of a wavelet represents the travel time of the acoustic energy output from the source <NUM> to an interface or surface and eventually to a receiver located along the streamer <NUM>. Differently patterned lines are added to represent wavefields that correspond to the example reflection events represented by corresponding differently patterned ray paths illustrated in <FIG>. For example, wavelets located along trace <NUM> correspond to the reflection events that reach the receiver <NUM> as represented by differently patterned lines in <FIG>. Wavelets located along dashed curve <NUM> represent a portion of the acoustic signal generated by the source <NUM> that travels directly to the receivers. Wavelets located along dashed curve <NUM> represents changes in pressure that correspond to acoustic energy reflected upward from the formation surface <NUM> as represented by ray path <NUM> in <FIG>. Wavelets located along dashed line <NUM> represents changes in pressure created by the head waves that travel just below the surface <NUM>, as represented by ray paths <NUM> and <NUM> in <FIG>. Wavelets located along dot-dashed curve <NUM> represent changes in pressure that correspond to acoustic energy reflected upward from the interface <NUM>, as represented by ray path <NUM> in <FIG>. Dotted-dashed line <NUM> represents changes in pressure created by the head waves that travel just below the interface <NUM>, as represented by ray paths <NUM> and <NUM> in <FIG>. Wavelets located along curve <NUM> represent changes in pressure that correspond to acoustic energy reflected upward from the interface <NUM>, as represented by ray path <NUM> in <FIG>. Wavelets located along curve <NUM> represent a diving wavefield created by a portion of the acoustic signal that is turned upward from a vertical velocity gradient of the layer <NUM> as represented by ray paths <NUM> and <NUM> in <FIG>. Note that for the sake of simplicity of illustration and discussion, the example traces in <FIG> only record a small number of the reflected wavefields and do not represent other reflections and various types of random and coherent noise that are typically recorded during a marine seismic survey, such as shot noise, swell noise, barnacle noise, streamer vibration, and bird noise.

Subterranean formations may also be surveyed using ocean bottom seismic techniques. In one implementation, these techniques may be performed with ocean bottom cables ("OBCs") laid on or near the water bottom. The OBCs are similar to towed streamers described above in that the OBCs include spaced-apart receivers, such as collocated pressure and/or particle motion sensors, deployed approximately every <NUM> to <NUM> meters. In other implementation, ocean bottom nodes ("OBNs") may be deployed along the formation surface. Each node may have collocated pressure and/or particle motion sensors. The OBCs and OBNs may be electronically connected to an anchored recording vessel that provides power, instrument command and control of the pressure and/or vertical velocity wavefield sent to recording equipment located on board the vessel. Traces of recorded seismic data using streamers, as described above, OBCs, or OBNs may processed as described below.

The variable density acoustic wave equation in terms of velocity and density is given by <MAT> where.

The collection of observation points <MAT> form the image domain. Acoustic wave impedance is a product of the seismic velocity and the density: <MAT> Using Equation (<NUM>) to substitute for the density in Equation (<NUM>) gives the acoustic wave equation in terms of the seismic velocity and impedance as follows: <MAT> Equation (<NUM>) may be expanded to obtain <MAT> Vector reflectivity is defined as <MAT> where.

The acoustic wave equation in Equation (<NUM>) may be rewritten in terms of velocity and vector reflectivity as follows: <MAT> The solution of Equation (<NUM>) is a complete pressure wavefield for steep reflection events (i.e., large dips). The time and space derivative operator on the left-hand side of Equation (<NUM>) models time and space propagation of seismic waves through various materials of the subterranean formation based on seismic velocities V(x) and reflectivity <MAT> of the various materials. For practical purposes, in most of the geological settings of economic interest that do not consider extreme steep dips, Equation (<NUM>) may be simplified by only considering vertical reflectivity in the z-direction. As a result, Equation (<NUM>) reduces to <MAT> where <MAT> is the z-component of the vector reflectivity <MAT>.

