Preamble

Research on Dynamic Three-Dimensional Terrain Correction Methods for Quantitative Inversion in Airborne Gamma-Ray Spectrometry

He-Xi Wu,¹ Wei-Cheng Li,²,† Rui Qiu,³,‡ Chao Xiong,¹ Yi-Ming Lyu,² Yi-Qiang Xing,² De-Hao Zhang,² Zong-Shuo Tao,² and Yang Wang²

¹School of Nuclear Science and Engineering, East China University of Technology, Nanchang 330013, China
²Institute of Energy, Hefei Comprehensive National Science Center (Anhui Energy Laboratory), Hefei 230031, China
³Department of Engineering Physics, Tsinghua University, Beijing 100084, China

Aerial surveying is a dynamic and continuous process, with varying ground elevation distributions across the measurement area that lead to issues such as overlapping measurement zones and inaccurate altitude corrections. Commonly employed terrain correction methods are based on the concept of finite elementization of ground surface radioactive sources, utilizing GPS coordinates, radar altitude, and ground elevation distribution data from aerial surveys, combined with sourceless efficiency calibration to construct a response matrix for inverting surface radionuclide concentrations. However, most sourceless efficiency calibration methods employ numerical calculations that treat the detector as a point detector, neglecting variations in intrinsic detection efficiency under different gamma-ray incident directions. Consequently, when altitude variations in the measurement area are substantial or the survey altitude is relatively low, such sourceless efficiency calibration methods tend to produce significant biases that affect terrain correction accuracy.

To address these limitations, this paper employs a novel sourceless efficiency calibration method based on Boolean operations of the ray deposition process, simplifying the traditional volumetric source measurement model to a surface source model to achieve rapid and accurate efficiency calibration. By discretizing the measurement process, the static measurement process is superimposed to approximate the dynamic measurement process, and the dynamic measurement response matrix is constructed and optimized based on this calibration method. Finally, the PSO-MLEM algorithm is used to solve the dynamic measurement response matrix, achieving dynamic terrain correction of aerial survey data. Analysis of the Baiyun'ebo test area reveals that after applying dynamic terrain correction, the inverted anomalies in uranium (eU), thorium (eTh), and potassium (K) concentrations are closer to ground measurements (within 5.72%–30.79%) and exhibit clearer anomaly boundaries compared to traditional height-based corrections. However, due to inherent statistical fluctuations and matrix inversion characteristics, higher measurement values tend to absorb lower ones, potentially enlarging anomalous regions. Nevertheless, the high-anomaly regions after inversion largely coincide with ground truth validation, demonstrating that the proposed method can effectively correct airborne gamma spectrometry data.

Keywords: Airborne gamma-ray spectrometry; Dynamic three-dimensional; Terrain correction

1. Introduction

Airborne gamma-ray spectrometry is a method for characterizing and quantitatively analyzing radionuclides by processing environmental gamma-ray measurement data \cite{1}. As a critical step in airborne gamma-ray spectrometry data processing, the inversion method for surface radionuclide content has a crucial impact on the accuracy of airborne measurement results \cite{2,3}.

In China's nuclear industry standards, height correction of energy window counts is typically performed by combining ground standard source calibration data with ray attenuation principles and elevation data acquired during flight \cite{4}. Conventional height correction relies solely on radar altitude parameters without considering the influence of actual ground undulation and non-uniform radionuclide distribution, resulting in eU content errors exceeding 30% in rough terrain \cite{5} and maximum eTh content errors of 21.5% across nine typical terrain types \cite{6}. Xiong Shengqing established a 2D terrain influence coefficient correction model by dividing the undulating surface into multiple oblique sources and accumulating their contributions \cite{7}, while Wan Jianhua obtained optimal results for the terrain influence coefficient method through body model segmentation after in-depth analysis \cite{8}, and Bai Yunfeng performed terrain correction using 3D spatial information subdivision of the aerial survey process via Google Earth \cite{9}. Subsequently, Brian Minty proposed using a response matrix for three-dimensional quantitative inversion, employing finite element concepts to correct interactions between survey areas and constructing an airborne gamma-ray spectrometry response matrix through radar altitude and terrain height, which significantly improved inversion results \cite{5}. This method was also validated by Md Moudud for inverting 137Cs activity in the environment \cite{10}. However, actual airborne gamma-ray surveys are continuous processes, and the energy spectrum at a recording point represents the synthesis of contributions from a continuous survey area segment during the measurement time. For this reason, Sun Kun and Liu Qiushi proposed a method combining aerial flight paths with sourceless efficiency calibration to construct a dynamic terrain-corrected response matrix, with validity verified through aerial surveying experiments in the LSS area of Gansu and the EGRY area of Inner Mongolia \cite{11–13}.

