## 3 Uranium Quantitative Inversion Method of Combined Logging  
As indicated by the integral expression (1), the key to solving for the uranium and radium content \(q_{k}(z)\) at any point within the uranium-bearing layer lies in the prior determination of the calibration parameters, including the conversion coefficient \(A_{k}\) and background response \(B_{k}\). This process involves selecting a saturated ore layer (standardized model well). Since the uranium and radium contents \(q_{k}(z)\) remain constant throughout the saturated ore layer, \(q_{k}(z)\) at any depth coordinate \(z\) can be regarded as a constant. This constant is represented by the value at the central point of the uranium-bearing layer (denoted as depth coordinate \(Z_{0}\)): \(q_{k}(z)=q_{k}(Z_{0})\). Therefore, the logging count rate \(N_{k}(Z)\) at any measurement point (depth coordinate \(Z\)) is given by  
\[N_{k}(Z)=B_{k}+A_{k}\int_{-H_{1}\rightarrow-\infty}^{H_{2}\rightarrow\infty} q_{k}(z)\cdot\frac{\alpha_{k}}{2}e^{-\alpha_{k}|z-z|}\cdot\mathrm{d}z=B_{k}+A_{k} \cdot q_{k}(Z_{0})=N_{k}(Z_{0})\quad(k=1,2). \tag{3}\]  
Consequently, the logging count rates at any measurement point within the central portion of the saturated ore layer are also uniformly equal and can be represented by the central point value, expressed as \(N_{k}(Z)=N_{k}(Z_{0})\). Based on the equation above, the calculation formula for the conversion coefficient \(A_{k}\) is derived as follows:  
\[A_{k}=\frac{N_{k}(Z_{0})-B_{k}}{q_{k}(Z_{0})}\quad(k=1,2). \tag{4}\]  
Clearly, the background response \(B_{k}\) must be determined using the logging count rates obtained from a saturated ore layer that contains no radioactive elements such as uranium  
Figure 1: Self-developed combined logging tool.  
or radium--referred to as a zero-value saturated ore layer. By substituting \(q_{k}(Z_{0})=0\) into Equation (3), the following expression is obtained:  
\[B_{k}=N_{k}(Z_{0})\quad(k=1,2). \tag{5}\]  
To minimize statistical errors, the calculation formula for the uranium-radium-radon balance coefficient can be derived using the calibration parameters--namely the conversion coefficient \(A_{k}\) and background response \(B_{k}\)--in conjunction with the measured epithermal neutron count rates \(N_{1}(Z_{j})\) and natural \(\gamma\)-ray total count rates \(N_{2}(Z_{j})\) at each measurement point \(Z_{j}\) within the target uranium-bearing layer [1]:  
\[K_{P}=\frac{\sum_{j}q_{2}(Z_{j})}{\sum_{j}q_{1}(Z_{j})}=\frac{\sum_{j}\frac{N_ {2}(Z_{j})-B_{2}}{A_{2}}}{\sum_{j}\frac{N_{1}(Z_{j})-B_{1}}{A_{1}}}=\frac{A_{1 }\sum_{j}\left(N_{2}(Z_{j})-B_{2}\right)}{A_{2}\sum_{j}\left(N_{1}(Z_{j})-B_{ 1}\right)}. \tag{6}\]  
For the natural \(\gamma\)-ray total count rates, inversion calculations of radium content \(q_{2}(Z_{j})\) at each measurement point are performed using Equation (1) or Equation (3), yielding quantitative estimates of the radium content. Then, applying Equation (6), the corrected true uranium content \(q_{1}(Z_{j})\) at each measurement point is determined using the following formula:  
\[q_{1}(Z_{j})=\frac{q_{2}(Z_{j})}{K_{p}}. \tag{7}\]  
Figure 2a illustrates the procedure for determining the calibration parameters--specifically, the conversion coefficient \(A_{k}\) and background response \(B_{k}\)--of the combination logging instrument. This is accomplished through calibration experiments conducted in both a zero-value saturated ore layer model well and a saturated ore layer model well. Figure 2b delineates the core workflow for uranium/radium quantitative interpretation, which comprises three sequential steps: the radium content inversion, uranium-radium-radon balance coefficient calculation, and true uranium content correction.  
Figure 2: (**a**) Steps of getting the scale parameters. (**b**) Steps of uranium and radium quantitative interpretation.