The acoustic wave equations in Equations (<NUM>) and (<NUM>) do not require a density field or high velocity contrasts to compute simulated reflections in the modeled data. Instead, the acoustic wave equations depend on a velocity model and the reflectivity (or image), which are available from previous steps in the velocity model building and imaging process. An acoustic wave traveling through a subterranean formation has a seismic velocity denoted by <MAT> and a vector reflectivity <MAT>. The seismic velocity <MAT> represents acoustic wave properties of a medium in terms of the speed at which acoustic waves travel within a subterranean formation. Each component of vector reflectivity <MAT> is the normalized change of impedance in a particular direction. For example, at a horizontal layered medium, the vertical component of the reflectivity is equivalent to the reflection coefficient. The seismic velocity and vector reflectivity depend on the observation point <MAT>, the composition of the medium varies from point to point. An observation point may represent a point located along a surface of a subterranean formation or represent a point along an interface between two different types of rock, sediment, or fluid within the subterranean formation. An observation point may also represent a point within a layer of fluid or solid with a homogeneous composition.

Although the following discussion describes building velocity and/or reflectivity models using Equation (<NUM>), in alternative implementations, Equation (<NUM>) may be substituted for Equation (<NUM>). The term reflectivity model refers to a vector reflectivity model <MAT> or a vertical reflectivity model with <MAT>.

Processes and systems described below are directed to generating velocity and reflectivity models of a subterranean formation from a pressure wavefield recorded in a marine survey of the subterranean formation. The velocity and reflectivity models are obtained with iterative FWI using the acoustic wave equation given by Equation (<NUM>) and may be used to identify features that correspond to oil and natural gas reservoirs. The velocity model by itself may be used in depth migration to improve the resolution of an image of the subterranean formation.

<FIG> shows a process for building velocity and reflectivity models of a subterranean formation from a pressure wavefield recorded in a marine seismic survey of a subterranean formation. Each block represents a different module of computer implemented machine-readable instructions stored in one or more data-storage devices and executed using one or more processors of a computer system. The construction of the velocity model may include additional modules or certain modules may be omitted or executed in a different order, depending on how the recorded seismic data is collected, conditions under which the recorded seismic data is collected, and depth of the body of water above the subterranean formation.

In <FIG>, block <NUM> represents retrieving a pressure wavefield recorded in a marine survey of a subterranean formation as described above with reference to <FIG>. Blocks <NUM>-<NUM> describe computational operations that precondition the pressure wavefield data. In block <NUM>, acquisition noise in the pressure wavefield, such swell noise and barnacle noise, is attenuated. In block <NUM>, the pressure wavefield is corrected for receiver motion created by moving streamers in the body of water. In block <NUM>, after receiver motion correction, the seismic data is resampled. Multiple reflections, or simply "multiples," are acoustic waves that have bounced more than once from subsurface reflectors and/or the free surface in contrast to primary reflections, or simply "primaries," which have reflected only once from the subterranean formation. In block <NUM>, multiples in the pressure wavefield are attenuated or suppressed. In block <NUM>, a "perform iterative full-waveform inversion ("FWI") to build a high-resolution velocity model Vf and a reflectivity model <MAT> of the subterranean formation" procedure is performed. An example implementation of the "perform iterative full-waveform inversion ("FWI") to build a high-resolution velocity model Vf and a reflectivity model <MAT> of the subterranean formation" procedure is described below with reference to <FIG>. In block <NUM>, the velocity model and the reflectivity model may be used to identify structural and lithological features in the subterranean formation that correspond to oil and gas reservoirs. The velocity and reflectivity models of a subterranean formation are displayed on monitors, projected onto screens, or displayed using other visual display devices. In some cases, the velocity and reflectivity models may be used to identify the composition of features and layers within a subterranean formation, such as oil and natural gas accumulations, and may be used for pre-drill prediction of pore pressure, enabling geoscientist to take steps that mitigate the risks and hazards associated with drilling into high-pressure petroleum reservoirs. Velocity and reflectivity models and images generated from the pressure wavefield recorded at different stages of oil or natural gas extraction of a reservoir may be used be used by geoscientist to monitor over time extraction of oil and natural gas from the reservoir.

<FIG> is a flow diagram illustrating an example implementation of the "perform iterative full-waveform inversion ("FWI") to build a high-resolution velocity model Vf and a reflectivity model <MAT> of the subterranean formation" procedure referenced in block <NUM> of <FIG>. In block <NUM>, an initial smooth velocity model, denoted by V<NUM>, and an initial reflectivity model, denoted by <MAT>, are received as input. The initial velocity model V<NUM> and the initial vector reflectivity model <MAT> may have been generated from previous velocity model building and imaging processes. In block <NUM>, traces of a recorded pressure wavefield denoted by <MAT> that have been obtained as described above with reference to <FIG> are received as input. Iterative FWI is executed by the computational operations represented by blocks <NUM>-<NUM>. Iterative FWI computes a final velocity model denoted by Vf and a final reflectivity model <MAT> that are output in block <NUM>.

