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.
f of the subterranean formation” procedure referenced in
f of the subterranean formation” procedure referenced in
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 8 km, 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 for 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, elastic 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 0.9-2.5 km/s, oil has seismic velocities in a range of about 1.2-1.25 km/s, sandstones have seismic velocities in a range of about 2.0 6.0 km/s, and granite and basalt have seismic velocities in a range of about 4.5-6.0 km/s. (See e.g.,
The novel acoustic wave equation described herein provides advantages over traditional acoustic wave equations used in velocity model building and seismic imaging: (1) 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. (2) The acoustic wave equation enables generation of a reflectivity model with a smooth velocity model in LSRTM. (3) 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.
The streamers may be towed to form a planar horizontal seismic data acquisition surface with respect to the free surface 112. 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
The streamers 106-111 are typically long cables containing power and data-transmission lines coupled to receivers (represented by shaded rectangles) such as receiver 118 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 102. Streamer depth below the free surface 112 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 300-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 112.
In
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 122, such as the reflection point 138, may receive a pressure disturbance from the source wavefield more quickly than a point within the subterranean formation 120, such as reflection points 140 and 142. Similarly, a reflection point on the formation surface 122 directly beneath the source 104 may receive the pressure disturbance sooner than a more distant-lying reflection point on the formation surface 122. Thus, the times at which waves are reflected from various reflection points within the subterranean formation 20 may be related to the distance, in three-dimensional space, of the reflection points from the activated source 104.
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 104 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 120 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 120, including information about the shapes, sizes, and locations of the various reflecting features within the subterranean formation 120 of interest to geoscientists.
Each receiver 118 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 1 to 5 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 104 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:
tr(r,
s, t)={A(
r,
s, tk)}k=1M (1)
where
The coordinate location r 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
s of the source 104 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 104. 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 (1) 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.
r,t). 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
(
r, t), where the vector
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
(
r, t). A particle motion sensor that measures particle acceleration (i.e., accelerometer) generates a trace of particle acceleration data denoted by
(
r, t). 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., =(0,0, z)) in which case gz(
r, t) is called vertical displacement wavefield, vz(
r, t) is called vertical velocity wavefield, and az(
r, t) is called vertical acceleration wavefield. Alternatively, each receiver 118 may include two additional particle motion sensors that measure particle motion in two other directions,
1 and
2, that are orthogonal to
(i.e.,
·
1=
·
2=0, where “·” is the scalar product) and orthogonal to one another (i.e.,
1·
2=0). In other words, each receiver 118 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 vz(
r, t), 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, vx(
r, t), and a particle motion sensor that measures the wavefield in the cross-line direction in order to obtain the cross-line velocity wavefield, vy(
r, t). 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
=(vx, vy, vz).
The streamers 106-111 and the survey vessel 102 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 104, absolute positions on the free surface 112, 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 102, where the seismic data may be stored on data-storage devices located onboard the survey vessel 102 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.”
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 25 to 50 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
where
is the gradient operator.
The collection of observation points form the image domain. Acoustic wave impedance is a product of the seismic velocity and the density:
Z()=V(
) ρ(
) (3)
Using Equation (3) to substitute for the density in Equation (2) gives the acoustic wave equation in terms of the seismic velocity and impedance as follows:
Equation (4) may be expanded to obtain
Vector reflectivity is defined as
where
The solution of Equation (7) 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 (7) models time and space propagation of seismic waves through various materials of the subterranean formation based on seismic velocities V() and reflectivity
(
) of the various materials. For practical purposes, in most of the geological settings of economic interest that do not consider extreme steep dips, Equation (7) may be simplified by only considering vertical reflectivity in the z-direction. As a result, Equation (7) reduces to
where Rz() is the z-component of the vector reflectivity
(
).
The acoustic wave equations in Equations (7) and (8) 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 V() and a vector reflectivity
(
). The seismic velocity V(
) 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
(
) 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
, 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 (7), in alternative implementations, Equation (8) may be substituted for Equation (7). The term reflectivity model refers to a vector reflectivity model (
) or a vertical reflectivity model with
(
)=(0,0, Rz(
)).
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. They velocity and reflectivity models are obtained with iterative FWI using the acoustic wave equation given by Equation (7) 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.
