Misfit Function Research Articles

Adaptive waveform inversion: Practice

Adaptive waveform inversion (AWI) reformulates the misfit function used to perform full-waveform inversion (FWI), so that it no longer contains local minima related to cycle skipping. It does this by finding a model that drives the ratio of the predicted and observed data sets to unity rather than driving the difference between these two data sets to zero as is the case for conventional FWI. We apply AWI to a 3D field data set acquired over a pervasive gas cloud in the North Sea, comparing its performance with that of conventional FWI in a variety of circumstances. When starting inversion from 3 Hz, and using a good starting model obtained from reflection tomography, FWI and AWI generate similar models although the FWI result contains edge artifacts that are not produced by AWI. However, when the starting frequency is increased to approximately 6 Hz, or when the starting model is less accurate, FWI fails to recover a good model whereas AWI continues to converge. When both of these conditions apply, FWI fails comprehensively, leading to a model that is significantly worse than the starting model, whereas the AWI result remains largely unaffected. We applied Kirchhoff depth migration to the fully-processed data using the FWI result obtained following reflection tomography, and using the AWI result obtained from a simple one-dimensional starting model. We use the resulting migrated volumes, together with measures of residual moveout throughout the volume, to show that the AWI result from a simple starting model is at least as good as the FWI result obtained following tomography. We conclude that AWI is robust in the presence of cycle skipping on this 3D field data set, and can proceed successfully from a less-accurate starting model, and from a higher starting frequency, in circumstances in which FWI fails completely.

Full waveform inversion for velocity and density with rock physical relationship constraints

The trade-off between parameters in inversion increases the non-linearity of multiparameter full wave-form inversion (FWI). Density is difficult to reconstruct in multiparameter FWI, because it has the same radiation pattern in narrow apertures with that of P- velocity. Inversion with constraint is a good method to reduce the solution space of inversion problem. Since the same rock formations satisfy consistent lithological properties, systematic relationships between parameters, for example, P-wave and density, can be established for many different sedimentary rocks which can be used to provide a good constraint for multiparameter FWI. Utilizing the relationship between density and velocity in FWI, we propose a novel method to mitigate the trade-off between density and velocity and to improve the resolution of the inverted results. We introduce three terms in the definition of the FWI misfit function: the data misfit itself, the Tikhonov regularization term which can be used to smooth the models and rock physical relationship term to mitigate the trade-off between density and velocity. The rock physical relationship term can be calculated from a prior information, for example, logging data, which can provide a constraint within the whole model area in multiparameter FWI. Numerical tests on Gaussian-anomaly model and Marmousi-2 model in different circumstances show that the proposed method can improve the resolution of inversion results.

A Comparative Study of Simulated Annealing and Genetic Algorithm Method in Bayesian Framework to the 2D-Gravity Data Inversion

The use of modern optimization method in geophysical inversion has effectively given a robust global solution in its application to solve a complex non-linearity problem. Here, we tested two artificial intelligent-based methods, the simulated annealing and the genetic algorithm to the gravity data. Using predicted anomalies geometries, these methods are addressed to invert a synthetical gravity data extracted from grav2d open source. Differences between these methods are observed in both single parameter inversion and simultaneous multi parameter inversion to evaluate the speed of computing and the use of Space in memory. The result give us an idea that the genetic algorithm are slower than the simulated annealing in solving a simple inversion problem (small data set and less parameter to be inverse) but efficient in a large data set. meanwhile, the simulated annealing faced some problem in locating a global minima of the misfit function for the large data. in the latter case, we simulated the methods in the Bayesian framework to see the distribution of posterior probability of the parameters.