<FIG> shows a high-level representation of iterative FWI executed by blocks <NUM>-<NUM> of <FIG>. Consider an example of a three-dimensional synthetic medium <NUM> that represents an initial approximation of multiple layers of a subterranean formation. This example synthetic medium <NUM> comprises eight isotropic layers. Each layer occupies a separate three-dimensional volume within the synthetic medium <NUM>. Initially, each layer has a uniform thickness in the z-direction and represents a homogeneous fluid or solid layer within the synthetic medium <NUM>. For example, top layer <NUM> may represent a body of water. Layers located below the top water layer <NUM> may represent different layers of rock, sediment, or fluid within the subterranean formation. Each layer of the synthetic medium <NUM> has an initial associated seismic velocity recorded in a velocity model. Each interface between layers has an initial associated reflectivity recorded in a reflectivity model. Seismic velocities of the layers in the synthetic medium <NUM> are denoted by <MAT>, where superscript "<NUM>" identifies the seismic velocities of the initial velocity model V<NUM> and subscript q = <NUM>,. ,<NUM> corresponds to the eight layers of the synthetic medium <NUM>. Reflectivity of the interfaces in the synthetic medium <NUM> are denoted by <MAT>, where superscript "<NUM>" identifies the reflectivity of the initial reflectivity model <MAT> and subscript q = <NUM>,. ,<NUM> corresponds to the interfaces between layers and formations of the synthetic medium <NUM>. Because the layers of the synthetic medium are homogeneous, observation points within the same layer of the synthetic medium <NUM> have the same seismic velocity. The seismic velocity at an observation point <MAT> in the q-th layer is denoted by <MAT>. Reflectivity at an observation point located at an interface is denoted by <MAT>. The synthetic medium <NUM> is a representative initial model of a subterranean formation, and for ease of illustration, has only eight layers and seven interfaces with corresponding seismic velocities and reflectivity. In other implementations, the number of layers may be more or less than eight. In <FIG>, iterative FWI <NUM> is represented by directional arrows <NUM> and <NUM>. The initial velocity model V<NUM> and initial reflectivity model <MAT> for the synthetic medium <NUM> are input to the iterative FWI <NUM>. Each iteration of the iterative FWI <NUM> updates the locations of reflectors (i.e., z-coordinate locations of the surface and interfaces) in the synthetic medium, updates velocities in the velocity model, and updates reflectivity of the interfaces in the reflectivity model. The velocity and reflectivity models generated after each iteration of iterative FWI <NUM> are denoted by Vj and <MAT>, respectively, where j is a non-negative integer used to denote the j-th iteration of iterative FWI <NUM>. <FIG> shows an example synthetic medium <NUM> and associated seismic velocities <NUM> of the j-th velocity model Vj and reflectivity <NUM> of the j-th reflectivity model <MAT> after completion of the j-th iteration. <FIG> shows an example final synthetic medium <NUM> and associated seismic velocities <NUM> of the final velocity model Vf and reflectivity <NUM> of the final reflectivity model <MAT> after completion of the final iteration of iterative FWI <NUM>. As shown in <FIG>, iterative FWI <NUM> changes the configuration of the layers and interfaces of the layers in the synthetic medium <NUM> to approximate the configuration of the layers and interfaces in the actual subterranean formation.

Returning to <FIG>, each iteration of iterative FWI begins with block <NUM>. In block <NUM>, forward modeling is performed to compute a synthetic pressure wavefield at each subsurface point as a function of time, <MAT> based on the j-th velocity model Vj updated in block <NUM> and the j-th vector reflectivity model <MAT> updated in block <NUM>. The traces of synthetic pressure data <NUM> at the receiver locations are denoted by <MAT>. In block <NUM>, forward modeling is performed using Equation (<NUM>): <MAT> where.