In f 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
f of the subterranean formation” procedure is described below with reference to
f of the subterranean formation” procedure referenced in block 506 of
0, are received as input. The initial velocity model V0 and the initial vector reflectivity model
0 may have been generated from previous velocity model building and imaging processes. In block 602, traces of a recorded pressure wavefield denoted by p(
r, t) that have been obtained as described above with reference to
f that are output in block 610.
q0, where superscript “0” identifies the reflectivity of the initial reflectivity model
0 and subscript q=1, . . . ,7 corresponds to the interfaces between layers and formations of the synthetic medium 700. Because the layers of the synthetic medium are homogeneous, observation points within the same layer of the synthetic medium 700 have the same seismic velocity. The seismic velocity at an observation point
in the q-th layer is denoted by Vq0(
). Reflectivity at an observation point located at an interface is denoted by Rq0(
). The synthetic medium 700 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
0 for the synthetic medium 700 are input to the iterative FWI 708. Each iteration of the iterative FWI 708 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 708 are denoted by Vj and
j, respectively, where j is a non-negative integer used to denote the j-th iteration of iterative FWI 708.
j after completion of the j-th iteration.
f after completion of the final iteration of iterative FWI 708. As shown in
Returning to , t) based on the j-th velocity model Vj updated in block 608 and the j-th vector reflectivity model
j updated in block 609. The traces of synthetic pressure data 611 at the receiver locations are denoted by pjsyn(
r, t). In block 603, forward modeling is performed using Equation (7):
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 pjsynth(, t) is the pressure wavefield at the observation point
in the medium at time t and is uniquely determined by the acoustic wave equation in Equation (9). The source wavefield Sj(
, t) is the source wavefield generated by the source 104 and may be obtained from near-field pressure measurements recorded using hydrophones located near the source 104 at the time the source 104 is activated or by modeling of the source array. Forward modeling with Equation (9) in block 603 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 pjsyn(
, t) 611 at each receiver location
r 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:
p
j
syn(r, t)=F(Vj,
j, S(
r,t)) (10)
where F represents a forward modeling operator.
In certain implementations, the source 104 may be regarded as a point source represented as follows:
S(r, t)=δ(
r−
s)S(t) (11)
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:
p
j
syn(r, t)=F(Vj,
j, S(t)) (12)
In block 604, a residual may be computed for each receiver coordinate and time sample as follows:
r
j(nr, tk)=pjsynth(
nr, tk)−p(
nr, tk) (13)
where
where ∥ ∥2 is an L2 norm.
Iterative FWI as represented in
ϕj<ε (15)
where ε is a residual magnitude threshold.
The output 610 comprises the final velocity model Vf, which is the j-th velocity model Vj, and the final reflectivity f, which is the j-th final reflectivity
j.
In block 606, adjoint migration is performed using Equation (9) in reverse time with the source term replaced by the superposition of the residual wavefield determined at each receiver location in Equation (13) as follows:
where
In block 607, an inverse scattering imaging condition (“ISIC. . . ”) kernel velocity is computed by
where
The ISIC. . . kernel velocity can substantially reduce or eliminate short-wavelength components of the velocity gradient and enhance macro velocity features. In Equation (17), the migrated residual wavefield Q (, t) is obtained by time reversing the back propagated residual wavefield. The illumination term is I(
)=ΣtT|S(
, t)|2Δt at each point
. 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:
where r is a trial weight and 0≤r≤1.
In block 608, the seismic velocity at each observation point in the velocity model Vj is updated as follows:
V
j+1()=Vj(
)+dvKVj(
) (18)
where dv is a constant called “velocity step length.”
In block 609, the reflectivity model j may be updated by mapping the reflectivity model to a new reflectivity model
j+1 based on the updated velocity model Vj+1 obtained in block 608. In block 609, the reflectivity model
j is converted to time coordinates using the velocity model Vj followed by a time to depth conversion using the updated velocity model Vj+1. The time to depth conversion may be performed trace by trace or by applying post-stack de-migration of the reflectivity
j using the velocity model from iteration Vj followed by a post-stack migration using the updated velocity model Vj+1. The ISIC− kernel velocity in Equation 17 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+1 output from block 608
Returning to f 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
f 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
f 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 608 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.
