Equations For Reflection Coefficients Research Articles

Classification of Hydrometeors During a Stratiform Precipitation Event in the Rainy Season of Liupanshan

This study conducted a classification analysis of hydrometeor types during a typical stratiform mixed cloud precipitation event in the rainy season using data from the Liupan Mountains micro rain radar power spectra. The primary research findings are as follows: (1) Utilizing the RaProM method synthesizes the information of particle falling velocity, equivalent radar reflection coefficient, particle scale characteristics at different stages, and the location of the bright zone in the zero-degree layer to classify hydrometeors during this precipitation process, and the results show that drizzle and raindrop distribution time periods do not match with the raindrop spectra and rain intensities observed by the DSG5 ground-based precipitation gauge. (2) Sensitivity experiments conducted on the RaProM method revealed that after modifying the discrimination thresholds for drizzle and raindrops, the distributions of drizzle and raindrops were more aligned with ground-based raindrop spectrum observations. Furthermore, these adjustments also showed better consistency with the radar reflectivity factor, Doppler velocity, and velocity spectrum width thresholds used by existing millimeter-wave cloud radars to discriminate between drizzle and raindrops. (3) Various kinds of hydrometeors show different vertical distribution characteristics in three precipitation stages: weak, strong, and weak. In the two weak precipitation stages, hydrometeors mainly existed in the form of snowflakes at altitudes above the zero-degree layer and in the form of drizzle at altitudes below the zero-degree layer. The vertical distribution disparity of hydrometeors between the mountain peak and base sites demonstrates that terrain significantly influences hydrometeors during the precipitation process.

Frequency-dependent nonlinear amplitude-variation-with-offset inversion for fluid mobility in viscoelastic media

Reservoir fluid mobility is defined as the ratio of rock permeability to fluid viscosity, which is an important attribute to guide the prediction of high-quality oil and gas reservoirs. Frequency-dependent amplitude-variation-with-offset (AVO) inversion, based on viscoelastic theory, is a useful tool for fluid identification. In conventional prestack inversion, linear approximations are usually used to calculate the reflection coefficients, whereas nonlinear inversions are only occasionally carried out. However, the nonlinear equation for the reflection coefficient, derived from the exact Zoeppritz equation, has higher accuracy and fewer assumptions than the linear approximations. Therefore, we derive a nonlinear frequency-dependent reflection coefficient equation of the PP wave in terms of P-wave velocity reflectivity, S-wave velocity reflectivity, and fluid mobility, based on the relationship among fluid mobility, incident angle, and seismic wave frequency. We consider viscoelastic media and frequency-dependent responses to improve the authenticity of the reflection coefficients. In addition, we develop a split-step inversion method for fluid mobility based on the artificial neural network inversion algorithm. Synthetic and field data examples demonstrate that our split-step inversion method outperforms traditional inversion methods that rely on approximate equations for predicting underground reservoirs, improving the stability of fluid mobility parameter inversion.

Reflection coefficients for non-welded interface between orthorhombic half-spaces under initial stress

SUMMARY Complex oil and gas reservoirs represented by orthorhombic shale usually have the characteristics of strong anisotropy, wide distribution of non-welded interfaces (i.e. particular boundaries caused by horizontal bedding or fracture development), and ubiquitous initial stress effects. The knowledge of their seismic response characteristics plays an essential role in hydrocarbon exploration in stressed shale reservoirs. However, seismic wave reflection in the orthorhombic media under initial stress remains unclear. To address this issue, we used the theories of acoustoelasticity and elastic anisotropy to derive a stress-dependent effective elastic stiffness tensor. At the same time, we obtain the wave velocity and polarization characteristics directly by solving the Christoffel equation. According to the linear slip theory, we further constructed the exact reflection and transmission coefficient equations for the non-welded interface between isotropic and orthorhombic half-spaces under the initial stress. The effects of the non-welded interface on seismic wave velocity and reflection and transmission coefficients were systematically analysed. Meanwhile, we characterize the law of P-wave reflection coefficient effected by elastic coefficients. Our equations and results potentially lay an equation foundation for orthorhombic reservoirs in high-stress fields and are essential in orthorhombic reservoirs in high-stress geophysical exploration.