An acoustic wave propagates in a medium by compressing and decompressing the medium such that a small volume of the material oscillates in the direction the acoustic wave is traveling. The synthetic pressure wavefield <MAT> is the pressure wavefield at the observation point <MAT> in the medium at time t and is uniquely determined by the acoustic wave equation in Equation (<NUM>). The source wavefield <MAT> is the source wavefield generated by the source <NUM> and may be obtained from near-field pressure measurements recorded using hydrophones located near the source <NUM> at the time the source <NUM> is activated or by modeling of the source array. Forward modeling with Equation (<NUM>) in block <NUM> may be performed with a finite differencing method, a pseudo-spectral method, a pseudo-analytic method, finite-element method, spectral-element method, or a finite-volume method to obtain the synthetic pressure wavefield <MAT> <NUM> at each receiver location <MAT> in the subterranean formation. The synthetic pressure wavefield obtained using forward modeling is a function of the velocity model, the vector reflectivity model, and the source wavefield: <MAT> where F represents a forward modeling operator. In certain implementations, the source <NUM> may be regarded as a point source represented as follows: <MAT> where S(t) is a source-time function. In this case, the synthetic pressure wavefield obtained using forward modeling is a function of the velocity model, the reflectivity model, and the source-time function: <MAT>.

In block <NUM>, a residual may be computed for each receiver coordinate and time sample as follows: <MAT> where.

The residual <MAT> is a difference between the trace of synthetic seismic data <MAT> and the trace of recorded pressure wavefield <MAT> for each of the N receivers and for each time sample. In block <NUM>, a residual magnitude is computed for the j-th iteration as follows: <MAT> where
<MAT>
is an L2 norm. Iterative FWI as represented in <FIG> stops when the residual magnitude satisfies the following condition <MAT> where ε is a residual magnitude threshold. The output <NUM> comprises the final velocity model Vf, which is the j-th velocity model Vj, and the final reflectivity <MAT>, which is the j-th final reflectivity <MAT>.

In block <NUM>, adjoint migration is performed using Equation (<NUM>) in reverse time with the source term replaced by the superposition of the residual wavefield determined at each receiver location in Equation (<NUM>) as follows: <MAT> where.

In block <NUM>, an inverse scattering imaging condition ("ISIC") kernel velocity is computed by <MAT> where.

The ISIC- kernel velocity can substantially reduce or eliminate short-wavelength components of the velocity gradient and enhance macro velocity features. In Equation (<NUM>), the migrated residual wavefield <MAT> is obtained by time reversing the back propagated residual wavefield. The illumination term is <MAT> at each point <MAT>. The dynamic weights are designed to optimally suppress the large- or small-scale components of the property updates in each case. The velocity dynamic weights are computed by minimization as follows: <MAT> <MAT> where r is a trial weight and <NUM> ≤ r ≤ <NUM>.

In block <NUM>, the seismic velocity at each observation point in the velocity model Vj is updated as follows: <MAT> where dv is a constant called "velocity step length. "
In block <NUM>, the reflectivity model <MAT> may be updated by mapping the reflectivity model to a new reflectivity model <MAT> based on the updated velocity model Vj+<NUM> obtained in block <NUM>. In block <NUM>, the reflectivity model <MAT> is converted to time coordinates using the velocity model Vj followed by a time to depth conversion using the updated velocity model Vj+<NUM>. The time to depth conversion may be performed trace by trace or by applying post-stack de-migration of the reflectivity <MAT> using the velocity model from iteration Vj followed by a post-stack migration using the updated velocity model Vj+<NUM>. The ISIC- kernel velocity in Equation <NUM> enhances updates of long-wavelength components of the velocity model Vj, which cannot be achieved with a velocity gradient obtained using conventional FWI. Once long-wavelength components of the velocity model Vj are updated with improved accuracy in later FWI iterations, a conventional FWI gradient may be used to correctly position short-wavelength features of the velocity model Vj, thereby further increasing resolution of the updated velocity model Vj+<NUM> output from block <NUM>.

<FIG> shows a plot of a conventional cross-correlation kernel produced for a model consisting of a single homogeneous layer overlying a half-space. The locations of a source and a receiver are correspondingly denoted by S and R. The cross-correlation kernel includes low wavenumber (i.e., large-scale) components <NUM> and high wavenumber (i.e., small-scale) components <NUM> that are related to a wavelength λ of an acoustic wave (i.e., k ∝ <NUM>/λ). The low wavenumber components <NUM> are the result of cross-correlation of the down-going wavefields and the backscattering produced by an interface. Low wavenumber components <NUM> correspond to low-wavenumber features of the velocity model. The high wavenumber components <NUM> may be referred to as a migration isochrone and correspond to specular reflections of the reflectivity model.