A robust waveform inversion using a global comparison of modeled and observed data

A high-resolution model of the subsurface is the product of a successful full-waveform inversion (FWI) application. However, this optimization problem is highly nonlinear, and thus, we iteratively update the subsurface model by minimizing a misfit function that measures the difference between observed and modeled data. The L2-norm misfit function provides a simple, sample-by-sample comparison between the observed and modeled data. However, it is susceptible to local minima in the objective function if the low-wavenumber components of the initial model are not accurate enough. We review an alternative formulation of FWI based on a more global comparison. A combination of Radon transform and utilizing a matching filter allows for comparisons beyond sample to sample. We combine two recent developments to suggest the following algorithm for optimal inversion: (1) we compute the matching filter between the observed and modeled data in the Radon domain, which helps reduce the crosstalk introduced in the deconvolution step of computing the matching filter, and (2) we use Wasserstein distance to measure the distance between the resulting matching filter in the Radon domain and a representation of the Dirac delta function, which provides us with the optimal transport between the two distribution functions. We use a modified Marmousi model to show how this Radon-domain optimal-transport-based matching-filter approach can mitigate cycle skipping. Starting from a rather simplified v(z) media as the initial model, the proposed method can invert for the Marmousi model with considerable accuracy, while standard L2-norm formulation is trapped in a local minimum. Application of the proposed method to an offshore data set further demonstrates its robustness and effectiveness.

Elastic reflection waveform inversion based on the decomposition of sensitivity kernels

Full-waveform inversion (FWI) has the potential to obtain an accurate velocity model. Nevertheless, it depends strongly on the low-frequency data and the initial model. When the starting model is far from the real model, FWI tends to converge to a local minimum. Based on a scale separation of the model (into the background model and reflectivity model), reflection waveform inversion (RWI) can separate out the tomography term in the conventional FWI kernel and invert for the long-wavelength components of the velocity model by smearing the reflected wave residuals along the transmission (or “rabbit-ear”) paths. We have developed a new elastic RWI method to build the P- and S-wave velocity macromodels. Our method exploits a traveltime-based misfit function to highlight the contribution of tomography terms in the sensitivity kernels and a sensitivity kernel decomposition scheme based on the P- and S-wave separation to suppress the high-wavenumber artifacts caused by the crosstalk of different wave modes. Numerical examples reveal that the gradients of the background models become sufficiently smooth owing to the decomposition of sensitivity kernels and the traveltime-based misfit function. We implement our elastic RWI in an alternating way. At each loop, the reflectivity model is generated by elastic least-squares reverse time migration, and then the background model is updated using the separated traveltime kernels. Our RWI method has been successfully applied in synthetic and real reflection seismic data. Inversion results demonstrate that the proposed method can retrieve preferable low-wavenumber components of the P- and S-wave velocity models, which are reliable to serve as a starting model for conventional elastic FWI. Also, our method with a two-stage inversion workflow, first updating the P-wave velocity using the PP kernels and then updating the S-wave velocity using the PS kernels, is feasible and robust even when P- and S-wave velocities have different structures.

Frequency Controllable Envelope Operator and Its Application in Multiscale Full-Waveform Inversion

Full-waveform inversion (FWI) attempts to find optimal models of subsurface by using full information of the observed data. One difficulty in conventional FWI is that the misfit function has many local minima because of cycle skipping. Envelope inversion (EI), which uses the envelope operator (EO)-based misfit function, has been proven to be effective in mitigating cycle skipping and recovering long-wavelength velocity model. However, EI ignores the fact that the information within different frequency bands plays different roles in inversion. In this paper, a frequency controllable EO, which can control the frequency components being used to construct envelope, is proposed. We propose a new misfit function and a multiscale FWI method. Using synthetic experiments based on the Marmousi model, we demonstrate that the proposed method is better than EI in mitigating cycle skipping and in building an accurate initial model for conventional FWI to significantly improve its final result. In addition, this method can tolerate a wide range of noise levels. Its effectiveness has also been successfully demonstrated using a field data set.

Seismic imaging and optimal transport

Seismology has been an active science for a long time. It changed character about 50 years ago when the earth's vibrations could be measured on the surface more accurately and more frequently in space and time. The full wave field could be determined, and partial differential equations (PDE) started to be used in the inverse process of finding properties of the interior of the earth. We will briefly review earlier techniques but mainly focus on Full Waveform Inversion (FWI) for the acoustic formulation. FWI is a PDE constrained optimization in which the variable velocity in a forward wave equation is adjusted such that the solution matches measured data on the surface. The minimization of the mismatch is usually coupled with the adjoint state method, which also includes the solution to an adjoint wave equation. The least-squares norm is the conventional objective function measuring the difference between simulated and measured data, but it often results in the minimization trapped in local minima. One way to mitigate this is by selecting another misfit function with better convexity properties. Here we propose using the quadratic Wasserstein metric as a new misfit function in FWI. The optimal map defining the quadratic Wasserstein metric can be computed by solving a Monge-Ampere equation. Theorems pointing to the advantages of using optimal transport over the least-squares norm will be discussed, and a number of large-scale computational examples will be presented.