Seismic scattering inversion for multiple parameters of overburden-stressed isotropic media

Subsurface reservoirs are compressed by overburden stress resulting from the gravity of overlying masses, and the resulting stress changes significantly affect the seismic reflection responses generated at the reservoir interfaces. Several exact reflection coefficient equations have been well established to delineate the role that in situ stress plays in altering the energy transition and amplitude of seismic reflection responses. These exact equations, however, cannot be effectively used in practice due to their intricate formulations and the difficult geophysical estimation for the third-order elastic constants (3oECs) embedded in reflection coefficients. Based on the theories of nonlinear elasticity and elastic wave inverse scattering, we derive an approximate seismic reflection coefficient equation for overburden-stressed isotropic media in terms of the P-wave modulus, shear modulus, density, and a defined stress-related parameter (SRP). The SRP is a combined quantity of elastic moduli, 3oECs, and overburden stress, which can be naturally treated as a dimensionless stress-induced anisotropy parameter. Its inclusion effectively eliminates the need for 3oECs information when using our equation to estimate the desired reservoir properties from seismic observations. By comparing our equation to the exact one, we confirm its validity within the moderate-stress range. Then, we introduce a Bayesian inversion approach incorporating the new reflection coefficient equation to estimate four model parameters. In our approach, the Cauchy and Gaussian distribution functions are used for the a priori probability and the likelihood distributions, respectively. The synthetic tests from two well-log data sets and a field example demonstrate that four parameters can be reasonably inverted using our approach with rather smooth initial models, which illustrates the feasibility of our inversion approach.

Simultaneous prediction of the geofluid and permeability of reservoirs in prestack seismic inversion

Geofluid discrimination and permeability prediction are indispensable steps in reservoir evaluation. From the perspective of prestack seismic inversion, predicting fluid indicators is an effective method for obtaining fluid properties directly from seismic data. In contrast, the direct prediction of permeability from observed seismic gathers is constrained by the difficulty in establishing a link between permeability and elastic parameters. However, we show that the pore structure parameters in seismic petrophysical theory are highly related to permeability, providing a new solution for predicting permeability using seismic data. Therefore, the correlation between the shear flexibility factor and permeability is first verified based on logging curves and laboratory data, and the results demonstrate that the shear flexibility factor can be an indicator of reservoir permeability. Second, an approximate reflection coefficient equation is derived for the direct characterization of the shear flexibility factor. In the developed equation, a novel fluid indicator, expressed as the ratio of Russell’s fluid indicator to the square of the shear flexibility factor, is defined for the simultaneous prediction of fluid types and permeability. With the validated response of the novel fluid indicator to geofluid types, we achieve simultaneous predictions of fluid types and reservoir permeability characteristics from prestack seismic data, using a boundary-constrained Bayesian inversion strategy. The model tests and the application of field data from a clastic reservoir confirm the effectiveness and applicability of the method.

An Improved Inversion Method of Reservoir Parameters Is Based on the Exact Reflection Coefficient Equation

An Improved Inversion Method of Reservoir Parameters Is Based on the Exact Reflection Coefficient Equation

PP-Wave Reflection Coefficient Equation for HTI Media Incorporating Squirt Flow Effect and Frequency-Dependent Azimuthal AVO Inversion for Anisotropic Fluid Indicator

PP-Wave Reflection Coefficient Equation for HTI Media Incorporating Squirt Flow Effect and Frequency-Dependent Azimuthal AVO Inversion for Anisotropic Fluid Indicator

Accurate P-wave reflection and transmission coefficients for non-welded interface incorporating elasto-plastic deformation