<FIG> shows a plot of ISIC kernel impedance produced with a dynamically weighted impedance sensitivity kernel for the same model as <FIG>. The impedance sensitivity kernel corresponds to Equation (<NUM>). The image illustrates that high wavenumber components <NUM> are preserved while low wavenumber components <NUM> are suppressed.

<FIG> shows a plot of ISIC- kernel velocity produced with a dynamically weighted velocity sensitivity kernel for the same model as <FIG>. The kernel velocity corresponds to Equation (17a). The image illustrates that high wavenumber components <NUM> present in <FIG> are suppressed while the low wavenumber components <NUM> are preserved.

Returning to <FIG>, in block <NUM>, the final velocity model Vf and final reflectivity model <MAT> may be used to identify compositions of the various features and layers within the subterranean formation. For example, the final velocity model Vf and the final reflectivity model <MAT> may be used to identify deposits such as natural gas and water, and identify the different types of rock, porous materials, and sediments in the layers of the subterranean formation. Reflectivity model <MAT> provides shape information of the different interfaces of subterranean formations and may be a strong indication of the potential structures that may be reservoirs of oil and natural gas. The velocity model may also be used to determine the pressure within a petroleum deposit, which enables petroleum engineers to reduce the risks and hazards of drilling into a high-pressure petroleum deposit.

Least-square reverse time migration described in the next subsection below may be applied to the recorded pressure wavefield using the velocity model obtained in block <NUM> to improve resolution of a reflectivity model of the subterranean formation. The image or reflectivity model of the subterranean formation may be displayed on a monitor or other display device to provide a visual representation of structures and features of the subterranean formation. The image of the subterranean formation may be a two-dimensional visual representation of a cross section of the subterranean formation. Alternatively, the image of the subterranean formation may be a three-dimensional visual representation of the subterranean formation.

Reverse time migration ("RTM") is a preferred migration method for modeling and imaging seismic data in subterranean formations that produce complex seismic wave phenomena because RTM is able to handle combinations of structural dip with high velocity contrasts, which are conditions common in salt basins and other geologic basins with complex structures and velocity distributions. However, even with an accurate velocity model of the subterranean formation, RTM alone still produces an approximation of the true reflectivity of the subterranean formation. In addition, RTM alone does not compensate for limitations associated with seismic data acquisition and variable acoustic illumination under complex overburden, such as salts or carbonates. By contrast, least-squares RTM ("LSRTM") overcomes problems that RTM or other conventional migration methods are not able to resolve and produces images with fewer artefacts, higher resolution, and more accurate amplitudes than conventional migration methods. In particular, LSRTM performs imaging as an inverse problem with an updated reflectivity model, thereby resulting in an image of a subterranean formation that is closer to the actual reflectivity of the subterranean formation.

<FIG> shows a process for building a reflectivity model of subterranean formation from a pressure wavefield recorded in a marine seismic survey of the subterranean formation using LSRTM. Each block represents a different module of computer implemented machine-readable instructions stored in one or more data-storage devices and executed using one or more processors of a computer system. Building the vector reflectivity model may include additional modules or certain modules may be omitted or executed in a different ordering, depending on how the recorded seismic data is collected, conditions under which the recorded seismic data is collected, and depth of the body of water above the subterranean formation.

In <FIG>, blocks <NUM>-<NUM> perform the same preconditioning operations represented by blocks <NUM>-<NUM> as described above with reference to <FIG>. In block <NUM>, a "perform iterative least-square reverse time migration ("LSRTM") to improve resolution of a reflectivity model of a subterranean formation" procedure is performed. An example implementation of the "perform iterative LSRTM to improve resolution of a reflectivity model of a subterranean formation" procedure is described below with reference to <FIG>.