Randomized Approach to Nonlinear Inversion Combining Random and Optimized Simultaneous Sources and Detectors

In partial differential equation-based (PDE-based) inverse problems with many measurements, many large-scale discretized PDEs must be solved for each evaluation of the misfit or objective function. In the nonlinear case, evaluating the Jacobian requires solving an additional set of systems. This leads to a tremendous computational cost, and this is by far the dominant cost for these problems. Several authors have proposed randomization and stochastic programming techniques to drastically reduce the number of system solves by estimating the objective function using only a few appropriately chosen random linear combinations of the sources. While some have reported good solution quality at a greatly reduced cost, for our problem of interest, diffuse optical tomography, the approach often does not lead to sufficiently accurate solutions. We propose two improvements. First, to efficiently exploit Newton-type methods, we modify the stochastic estimates to include random linear combinations of detectors, drastically reducing the number of adjoint solves. Second, after solving to a modest tolerance, we compute a few simultaneous sources and detectors that maximize the Frobenius norm of the sampled Jacobian to improve the rate of convergence and obtain more accurate solutions. We complement these optimized simultaneous sources and detectors by random simultaneous sources and detectors constrained to a complementary subspace. Our approach leads to solutions of the same quality as obtained using all sources and detectors but at a greatly reduced computational cost, as the number of large-scale linear systems to be solved is significantly reduced.

Analysis of Regularized Kantorovich--Rubinstein Metric and Its Application to Inverse Gravity Problems

We propose an $L^2$ regularized Kantorovich--Rubinstein (KR) metric as a novel misfit function for the level-set based inverse gravity problems without assuming mass conservation. We study the well...

Multiscale Direct Envelope Inversion: Algorithm and Methodology for Application to the Salt Structure Inversion

AbstractFor the full‐waveform inversion, obtaining a globally optimal solution is difficult when the seismic data do not contain sufficient low‐frequency information and the inversion object is a salt structure with a large‐scale salt dome. The large salt dome presents difficulty for the inversion of the salt structure; however, it also confers a convenience that can be utilized: A large salt dome makes the events in the seismic record sparse and isolated. We use the envelope to obtain the arrival time of the events in the seismic data and apply multiscale strategies to extract the low‐frequency information of the seismic data that relates to the long‐wavelength components of the subsurface. In calculating the gradient, the envelope Fréchet derivative is used to take advantage of the multiscale envelope for the characteristics of the salt structure. Therefore, the multiscale direct envelope inversion using the window‐averaged envelope and the direct envelope Fréchet derivative is proposed. To further improve the computational efficiency and inversion quality of the multiscale direct envelope inversion, several auxiliary strategies are proposed: A joint misfit function is designed to make the inversion method more adaptable, and an offset weighting factor is introduced to implement the multioffset inversion method to improve the inversion quality of subsalt. We analyze the antinoise performance of the multiscale direct envelope inversion. The SEG/EAGE salt model is used to test the application effect of the multiscale direct envelope inversion. The antinoise ability and insensitivity to low‐frequency data of the method are verified in the numerical experiments.

