## 2 Uranium Quantitative Forward Equation of the Combined Logging  
Typically, \(\gamma\)-rays generate negligible count rates in neutron detectors such as \({}^{3}\)He proportional counters, whereas neutrons can induce inelastic scattering and capture reactions in surrounding formations and borehole materials, resulting in the emission of secondary \(\gamma\)-rays. However, most prompt neutron logging instruments are equipped with natural \(\gamma\)-ray total logging (or spectral logging) capabilities. When the distance between \(\gamma\)-ray detectors and neutron detectors is less than 1.5 m, these inelastic and capture \(\gamma\)-rays can be detected by the \(\gamma\)-ray detectors, thereby contributing background count rates to the \(\gamma\)-ray logging signal. As a result, the natural \(\gamma\)-ray logging function integrated into prompt neutron logging instruments is not directly used for uranium quantification but serves primarily to identify whether the instrument has entered a uranium-bearing layer. In contrast, prompt neutron time spectrum logging typically exhibits count rates one to two orders of magnitude lower than those of natural \(\gamma\)-ray total logging. This low logging efficiency, combined with the high operational cost and limited lifespan of neutron tubes, has hindered the broader adoption of this method. To overcome these limitations, this study proposes a uranium quantification method that leverages natural \(\gamma\)-ray total logging as the primary measurement and employs prompt neutron time spectrum logging to correct the results. This approach not only enhances the logging speed but also achieves rapid low-cost uranium quantification. Common prompt neutron time spectrum logging instruments typically position the \(\gamma\)-ray detector in the upper section and the neutron detector in the lower section of the tool. Logging data are acquired during the instrument's upward movement [16, 22, 23]. Using the center of the \(\gamma\)-ray detector as the reference point and assigning its depth coordinate accordingly, the logging data from both detectors, when each reaches this reference position, are recorded as co-located depth point data. This method achieves the spatial integration of the measurement points. The self-developed combination logging instrument (structural schematic shown in Figure 1) consists of three main components:  
1. Neutron Source: D-T neutron tube and neutron generator;
2. Detection Units: two \({}^{3}\)He neutron detectors and one \(\gamma\)-ray detector;
3. Electronics Units: dual energy neutron time spectrum unit, \(\gamma\)-ray energy spectrum unit, and communication control unit. Regarding proximity to the neutron source, the near \({}^{3}\)He neutron detector, shielded with cadmium, detects epithermal neutrons, while the far \({}^{3}\)He neutron detector measures thermal neutrons within the borehole. The lanthanum bromide (LaBr\({}_{3}\)) \(\gamma\)-ray detector is positioned distally to minimize interference from neutron-induced \(\gamma\)-rays and is specifically used to quantify uranium-derived \(\gamma\)-rays for uranium quantification within the uranium-bearing layer. Since the count rates of the prompt neutron time spectrum logging exhibit proportionality solely to uranium content within the uranium-bearing layer [24], and considering that uranium quantification may neglect or subtract contributions from thorium-series and potassium \(\gamma\)-rays (including uranium-series \(\gamma\)-rays), the count rates of natural \(\gamma\)-ray total logging primarily correlate with radium content in the uranium-bearing layer [25].