<FIG> is a flow diagram illustrating an example implementation of the "perform iterative LSRTM to improve resolution of a reflectivity model of a subterranean formation" procedure referenced in block <NUM> of <FIG>. In the example of <FIG>, the procedure is performed in the data domain. The data domain comprises time, shot coordinate locations, and receiver coordinate locations (i.e., t, <MAT>). In block <NUM>, an initial velocity model V<NUM>, and an initial reflectivity model <MAT> are received as input. The initial velocity model V<NUM> and the initial reflectivity model <MAT> may be simple approximations of seismic velocities and reflectivity of the subterranean formation as described above with reference to <FIG>. In block <NUM>, traces of the recorded pressure wavefield <MAT> are received as input. Iterative LSRTM is executed in computational operations represented by blocks <NUM>-<NUM>. Each iteration of iterative LSTRM begins with block <NUM>. Forward modeling is performed in block <NUM> to compute the synthetic wavefield in block <NUM> and consequently the traces of synthetic pressure data at the receiver locations, denoted by <MAT>. The forward modeling is based on the initial velocity model V<NUM> and the j-th reflectivity model <MAT> updated in block <NUM>. Note that in the inversion procedure, for each iteration, the reflectivity model is updated while the velocity model is not updated. Forward modeling is performed with Equation (<NUM>) based on the novel parameterization to determine a synthetic pressure wavefield <MAT>: <MAT> The source wavefield <MAT> is the source wavefield generated by the source <NUM> and may be obtained from near-field pressure measurements recorded using hydrophones located near the source <NUM> or may be computed from modeling as described above with reference to <FIG>. Forward modeling with Equation (<NUM>) in block <NUM> may be performed with a finite differencing method, a pseudo-analytic method, a pseudo-spectral method, a finite-element method, spectral-element method, or a finite-volume method to obtain the synthetic pressure wavefield <MAT> in block <NUM> at each receiver coordinate location <MAT> in the subterranean formation. In block <NUM>, a residual is computed for each receiver coordinate and time sample as described above with reference to Equation (<NUM>). In block <NUM>, a residual magnitude is computed for the j-th iteration using Equation (<NUM>). Iterative LSRTM stops when the residual magnitude satisfies the condition in Equation (<NUM>). In block <NUM>, adjoint migration is performed using Equation (<NUM>) in reverse time, in which the source term is given by the superposition of the residual wavefield determined at each receiver location as described in Equation (<NUM>): <MAT> In block <NUM>, an ISIC kernel impedance is computed by <MAT> where impedance dynamic weights <MAT> and <MAT> are computed as follows: <MAT> <MAT> In block <NUM>, the reflectivity estimation at each observation point in the vector reflectivity model <MAT> is updated by <MAT> where dp is the corresponding "reflectivity step length.

Returning to <FIG>, in block <NUM>, the final reflectivity model <MAT> may be used to identify compositions of the various features and layers within the subterranean formation. In particular, the final reflectivity model <MAT> may be used to identify subsurface structures that may contain deposits such as, for example, oil and natural gas, water, and different types of rocks.

LSRTM in the image domain is used to improve the resolution and amplitude fidelity of an image of a subterranean formation. The acoustic wave equation in Equation (<NUM>) may be used to compute a synthetic pressure wavefield from a velocity model and a reflectivity model containing point diffractors. The synthetic pressure wavefield is migrated using forward modeling to construct a model point spread function ("PSF") for an image of the subterranean formation. The model PSF contains a degree of blurring of the image and may contain factors that contribute to degradation of the image. The model PSF is deconvolved from the image to obtain a corrected image of the subterranean formation with increased resolution of reflection events, interfaces, layers, and other features displayed in the image.

<FIG> is a flow diagram of an unclaimed illustrative example method for increasing resolution of an image of a subterranean formation in the image domain. Block <NUM> represents a velocity model V<NUM> and a reflectivity model <MAT> with point diffractors. In block <NUM>, forward modeling is performed with Equation (<NUM>) to determine a synthetic pressure wavefield <MAT>: <MAT> where <MAT> is a source wavefield at the observation point <MAT> and time t in the synthetic medium as described above with reference to Equations (<NUM>) and (<NUM>). In block <NUM>, the synthetic pressure wavefield is used to construct a model PSF <NUM> using RTM based on Equations <NUM> to <NUM>, except that the ISIC impedance kernel in Equation (<NUM>) is transformed from the space-time domain to space-frequency domain using a Fourier transform. As a result, the resulting model PSF, <MAT>, in block <NUM> is constructed for individual frequencies rather than for a frequency bandwidth in the time domain. The model PSF is a superposition of images computed for individual frequencies. This is done for performing deconvolution in the space-frequency domain. In block <NUM>, the recorded pressure wavefield and velocity model V<NUM> are received as input. In block <NUM>, RTM is applied to the recorded pressure wavefield and velocity model <NUM> to obtain a field data image <MAT> <NUM> of a subterranean formation using the same procedure for individual frequencies as performed in block <NUM>. In block <NUM>, a corrected image of the subterranean formation is obtained by deconvolving the model PSF <MAT> from the image <MAT>: <MAT> where ε is a non-zero stabilization constant. By deconvolving the model PSF <MAT> from the field data image <MAT> in block <NUM>, the blurring and degradation are removed from the field data image to obtain a corrected image <MAT> <NUM>. The resulting corrected image has improved representation of reflectivity in the subterranean formations, has enhanced resolution, and is corrected for illumination effects.