P-wave reflection and transmission coefficients for non-welded interface play crucial roles in broad practical engineering productions, involving fracture properties prediction and seismic inversion. However, the existing reflection coefficient equations for non-welded interface in elasto-plastic media are seldom studied, although the elasto-plastic deformation is frequently encountered in the Earth’s subsurface due to artificial and tectonic activities. In this study, we proposed the accurate reflection and transmission coefficients equation for a non-welded interface embedded in an elasto-plastic deformed medium based on the elasto-plastic acoustoelastic and linear-slip theory. In detail, this paper uses elasto-plastic acoustoelastic theory to derive the reflection and transmission coefficients equation. The reflection and transmission coefficients matrix are solved using the linear-slip theory as the boundary condition. Moreover, we use the hardening parameter and plastic deformation to represent the plastic properties of the rock, which is a function of stress and plastic deformation. Through Numerical analysis, the deformation caused by static stress has significantly changed the amplitude and the slope of the reflection and transmission coefficients amplitude. As the stress increases, the rock’s velocity becomes higher, and all reflection and transmission coefficients (i.e., RPP, RPS, TPP, TPS) abruptly change at the critical angle. Furthermore, with the increase in plastic deformation, the critical angle of the incident P-wave and the hardening parameter becomes larger than the unstressed state. The non-welded interface exhibits a low-pass frequency filter for reflected SV-waves and a high-pass frequency filter for reflected P-waves and transmitted P and SV waves. In addition, we can observe that static vertical stress can weaken the anomalous reflections caused by non-welded formations, but the effect is insignificant. On the other hand, the effect of fracture normal compliance to reflection and transmission is detailly investigated. When N<2.5*10-10(MPa-1), The non-welded interface is close to the welded interface, while N>2.5*10-5(MPa-1), the non-welded interface is close to the solid-air interface.

Generalized multi-cavity laser self-mixing interferometry based on scattering theory.

We present a generalized mathematical model and algorithm for the multi-cavity self-mixing phenomenon based on scattering theory. Scattering theory, which is extensively used for travelling wave is exploited to demonstrate that the self-mixing interference from multiple external cavities can be modelled in terms of individual cavity parameters recursively. The detailed investigation shows that the equivalent reflection coefficient of coupled multiple cavities is a function of both attenuation coefficient and the phase constant, hence propagation constant. The added benefit with recursively model is that it is computationally very efficient to model large number of parameters. Finally, with the aid of simulation and mathematical modelling, we demonstrate how the individual cavity parameters such as cavity length, attenuation coefficient, and refractive index of individual cavities can be tuned to get a self-mixing signal with optimal visibility. The proposed model intends to leverage system description for biomedical applications when probing multiple diffusive media with distinct characteristics, but could be equally extended to any setup in general.

Monitoring the change in horizontal stress with multi-wave time-lapse seismic response based on nonlinear elasticity theory

Monitoring the change in horizontal stress from the geophysical data is a tough challenge, and it has a crucial impact on broad practical scenarios which involve reservoir exploration and development, carbon dioxide (CO2) injection and storage, shallow surface prospecting and deep-earth structure description. The change in in-situ stress induced by hydrocarbon production and localized tectonic movements causes the changes in rock mechanic properties (e.g. wave velocities, density and anisotropy) and further causes the changes in seismic amplitudes, phases and travel times. In this study, the nonlinear elasticity theory that regards the rock skeleton (solid phase) and pore fluid as an effective whole is used to characterize the effect of horizontal principal stress on rock overall elastic properties and the stress-dependent anisotropy parameters are therefore formulated. Then the approximate P-wave, SV-wave and SH-wave angle-dependent reflection coefficient equations for the horizontal-stress-induced anisotropic media are proposed. It is shown that, on the different reflectors, the stress-induced relative changes in reflectivities (i.e., relative difference) of elastic parameters (i.e., P- and S-wave velocities and density) are much less than the changes in contrasts of anisotropy parameters. Therefore, the effects of stress change on the reflectivities of three elastic parameters are reasonably neglected to further propose an AVO inversion approach incorporating P-, SH- and SV-wave information to estimate the change in horizontal principal stress from the corresponding time-lapse seismic data. Compared with the existing methods, our method eliminates the need for man-made rock-physical or fitting parameters, providing more stable predictive power. 1D test illustrates that the estimated result from time-lapse P-wave reflection data shows the most reasonable agreement with the real model, while the estimated result from SH-wave reflection data shows the largest bias. 2D test illustrates the feasibility of the proposed inversion method for estimating the change in horizontal stress from P-wave time-lapse seismic data.

