## 3 Methods and Materials
### Joint Inversion as a Tool for Velocity Models Validation  
The SJI procedure applied to the validation of the VID was carried on according to the scheme shown in Figure 4, which demonstrates the general operational principle of the discussed algorithm. In this algorithm, two independent geophysical datasets, i.e., the residual values of the gravity field and the times of first breaks for 0-6 km offsets were synchronized to simultaneous inversion. Precisely, the mentioned above datasets were processed simultaneously with a single, coherent algorithm. The application of the SJI procedure was mainly aimed at generating a credible distribution model of petrophysical parameters of rock formations based on available data not analyzed previously in a simultaneous mode. In this SJI solution, the link function of two independent datasets was the spatial distribution of the Cross Gradient (CG) for two different model vectors (Equation (1)) [1; 2; 3; 9]. \[\text{CG}(m_{1},m_{2})=\
abla m_{1}\times\
abla m_{2} \tag{1}\]  
Figure 4: Scheme of application of SJI procedure for PVM model validation. Scheme has been based on published instructions [10; 30; 31]. The testing loop is presented as tool for VID models improving and their verification. A simplified explanation of the simultaneous inversion methodology is demonstrated in the form of Equation (2) of a Join Objective Function (JOF). This formula represents how the minimization in the optimization algorithm is being released below, where the \(\
u\Phi\)_coupling_ (\(m\)) parameter is responsible for the integration element [3; 30; 31]. This parameter defines mathematically the relationship between the parameters of the model which can be related to CG:  
\[\Phi_{joint}\left(m\right)=\Phi_{d}\left(m\right)+\lambda\Phi\text{{reg}} \left(m\right)+\
u\Phi\text{{coupling}}\left(m\right) \tag{2}\]  
where:  
\(\Phi_{joint}\left(m\right)\)--join objective function  
\(\Phi_{data}\left(m\right)\)--fitting coefficient of calculated and measured data  
\(\Phi_{reg}\left(m\right)\)--regularization condition  
\(\left(m\right)\)--model vector  
\(\lambda\)--lagrange multiplier used in regularization process  
\(\
u\Phi\)_coupling_ (\(m\))--parameter, which defines mathematically the relationship between parameters of the models used in calculations. Application of the SJI with the CG resulted in the generation of two models, which determined the spatial variability of velocity and density. An important problem in the application of the SJI procedure with the previously described datasets for greater depths of rock formations was the incommensurability of sensitivities of gravimetric and seismic refraction methodologies. The depth range of gravity data is much greater but after exceeding the depth range of sensitivity for the refraction method, the joint inversion becomes a cooperative inversion. In each calculation loop, the average fitting error of calculated and measured values for both parameters was estimated together with the average value of the cross-gradient for both models. Thus, the data quality of obtained results could be evaluated and the range of input model refining could be determined. The aim of the inversion procedure was to minimize the fitting error down to the lowest possible value. In the case of geophysical data, the goal was to reconstruct the measured physical field from a series of simple models. The minimization of cross-gradient values meant obtaining structurally convergent geophysical models [3; 9; 30]. Therefore, the uncertainty was reduced, caused by the reconstruction of rock formations using the developed geophysical models. The introduction of data recorded independently on the leading methodology enabled the researchers to verify one method with the results of the other [3; 9; 10; 15]. According to the assumed sensitivity reduction of gravity method with depth, the applied weighting algorithm allowed the researchers to filter out the effects resulting from the projection of anomalies distribution at shallower depths.