<FIG> shows an example of a computer system that executes an efficient process for generating a velocity model and therefore represents a geophysicalanalysis system. The internal components of many small, mid-sized, and large computer systems as well as specialized processor-based storage systems can be described with respect to this generalized architecture, although each system may feature many additional components, subsystems, and similar, parallel systems with architectures similar to this generalized architecture. The computer system contains one or multiple central processing units ("CPUs") <NUM>-<NUM>, one or more electronic memories <NUM> interconnected with the CPUs by a CPU/memory-subsystem bus <NUM> or multiple busses, a first bridge <NUM> that interconnects the CPU/memory-subsystem bus <NUM> with additional busses <NUM> and <NUM>, or other types of high-speed interconnection media, including multiple, high-speed serial interconnects. The busses or serial interconnections, in turn, connect the CPUs and memory with specialized processors, such as a graphics processor <NUM>, and with one or more additional bridges <NUM>, which are interconnected with high-speed serial links or with multiple controllers <NUM>-<NUM>, such as controller <NUM>, that provide access to various different types of computer-readable media, such as computer-readable medium <NUM>, electronic displays, input devices, and other such components, subcomponents, and computational resources. The electronic displays, including visual display screen, audio speakers, and other output interfaces, and the input devices, including mice, keyboards, touch screens, and other such input interfaces, together constitute input and output interfaces that allow the computer system to interact with human users. Computer-readable medium <NUM> is a data-storage device and may include, for example, electronic memory, optical or magnetic disk drive, USB drive, flash memory and other such data-storage devices. The computer-readable medium <NUM> can be used to store machine-readable instructions that encode the computational processes described above and can be used to store encoded data, during store operations, and from which encoded data can be retrieved, during read operations, by computer systems, data-storage systems, and peripheral devices.

<FIG> shows two gathers, <NUM> and <NUM>, of actual seismic data that was recorded in a marine survey of a subterranean formation. <FIG> shows two synthetic gathers, <NUM> and <NUM>, simulated by using the proposed acoustic wave equation (<NUM>) and velocity and reflectivity models of the subterranean formation as described above. <FIG> shows two synthetic gathers, <NUM> and <NUM>, of seismic data that were simulated using a traditional acoustic wave equation with a constant density model of the subterranean formation. Gathers <NUM> and <NUM> better match those corresponding to the field data, in gathers 1301and <NUM>, rather than those using the traditional acoustic wave equation. The reflection events enclosed by the circle in the gather <NUM> reveals that the simulated seismic data obtained using the traditional acoustic wave equation with the constant density model only faintly displays the reflection events enclosed by the circle in the gather <NUM>. By contrast, the reflection events enclosed by the circle in the gather <NUM> reveals that the simulated seismic data obtained using the acoustic wave equation in Equation (<NUM>) closely matches the reflection events enclosed by the circle in the gather <NUM>.

Claim 1:
In a computer-implemented process for determining properties of a subterranean formation located beneath a body of water using a pressure wavefield (<NUM>, <NUM>) recorded during a marine survey of the subterranean formation, an improvement characterised in that it comprises:
determining a velocity model and a reflectivity model of the subterranean formation based on the recorded pressure wavefield (<NUM>, <NUM>) and using an acoustic wave equation that models acoustic wavefields and depends on velocities and reflectivity of materials comprising the subterranean formation (<NUM>); and
using the velocity model and the reflectivity model to identify properties of features in the subterranean formation. (<NUM>)