# Petrophysics Parameter Inversion and Its Application Based on the Transient Electromagnetic Method
## 2 Theory
### Inversion Algorithm  
When performing a one-dimensional inversion of the loop source transient electromagnetic method, not only the fitting error between the observed data and the model data needs to be minimized, but also the roughness of the inversion model needs to be minimized. The objective function of the inversion is [33]:  
\[\Phi(\mathbf{m})=\Phi_{d}(\mathbf{m})+\beta\Phi_{m}(\mathbf{m}) \tag{10}\]  
Here, \(\Phi_{d}\) represents the data misfit term, which quantifies the discrepancy between the predicted and observed data; \(\Phi_{m}\) denotes the model regularization term, which imposes constraints on the model complexity; and \(\beta\) is the regularization parameter that balances the trade-off between data fitting and model smoothness. The objective of inversion is to reconstruct the spatial distribution of subsurface physical parameters (e.g., resistivity, saturation, and porosity) and iteratively update the model during the inversion process so that its predicted response increasingly aligns with the observed data. The data misfit or data fitting error quantifies the discrepancy between the surface response generated by the model and the actual measured data. It is defined as:  
\[\Phi_{d}(\mathbf{m})=\frac{1}{2}\|\mathbf{W}_{d}(F[\mathbf{m}]-\mathbf{d}_{ obs})\|_{2}^{2} \tag{11}\]  
where \(F[\mathbf{m}]\) denotes the predicted data obtained through forward modeling; \(\mathbf{d}_{obs}\) represents the observed data; \(\mathbf{W}_{d}\) is a diagonal weighting matrix, where \(\mathbf{W}_{dii}=1/\epsilon_{i}\), and \(\epsilon_{i}\) is the standard deviation of the observation error associated with the _ith_ data point. Since the number of field measurements is significantly smaller than the number of unknowns in the inversion model, directly minimizing the data misfit may lead to non-unique solutions. To obtain a more geologically reasonable model and reduce the ambiguity of the inversion results, this study employs the Gauss-Newton iterative optimization algorithm. The objective function, which incorporates an L\({}_{2}\) regularization term, is expressed as follows:  
\[\Phi(\mathbf{m})=\frac{1}{2}\|\mathbf{W}_{d}(F[\mathbf{m}]-\mathbf{d}_{obs})\| _{2}^{2}+\frac{1}{2}\beta\Big{\|}\mathbf{W}_{m}\Big{(}\mathbf{m}-\mathbf{m}_{ ref}\Big{)}\Big{\|}_{2}^{2} \tag{12}\]  
Approximately solve a linear system via the conjugate gradient method as follows:  
\[\Big{(}\mathbf{J}_{k}^{\top}\mathbf{W}_{\mathbf{d}}^{\top}\mathbf{W}_{ \mathbf{d}}\mathbf{J}_{k}+\beta\mathbf{W}_{\mathbf{m}}^{\top}\mathbf{W}_{ \mathbf{m}}\Big{)}\Delta\mathbf{m}=-\mathbf{J}_{k}^{\top}\mathbf{W}_{\mathbf{ d}}^{\top}\mathbf{W}_{\mathbf{d}}\mathbf{r}_{k}-\beta\mathbf{W}_{\mathbf{m}}^{ \top}\mathbf{W}_{\mathbf{m}}(\mathbf{m}_{k}-\mathbf{m}_{\text{ref}}) \tag{13}\]  
where \(\mathbf{J}_{k}\) is the Jacobian matrix, and \(\mathbf{r}_{k}\) is the residual vector. The subsurface resistivity distribution was obtained using the forward modeling and inversion techniques described above. To facilitate geological interpretation, the inversion results were further transformed from resistivity into porosity and water saturation by applying the constitutive relationships between resistivity and the rock's physical properties. This approach enables the inversion of formation porosity and water saturation directly from the transient electromagnetic data. To enable the inversion of rock physical parameters, it is first necessary to establish a model that relates geophysical parameters to petrophysical properties. In this study, the relationship between resistivity, porosity, and water saturation is described using [PERSON]'s equation as follows:  
\[\rho=\rho_{w}\phi^{-m}S_{w}^{-n} \tag{14}\]  
where \(\rho\) is the resistivity of the rock formation, \(\rho_{w}\) is the resistivity of the pore water, \(\phi\) is the porosity, and \(S_{w}\) is the water saturation. The parameter \(m\) is the cementation exponent, which typically ranges from 1.3 to 3.0, and \(n\) is the saturation exponent, which is commonly taken as 2 [34].