A method of geo-acoustic parameter inversion in shallow sea by the Bayesian theory and the acoustic pressure field

A method of geo-acoustic parameter inversion based on the Bayesian theory is proposed for the acquisition of acoustic parameters in shallow sea with the elastic seabed. Firstly, the theoretical prediction value of the sound pressure field is calculated by the fast field method (FFM). According to the Bayesian theory, we establish the misfit function between the measured sound pressure field and the theoretical pressure field. It is under the assumption of Gaussian data errors which are in line with the likelihood function. Finally, the posterior probability density (PPD) of parameters is given as the result of inversion. Our research is conducted in the light of Metropolis sample rules. Apart from numerical simulations, a scaled model experiment has been taken in the laboratory tank. The results of numerical simulations and tank experiments show that sound pressure field calculated by the result of inversion is consistent with the measured sound pressure field. Besides, s-wave velocities, p-wave velocities and seafloor density have fewer uncertainties and are more sensitive to complex sound pressure than s-wave attenuation and p-wave attenuation. The received signals calculated by inversion results are keeping with received signals in the experiment which verify the effectiveness of this method.

Seismic deconvolution and inversion with erratic data

ABSTRACTIf there are some erratic data (e.g. outliers), which may arise from measurement error, or other reasons, in seismic data, the seismic deconvolution and inversion need to be implemented in a way that minimizes their effects. However, the deconvolution and inversion methods based on L2‐norm misfit function are highly sensitive to these erratic seismic observations. As an alternative, L1‐norm misfit functions are more robust and erratic‐resistant. In order to find the solution of the inverse problem constrained by an L1‐norm misfit function, an iteratively re‐weighted least squares algorithm is used frequently. However, it is relatively time consuming. In this paper, we propose a new method based on the sparse signal representation theory. The overcomplete dictionary used for the sparse representation of seismic data with erratic data is composed of two bases: a wavelet basis used for representing the seismic data to implement deconvolution and a Dirac basis used for representing the erratic data. In addition, at the stage of seismic inversion after deconvolution, total variation and a priori model are used as the regularization constraint terms to estimate inversion results with a blocky and laterally continuous structure. The new method is successfully tested on the noisy synthetic seismic data with erratic data. Finally, the proposed method is performed on a real seismic data section, and the inversion results are reasonable, i.e. consistent with the geologic structure of the original seismic data. Compared to the conventional sparse deconvolution and inversion method, the proposed method not only eliminates the effect of outliers, but also has highly improved computational efficiency.

A New Approach to Guided Wave Ray Tomography for Temperature-Robust Damage Detection Using Piezoelectric Sensors.

In this paper, a new approach to guided wave ray tomography for temperature-robust damage detection with time-of-flight (TOF) temperature compensation is developed. Based on the linear relationship between the TOF of a guided wave and temperature, analyses show that the TOF of the baseline signal can be compensated by the temperature measurement of the inspected materials without estimating the temperature compensation parameters. The inversion is based on the optimization of the TOF misfit function between the inspected and compensated baseline TOFs of the guided waves, and is applied by the elastic net penalty approach to perform thickness change mapping in a structural health monitoring (SHM) application. Experiments that are conducted in isotropic plates by piezoelectric sensors demonstrate the effectiveness of the proposed method. According to the results, our approach not only eliminates the artefacts that are caused by a temperature variation from 25 °C to 70 °C but also provides more accurate and clearer imaging of damage than conventional ray tomography methods.

An optimal transport approach to linearized inversion of receiver functions

Receiver function analysis is widely used to make quantitative inferences about the structure below a seismic station. As these observables are mainly sensitive to traveltimes of phases converted and reflected at seismic discontinuities, the resulting inverse problem is highly non-linear, the solution non-unique, and there are strong trade-offs between the depth of discontinuities and absolute velocities. To overcome this difficulty, we propose to measure the misfit between the predicted and observed data with an optimal transport distance instead of the conventional least-squares distance, a strategy that has shown its assets in the context of full waveform inversion. This approach views a seismogram as a distribution of ‘mass’. The optimal transport distance between two waveforms is the minimal cost of transporting one waveform onto the other. We test the optimal transport approach on the inversion of a radial P-wave receiver function. We also show how it can be applied to measure the cross-convolution distance between the radial and vertical components, thus avoiding the need for deconvolution associated with the calculation of the receiver function. The resulting misfit function is minimized with a local optimization algorithm to constrain the receiver-side structure. The benefits of this methodology are studied in simple synthetic tests and with real data. In particular, we show that with its increased sensibility to time-shifts, the optimal transport distance reduces the number of local minima in the misfit function, which, in the case of a linearized inversion, significantly reduces the dependency to the starting model and results in a better convergence towards the solution model. A joint inversion of the P-wave receiver function and surface wave dispersion curves is performed at the Hyderabad station in India.