Investigation of the electromagnetic properties of a transverse insert based on a planar layer of a chiral metamaterial in a rectangular waveguide

The paper considers the solution of the problem of diffraction of the fundamental wave of a rectangular waveguide H10 on a planar transverse insert based on a chiral metamaterial created on the thin-wire conducting helices. To describe the chiral layer, a particular mathematical model is constructed that takes into account the properties of heterogeneity and dispersion of the permittivity and the chirality parameter of the artificial media. The well-known in physics model of Maxwell Garnett was used to take into account the heterogeneity property. To take into account the permittivity dispersion the Drude–Lorentz formula was applied and for the chirality parameter was used the Condon formula. The problem of diffraction of the rectangular waveguide main wave on a planar layer of a chiral metamaterial was solved by the partial regions method and was reduced to a system of linear algebraic equations for unknown reflection and transmission coefficients. It is shown that in the presence of a transverse chiral layer in the waveguide structure, a wave of the H01 type cross-polarized with respect to the main one arises. An analysis of the frequency dependences of the moduli of the reflection and transmission coefficients of the fundamental H10 and cross-polarized H01 showed that in some narrow frequency intervals in the single-mode gap, situations arise when the fundamental wave type is replaced from H10 to H01 near resonant frequencies. The transmission line under consideration can find application in the creation of frequency selective filters and polarization converters in the microwave range.

Investigation of the microwave chiral metamaterial based on a uniform set of C-shaped conductive inclusions

The paper considers an artificial chiral metamaterial created on a homogeneous container basefrom foamed dielectric, in which flat conducting S-shaped microelements are evenly placed and arbitrarily oriented. To describe the metamaterial, a particular mathematical model was constructed that takes into account chirality, dispersion, and heterogeneity of the structure. The Maxwell Garnett model was used to account for heterogeneity. To take into account the dispersion of the chirality parameter, the Condon model known from the theory of optically active media was used. The partial domain method was used to solve the problem of the incidence of a plane electromagnetic wave of linear polarization on a planar layer created on the base of the investigated chiral metamaterial. The solution of the problem was reduced to an inhomogeneous system of linear algebraic equations for unknown reflection and transmission coefficients, taking into account the cross-polarization of the electromagnetic field. An analysis of the numerical results showed that the structure has pronounced frequency selective properties, in particular, as in the case of chiral metamaterial based on three-dimensional conductive elements, discrete frequencies were determined at which the structure is transparent to microwave radiation. Chiral metamaterial based on C-shaped microelements can be used to create narrow-band frequency-selective microwave energy concentrators of planar type.

Direct Exact Nonlinear Broadband Seismic Amplitude Variations With Offset Inversion for Young’s Modulus

Young’s modulus and the Poisson ratio are critical for shale reservoir identification, and oil and natural gas detection since they are elastic parameters that can reflect the fracturing properties of subsurface rocks. The broadband inversion approach can make full use of the seismic ultralow-frequency information by combining it with the inversion results in the complex frequency domain. This plays an important role in the stability and reliability of the inversion, improving its resolution. Nevertheless, at present, the amplitude variation with offset (AVO) inversions of these parameters is mostly in the form of linear approximations, and there are few inversion methods of exact nonlinear equations or broadband. Considering that the nonlinear equation of reflection coefficient compared with the linear approximation has higher precision and fewer assumption conditions, and the low-frequency information of the complex frequency-domain inversion can improve the reliability of the inversion results, we derive an exact reflection coefficient equation for Young’s modulus, the Poisson ratio coefficient, and the density based on the exact Zoeppritz equation and apply it to the broadband complex domain nonlinear prestack inversion. Young’s modulus and the Poisson ratio coefficients estimated by the new inversion method provide a theoretical basis for evaluating the reservoir brittleness and compressibility, as well as a new approach for shale gas reservoir prediction and sweet spot identification. We tested the accuracy and rationality of this method with both synthetic and field data examples.

