## 2 Uranium Quantitative Forward Equation of the Combined Logging
By conceptually dividing the uranium-bearing layer along the borehole into infinitely thin layers, where \(q_{k}(z)\) denotes uranium or radium content and \(N_{k}(z)\) represents the corresponding count rates of natural \(\gamma\)-ray total logging and prompt neutron time spectrum logging, a functional relationship is established:  
\[N_{k}(Z)=B_{k}+A_{k}\int_{-H_{1}\rightarrow-\infty}^{H_{2}\rightarrow\infty}q_ {k}(z)\cdot\frac{\alpha_{k}}{2}e^{-\alpha_{k}|z-z|}\cdot\mathrm{d}z\quad(k=1,2) \tag{1}\]  
This expression is referred to as the integral form of the forward equation for uranium quantification. Here, \(q_{1}(z)\) and \(N_{1}(z)\) denote the uranium content and prompt neutron time spectrum logging count rates within infinitesimally thin uranium-bearing layers, respectively, while \(q_{2}(z)\) and \(N_{2}(z)\) represent the radium content and natural \(\gamma\)-ray total logging count rates within the same layers. Additionally, the thickness of a thin uranium-bearing layer at borehole depth coordinate \(z\) is represented by the differential element dz. The differential expression for logging count rates generated by this thin uranium-bearing layer at depth coordinate \(Z\) is  
\[dN_{k}(Z)=A_{k}\cdot q_{k}(z)\cdot\frac{\alpha_{k}}{2}e^{-\alpha_{k}|z-z|} \cdot\mathrm{d}z\quad(k=1,2). \tag{2}\]  
In the formula, \(Z\) and \(z\) denote the depth coordinates of any measurement point in the borehole and any point within the uranium-bearing layer (i.e., a thin uranium-bearing layer), respectively. Here, \(|Z-z|\) represents the distance from any measurement point \(Z\) to a uranium-bearing thin layer at coordinate \(z\). \(H_{1}\) and \(H_{2}\) denote the thicknesses of the upper and lower segments of the uranium-bearing layer when divided at point \(z\) along the borehole direction, where \(H=H_{1}+H_{2}\) represents the total thickness of the uranium-bearing layer. \(q_{k}(z)\) represents the uranium or radium content in a thin uranium-bearing layer at depth coordinate \(z\), where the subscript \(k\) (=1, 2) distinguishes uranium (\(k=1\)) from radium (\(k=2\)). \(N_{k}(Z)\) represents the logging count rates at measurement points (depth coordinate \(Z\)) within the uranium-bearing layer, where the subscript \(k\) (=1, 2) distinguishes between \(k=1\) for the prompt neutron time spectrum logging count rates (uranium quantification), and \(k=2\) for the natural \(\gamma\)-ray total logging count rates (radium quantification). \(\varphi_{k}(Z-z)=\frac{\alpha_{k}}{2}e^{-\alpha_{k}|Z-z|}\) is designated as the geologic pulsed response function, characterizing the variation pattern of logging count rates responsive to any thin uranium-bearing layer. Specifically, the value of \(\varphi_{k}(Z-z)\) decreases with the increasing distance \(|Z-z|\) between measurement points and the thin uranium-bearing layer. This decay behavior is approximated by a negative exponential function, where \(\alpha_{k}\) is termed the characteristic parameter representing the decay rate. The magnitude of \(\alpha_{k}\) depends on the radiation type and interaction mechanisms between \(\gamma\)-rays and the uranium-bearing layer. \(A_{k}\) is termed the conversion coefficient, representing the logging count rate solely produced by the unit uranium content (or unit radium content) at the center of a saturated ore layer. Here, a saturated ore layer refers to an \"infinitely thick\" saturated ore layer, practically defined as having both upper segment thickness \(H_{1}\) and lower segment thickness \(H_{2}\) exceeding 0.6 m (measured from the central position of the saturated layer). Note that the radium content is conventionally expressed as equilibrium uranium content. In saturated ore layers composed of equilibrium uranium-series elements, uranium content \(q_{1}(z)\) and radium content \(q_{2}(z)\) are numerically equal, i.e., \(q_{1}(z)=q_{2}(z)\). However, the corresponding logging count rates differ, indicating that \(A_{k}\) remains a constant parameter determined by the ore composition and uranium-bearing layer structure.