Regularized inversion of amplitude-versus-incidence angle (AVA) based on a piecewise-smooth model

Different from the stacked seismic data, pre-stack data includes abundant information about shear wave and density. Through inversing the shear wave and density information from the pre-stack data, we can determine oil-bearing properties from different incident angles. The state-of-the-art inversion methods obtain either low vertical resolution or lateral discontinuities. However, the practical reservoir generally has sharp discontinuities between different layers in vertically direction and is horizontally smooth. Towards obtaining the practical model, we present an inversion method based on the regularized amplitude-versus-incidence angle (AVA) data to estimate the piecewise-smooth model from pre-stack seismic data. This method considers subsurface stratum as a combination of two parts: a piecewise smooth part and a constant part. To fix the ill-posedness in the inversion, we adopt four terms to define the AVA inversion misfit function: the data misfit itself, a total variation regularization term acting as a sparsing operator for the piecewise constant part, a Tikhonov regularization term acting as a smoothing operator for the smooth part, and the last term to smoothly incorporate a priori information for constraining the magnitude of the estimated model. The proposed method not only can incorporate structure information and a priori model constraint, but also is able to derive into a convex objective function that can be easily minimized using iterative approach. Compared with inversion results of TV and Tikhonov regularization methods, the inverted P-wave velocity, S-wave velocity and density of the proposed method can better delineate the piecewise-smooth characteristic of strata.

Estimation of elastic anisotropy from three-component ultrasonic measurements using laser Doppler interferometry

Ultrasonic measurements using laser Doppler interferometry (LDI) have been reported to provide robust estimates of elastic anisotropy of rock samples. In this approach, an ultrasonic wave is emitted by a piezo-electric source and detected by the LDI, which can be configured to measure three components of the particle velocity in a very small area (~1 mm2) of the sample. Repeating these measurements for a dense array of points on the sample’s surface gives a distribution of traveltimes and polarisation fields on the surface. Anisotropy is then obtained by inverting these fields using analytical expressions or numerical algorithms for computing phase and group velocities. The existing implementation of this approach involves the inversion of direct compressional (P) and shear (S) wave arrivals only. A previous study showed that this approach produces stable results if only a small range of source–receiver offsets is included in the inversion. This limitation resulted in a relatively large uncertainty of the result. This uncertainty can be reduced by inverting the entire traveltime field. To this end, we numerically simulate the wavefield in the sample. Analysis of the computed wavefield reveals the presence of P- and S-waves as well as a critically refracted converted PS-wave. Hence, the inversion of the entire traveltime field must include these three waves. We implement this inversion using global minimisation of the traveltime misfit function, coupled with numerical computation of ray velocities. Application of this algorithm to laboratory LDI measurements on a transversely isotropic phenolic sample provides stable anisotropy estimates consistent with previous studies.

Optimal transport for mitigating cycle skipping in full-waveform inversion: A graph-space transform approach

Optimal transport distance has been recently promoted as a tool to measure the discrepancy between observed and seismic data within the full-waveform-inversion strategy. This high-resolution seismic imaging method, based on a data-fitting procedure, suffers from the nonconvexity of the standard least-squares discrepancy measure, an issue commonly referred to as cycle skipping. The convexity of the optimal transport distance with respect to time shifts makes it a good candidate to provide a more convex misfit function. However, the optimal transport distance is defined only for the comparison of positive functions, while seismic data are oscillatory. A review of the different attempts proposed in the literature to overcome this difficulty is proposed. Their limitations are illustrated: Basically, the proposed strategies are either not applicable to real data, or they lose the convexity property of optimal transport. On this basis, we introduce a novel strategy based on the interpretation of the seismic data in the graph space. Each individual trace is considered, after discretization, as a set of Dirac points in a 2D space, where the amplitude becomes a geometric attribute of the data. This ensures the positivity of the data, while preserving the geometry of the signal. The differentiability of the misfit function is obtained by approximating the Dirac distributions through 2D Gaussian functions. The interest of this approach is illustrated numerically by computing misfit-function maps in schematic examples before moving to more realistic synthetic full-waveform exercises, including the Marmousi model. The better convexity of the graph-based optimal transport distance is shown. On the Marmousi model, starting from a 1D linearly increasing initial model, with data without low frequencies (no energy less than 3 Hz), a meaningful estimation of the P-wave velocity model is recovered, outperforming previously proposed optimal-transport-based misfit functions.