Frequency-Dependent Nonlinear AVO Inversion for Q-Factors in Viscoelastic Media

Viscoelastic theory-based frequency-dependent amplitude variation with offset (AVO) inversions are a useful tool for identifying fluids based on their dispersion properties. When used to identify reservoir oil and gas qualities, the quality factors ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q$ </tex-math></inline-formula> -factors) derived from the inversion have produced good results. But currently, the majority of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q$ </tex-math></inline-formula> -factor inversions are linear inversions based on linear approximations, whereas nonlinear inversions based on exact equations with higher precision and fewer assumptions are only occasionally carried out. Meanwhile, in broadband seismic inversion, the low-frequency model estimated by complex frequency inversion can fully utilize seismic data’s low-frequency information and provide improved robustness and rationality for inversion. Therefore, we derive a frequency-dependent reflection coefficient equation that varies with angle by replacing the nearly constant <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q$ </tex-math></inline-formula> model suggested by Aki and Richards into the exact Zoeppritz equation and simplifying. Additionally, using the Bayesian framework as a foundation, we created a novel two-stage broadband frequency-dependent nonlinear inversion approach for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q$ </tex-math></inline-formula> -factors that can estimate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q$ </tex-math></inline-formula> -factors, produce accurate fluid identification results, and serve as a solid foundation for oil and gas exploration and reservoir identification. Utilizing examples from both synthetic and real-world data, we verify the applicability of oil and gas indicators.

Seismic inversion and fracture prediction in tilted transversely isotropic media

Abstract The conventional Amplitude variation with offset (AVO) inversion has been mainly developed for the isotropic media and therefore it is generally inapplicable to the anisotropic fractured formations. A set of tilted fractures in isotropic medium can be regarded as a transversely isotropic (TTI) medium. The reflection coefficient equation in TTI media contains many parameters such as anisotropic parameters, velocities, azimuth and dip angles. The selection of objective functions can significantly affect its performance in searching for the optimal solution. The seismic inversion of the TTI medium remains challenging because of its many parameters, and is very complicated after conversion into solving function problems. The genetic algorithm (GA) provides a universal framework to solve the optimization problem in complex systems, and it is independent of the types and disciplines of problems. In this study, the GA-based seismic prestack inversion of the TTI medium is implemented by constructing the objective functions of anisotropic parameters based on the TTI medium reflection coefficient approximation. The inversion performance is shown to be satisfactory on the one-dimensional well logging model, modified Hess model and the field datum. Furthermore, fracture filling materials are well predicted according to the theories of the equivalent medium for fractured materials and fracture weakness. The inversion results demonstrate the proposed method to be feasible and robust. The findings of this research are expected to provide theoretical guidance for the inversion of elastic parameters of anisotropic media and fracture type identification in hydrocarbon reservoirs.

A VTI anisotropic media inversion method based on the exact reflection coefficient equation

Anisotropy is widespread in the Earth’s crust, and VTI (vertical axis symmetry transverse isotropy) anisotropy is common due to stratigraphic pressure. Disregarding anisotropy leads to inaccurate inversion results in VTI media. To estimate accurate elastic parameters, the exact reflection coefficient equation of VTI media should be used. This equation is nonlinear and more accurate than the commonly used linear reflection coefficient equation. Although the inversion based on the VTI anisotropy exact reflection coefficient equation is a complex nonlinear inversion problem, it is still computable. Therefore, for VTI media, we derive the objective function combining Bayesian theory and use the iterative reweighted least squares method for a fast and stable solution. Adding Bayesian theory can improve the robustness of the algorithm. The accuracy and noise immunity of the method is tested with synthetic data. Finally, the method is applied on field data and obtains accurate elastic parameters. The results can provide an understanding of subsurface formations and serve as the base data for calculating fluid and fracture distribution.

Estimation of in situ stresses from PP-wave azimuthal seismic data in fracture-induced anisotropic media

Horizontally transverse isotropy (HTI) induced by vertical or subvertical aligned fractures is common for unconventional fractured porous shale oil or gas reservoirs. Compared with the unfractured rocks, the seismic response characteristics of PP-wave azimuthal amplitudes are usually disturbed by the fractures and the in situ stresses. Knowledge of fracture properties, as well as in situ stresses, is required to optimize horizontal well planning and hydraulic fracturing during production, and the seismic inversion for in situ stresses from the PP-wave azimuthal amplitude data in fracture-induced anisotropic media is an essential step. Using the linear-slip theory and the effective stress law, we derive the fluid-saturated effective elastic stiffness tensors parameterized by background elastic moduli, the effective stress coefficient of isotropic host rocks, fluid modulus, porosity, and fracture parameters based on the anisotropic Gassmann’s fluid substitution equation. Combining the perturbations in saturated stiffness tensors and scattering theory, we formulate the reflection coefficient equation of PP-wave data as a function of background porosity-related stress parameter and two (i.e., normal and shear) fracture weakness parameters. Following Bayes’ rule, we estimate the porosity-related stress parameter and fracture weaknesses using the inversion method of azimuthal Fourier coefficients. Finally, we compute the effective horizontal and vertical in situ stresses using the estimated elastic moduli, porosity-related stress parameter, and fracture weaknesses. Synthetic and real data sets demonstrate that our proposed inversion approach based on the derived reflection equation provides us another way to obtain reasonable estimates of in situ stresses in a complex-fractured porous shale reservoir.