In
s,
r). In block 1001, an initial velocity model V0, and an initial reflectivity model
0 are received as input. The initial velocity model V0 and the initial reflectivity model
0 may be simple approximations of seismic velocities and reflectivity of the subterranean formation as described above with reference to
r, t) are received as input. Iterative LSRTM is executed in computational operations represented by blocks 1003-1008. Each iteration of iterative LSTRM begins with block 1003. Forward modeling is performed in block 1003 to compute the synthetic wavefield in block 1010 and consequently the traces of synthetic pressure data at the receiver locations, denoted by pjsyn(
r, t). The forward modeling is based on the initial velocity model V0 and the j-th reflectivity model
j updated in block 1008. 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 (7) based on the novel parameterization to determine a synthetic pressure wavefield pjsyn(
t):
The source wavefield Sj(, t) is the source wavefield generated by the source 104 and may be obtained from near-field pressure measurements recorded using hydrophones located near the source 104 or may be computed from modeling as described above with reference to
r, t) in block 1010 at each receiver coordinate location
r the subterranean formation. In block 1004, a residual is computed for each receiver coordinate and time sample as described above with reference to Equation (13). In block 1005, a residual magnitude is computed for the j-th iteration using Equation (14). Iterative LSRTM stops when the residual magnitude satisfies the condition in Equation (15). In block 1006, adjoint migration is performed using Equation (19) 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 (13):
In block 1007, an ISIC impedance kernel impedance is computed by
where impedance dynamic weights W3(, t) and W4(
, t) are computed as follows:
In block 1008, the reflectivity estimation at each observation point in the vector reflectivity model j is updated by
j+1()=
j(
)+dp
log KZj(
) (22)
where dp is the corresponding “reflectivity step length.”
Returning to f may be used to identify compositions of the various features and layers within the subterranean formation. In particular, the final reflectivity model
f 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 (7) 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.
0 with point diffractors. In block 1102, forward modeling is performed with Equation (7) to determine a synthetic pressure wavefield psyn(
, t):
where S(, t) is a source wavefield at the observation point
and time t in the synthetic medium as described above with reference to Equations (9) and (11).
In block 1103, the synthetic pressure wavefield is used to construct a model PSF 1106 using RTM based on Equations 19 to 21, except that the ISIC impedance kernel in Equation (21) is transformed from the space-time domain to space frequency domain using a Fourier transform. As a result, the resulting model PSF, QPSF(, ω), in block 1106 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 1104, the recorded pressure wavefield and velocity model V0 are received as input. In block 1105, RTM is applied to the recorded pressure wavefield and velocity model 1104 to obtain a field data image Im(
, ωw) 1107 of a subterranean formation using the same procedure for individual frequencies as performed in block 1103. In block 1108, a corrected image of the subterranean formation is obtained by deconvolving the model PSF QPSF(
, ω) from the image Im(
, ω):
where ε is a non-zero stabilization constant.
By deconvolving the model PSF QPSF(, ω) from the field data image Im(
, ω) in block 1108, the blurring and degradation are removed from the field data image to obtain a corrected image Imcor(
) 1109. The resulting corrected image has improved representation of reflectivity in the subterranean formations, has enhanced resolution, and is corrected for illumination effects.
The processes and systems disclosed herein may be used to form a geophysical data product indicative of certain properties of a subterranean formation. The geophysical data product may be manufactured by using the processes and systems described herein to generate geophysical data and storing the geophysical data in the computer readable medium 1228. The geophysical data product may include geophysical data such as pressure wavefield data, particle motion data, particle velocity data, particle acceleration data, upgoing and downgoing pressure wavefield data, velocity models, reflectivity models, of a subterranean formation computed from using the processes and systems described herein. The geophysical data product may be produced offshore (i.e., by equipment on the survey vessel 102) or onshore (i.e., at a computing facility on land), or both.
It is appreciated that the previous description of the disclosed embodiments is provided to enable any person skilled in the art to make or use the present disclosure. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the generic principles defined herein may be applied to other embodiments without departing from the spirit or scope of the disclosure. Thus, the present disclosure is not intended to be limited to the embodiments shown herein but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.
This application claims the benefit of Provisional Application 62/911,464, filed Oct. 7, 2019, which application is hereby incorporated by reference as if entirely set forth herein.
Number | Date | Country | |
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62911464 | Oct 2019 | US |