Wave‐Equation Dispersion Inversion of Guided P Waves in a Waveguide of Arbitrary Geometry

AbstractWe present a dispersion‐inversion method which inverts for the P‐velocity model from guided waves propagating in wave guides of arbitrary geometry. Its misfit function is the squared summation of differences between the predicted and observed dispersion curves of guided P waves, and the inverted result is a high‐resolution estimate of the near‐surface P‐velocity model. We denote this procedure as wave‐equation dispersion inversion of guided P waves (WDG), which is valid for near‐surface waveguides with irregular layers and does not require a high‐frequency approximation. It is more robust than full waveform inversion and can sometimes provide velocity models with higher resolution than wave‐equation traveltime tomography. Both the synthetic‐data and field data results demonstrate that WDG for guided P waves can accurately invert for complex P‐velocity models at the near surface of the Earth.

3D gravity inversion of basement relief for a rift basin based on combined multinorm and normalized vertical derivative of the total horizontal derivative techniques

The basement of a rift sedimentary basin, often possessing smooth and nonsmooth shapes, is not easily recovered from gravity data by current inversion methods. We have developed a new 3D gravity inversion method to estimate the basement relief of a rift basin. In the inversion process, we have established the objective function by combining the gravity data misfit function, the known depth constraint function, and the model constraint function composed of the [Formula: see text]-norm and [Formula: see text]-norm, respectively. An edge recognition technology based on the normalized vertical derivative of the total horizontal derivative for gravity data is adopted to recognize the discontinuous and continuous parts of the basin and combine the two inputs to form the final model constraint function. The inversion is conducted by minimizing the objective function by the nonlinear conjugate gradient algorithm. We have developed two applications using synthetic gravity anomalies produced from two synthetic rift basins, one with a single graben and one with six differently sized grabens. The test results indicate that the inversion method is a feasible technique to delineate the basement relief of a rift basin. The inversion method is also tested on field data from the Xi’an depression in the middle of the Weihe Basin, Shaanxi Province, China, and the result illustrates its effectiveness.

Crustal earthquakes in the Cook Inlet and Susitna region of southern Alaska

Several large (M ≥ 6) earthquakes have occurred in the vicinity of Anchorage, Alaska, within the past century. The presence of the underlying subducting Pacific plate makes it difficult to determine the origin of these older earthquakes as either crustal, slab, or the subduction plate interface. We perform a seismological study of historical and modern earthquakes within the Cook Inlet and Susitna region, west of Anchorage. We first estimate hypocenters for historical large earthquakes in order to assess their likelihood of origin as crustal, slab, or plate interface. We then examine modern crustal seismicity to better understand the style of faulting and the location of active structures, including within (and beneath) the Cook Inlet and Susitna basins. We perform double-couple moment tensor inversions using high frequency body waves (1–10 Hz) for small to moderate (M ≥ 2.5) crustal earthquakes (depth ≤ 30 km) occurring from 2007 to 2017. Our misfit function combines both waveforms differences as well as first-motion polarities in order to obtain reliable moment tensor solutions. The three focus regions—Beluga, upper Cook Inlet, and Susitna—exhibit predominantly thrust mechanisms for crustal earthquakes, indicating an overall compressive regime within the crust that is approximately consistent with the direction of plate convergence. Mechanisms within upper Cook Inlet have strike directions aligned with active anticlines previously identified in Cook Inlet from active-source seismic data. Our catalog of moment tensors is helpful for identifying and characterizing subsurface faults from seismic lineaments and from faults inferred from subsurface images from active-source seismic data.