Fluid discrimination incorporating amplitude variation with angle inversion and squirt flow of the fluid

Pre-stack seismic inversion is an important method for fluid identification and reservoir characterization in exploration geophysics. In this study, an effective fluid factor is initially established based on Biot poroelastic theory, and a pre-stack seismic inversion method based on Bayesian framework is used to implement the fluid identification. Compared with conventional elastic parameters, fluid factors are more sensitive to oil and gas. However, the coupling effect between rock porosity and fluid content is not considered in conventional fluid factors, which may lead to fuzzy fluid identification results. In addition, existing fluid factors do not adequately consider the physical mechanisms of fluid content, such as squirt flow between cracks and pores. Therefore, we propose a squirt fluid factor (SFF) that minimizes the fluid and pore mixing effects and takes into account the squirt flow. On this basis, a novel P-wave reflection coefficient equation is derived, and the squirt fluid factor is estimated by amplitude variation with offset (AVO) inversion method. The new reflection coefficient equation has sufficient accuracy and can be utilized to estimate the parameters. The effectiveness and superiority of the proposed method in fluid identification are verified by the synthetic and field examples.

P-wave reflectivity parameterization and nonlinear inversion in terms of Young’s modulus and Poisson ratio

Young’s modulus and Poisson ratio are essential parameters for rock brittleness evaluation. At present, prestack inversions of these parameters mostly adopt linear approximation, whereas the nonlinear inversion methods that are based on the exact Zoeppritz equation are seldomly conducted. Nevertheless, in the case of large incident angles and high-impedance contrast, the nonlinear equation has higher accuracy than the linear approximation. It is more in line with the nature of geophysical inversion problems, and it describes the variation of formations’ elastic parameters better than the linear approximation does. Therefore, based on the exact solution of the Zoeppritz equation, we have parameterized and derived a new nonlinear reflection coefficient equation using nonlinear Young’s modulus, Poisson ratio, and density (YPD) as variables. Subsequently, by combining the inverse operator algorithm with the YPD equation, we develop a new nonlinear amplitude-variation-with-offset inversion scheme. This approach can robustly extract Young’s modulus and Poisson ratio from seismic data and identify shale-gas sweet spot locations within the reservoir, as well as predict reservoir fluid properties. We have tested the accuracy and applicability of this approach with synthetic and real data examples.

Amplitude-variation-with-offset inversion using P- to S-wave velocity ratio and P-wave velocity

The estimation of P- to S-wave velocity ratio (PSR) in the subsurface has many applications in gas-bearing reservoir prospecting, lithology discrimination, and anomalous pore-pressure prediction. Conventionally, it is estimated with the P- and S-wave velocities/moduli/impedances that are directly obtained from prestack seismic data using the existing reflection coefficient equation (e.g., the Aki-Richards approximation). However, this indirect inversion method creates cumulative errors for the estimated PSR results. To eliminate cumulative errors, we first develop a novel generalized elastic impedance that has more explicit physical meaning compared to conventional elastic impedance. Then, we derive a linear P-wave reflection coefficient equation in terms of the PSR, P-wave velocity, and density under the assumption of weak contrast. Furthermore, a robust amplitude-variation-with-offset inversion method is constructed with the proposed reflection coefficient equation in the Bayesian framework. Cauchy and Gaussian probability distributions are used as the prior probability distribution of the model parameter and likelihood function, respectively, to predict the maximum posterior probability solution for PSR, P velocity, and density. Synthetic and field examples illustrate that the proposed direct inversion method performs with higher accuracy compared with the indirect method that inverts for the P- and S-wave velocities and also illustrate the feasibility of our method for inverting the three parameters even with strong noise. Our inverted results conform to the drilling results, which validate the robustness of the proposed direct inversion method.