## 3 Methodology
### Rock Physics Workflow
#### 3.1.4 Petrophysical Seismic Inversion  
The linearized Bayesian AVO inversion, as described in [37], formulates the challenge within a probabilistic and statistical framework. In this approach, prior knowledge sourced from independent references, such as well-log data, is integrated with AVO equations or the likelihood function. Reservoir coupling information, serving as prior data, plays a pivotal role in constraining the AVO inverse problem while operating under the assumption of a Gaussian probability density function. \"Equation (1) represents the forward linear model, where '_dobs_' denotes the observed seismic data, \(G\) stands for the kernel matrix, \(W\) is a block diagonal matrix containing angle-dependent wavelets, \(C\) is a sparse matrix containing AVO coefficients, '_m_' comprises the model parameters, and \(D\) serves as a derivative operator. In this context, '_m_' takes the form of a column vector encompassing compressional wave velocity (\(V_{p}\)), shear wave velocity (\(V_{s}\)), and density (\(\rho\)). The likelihood function employed is the linearized version of the [PERSON] equation [38]. \[dobs=G_{m}+Noise\ (ab) \tag{1}\]  
\[G=W\ C\ D\ (ab)\]  
The utilization of linearized forward equations enhances computational efficiency within the Bayesian model. The three-term AVO equation characterizes the relationship between reflection amplitude and offset (angle). Equation (2) represents a continuous-time formulation of [PERSON] and [PERSON] equation, as detailed in [39]. \[R_{pp}(t,\theta)=C_{1}(\theta,t)\frac{\partial}{\partial t}InV_{p}+C_{2}( \theta,t)\frac{\partial}{\partial t}InV_{s}+C_{3}(\theta,t)\frac{\partial}{ \partial t}In\rho \tag{2}\]  
In this context, \(R_{pp}\) refers to the offset-dependent compressional reflectivity and \(C_{1}\) to \(C_{3}\) represent the AVO coefficients, with \(\theta\) denoting the offset angle. [PERSON] theorem provides a mathematical framework for incorporating prior information related to the elastic parameters, including density, compressional, and shear wave properties, which are to be derived from the seismic data using the forward model (AVO equations). This approach allows for the generation of probabilistic estimates, facilitating the assessment of the uncertainties associated with these parameters. The theorem can be expressed as follows:  
\[Q(m|d)=\frac{P(d|m)P(m)}{P(d)} \tag{3}\]  
Here, \(Q(m|d)\) represents the posterior distribution, which provides a probabilistic solution to the inverse problem; \(P(d|m)\) denotes the AVO equation; \(P(d)\) is the data distribution, which is typically disregarded when focusing on the posterior shape; and \(P(m)\) stands for the prior model employed to constrain the solution. The petrophysical seismic inversion is executed in steps. The initial step entails using a Bayesian algorithm to invert angle-varying seismic amplitudes [37]. The closer counterpart in the deterministic domain is the simultaneous inversion that employs linear intra-parameter relations as constraints to couple elastic properties [40]. The linear constraint is motivated by the linear trend between the logarithm of acoustic impedance and shear impedance [34] and acoustic impedance with density [41] for brine-saturated sediments. The Bayesian framework is favored because it offers better constraints that curb the non-uniqueness problem in the seismic inversion processes. Because of the bandwidth restrictions of the seismic data, the generated elastic earth model is not unique and is exacerbated further in the case history by insufficient well coverage to calculate the earth model. In addition, the Bayesian formulation mitigates the effect of noisy traces. The inverse process is automatically constrained by the a priori model, such that when the signal-to-noise ratio (SNR) is low, the priors dominate the inversion model. However, when the SNR is high, the seismic signals dominate.