Through the research of many scholars, dynamic terrain correction theory has gradually matured, and the primary challenge of terrain correction methods now lies in the detection efficiency calculation of radioactive finite elements \cite{14}. Sourceless efficiency calibration methods can be mainly divided into three categories: numerical calculation methods, Monte Carlo simulation methods, and hybrid approaches \cite{15}. Wu Hexi applied numerically calculated sourceless efficiency calibration to airborne gamma-ray spectrometry and found good agreement with measured values after comparison \cite{16}. Xiong Chao derived a sourceless efficiency calibration method based on numerical integration through the axial symmetry of detector and object geometry and the principle of gamma field superposition, demonstrating accuracy in measurement experiments with NaI and HPGe detectors \cite{17}. Zhang Jian constructed a numerical model of ray deposition by studying the deposition process of rays in the detector sensitive volume combined with reaction probabilities, which was verified through point source experiments with a CeBr3 detector \cite{18}. Numerical calculations offer speed and convenience for sourceless efficiency calibration; however, they struggle to account for the effect of multiple scattering within the detector on photopeaks, and for complex measurement systems, numerical methods often cannot provide analytical equations or require significant model optimization, reducing credibility and complicating calculations.

The second category of sourceless efficiency calibration methods, based on Monte Carlo simulation, is frequently applied to simulate difficult measurement conditions due to its modeling flexibility \cite{19}. Kalus Noack explored reducing errors caused by deep penetration in Monte Carlo simulation and found that bias methods can effectively reduce simulation errors \cite{20}. Ghassoun investigated the relationship between point source and surface source detection efficiency and measurement distance/detector size using Monte Carlo simulation, providing a theoretical basis for optimizing large-volume source simulations \cite{21}. Zhao Jun proposed a geometric transformation method for converting a body source into a line source for Monte Carlo simulation of large body source detection efficiency, which is useful for reducing variance in large source simulations \cite{22}. Bao-Lu Yang tested the accuracy of LabSOCS and ANGLE sourceless efficiency calibration performance under a 241Am source, demonstrating validity for HPGe gamma-ray measurements \cite{23}. Frosio optimized the ISOCS/LabSOCS model by analyzing the effect of each geometry in the measurement model on radioactive source measurement uncertainty, showing that the method can reduce uncertainty by up to 8 times \cite{24}. Although Monte Carlo simulation can eliminate the effect of multiple scattering on photopeaks, its computational efficiency is difficult to meet practical needs when simulating complex measurement systems.

The third category, the combined method, utilizes the advantages of numerical computation for solving incident ray penetration distances at different angles and Monte Carlo simulation for handling complex particle transport processes within the detector, achieving complementary advantages. Qingxian Zhang obtained the angular injection distribution of rays from an infinite body source through numerical computation and combined it with Monte Carlo simulation to determine detection efficiency under different angular distributions, energies, and heights. The results demonstrated high accuracy for natural radionuclides such as U-series, Th-series, and K after ground and flight calibration experiments in Shijiazhuang \cite{25}. Building on this method, we improved the Monte Carlo simulation component by using Geant4 to simulate the gamma-ray deposition process in the detector combined with Boolean operations to rapidly obtain detection efficiency for different incidence modes. This improved the method's efficiency by at least 162 times compared with traditional Monte Carlo simulation while maintaining errors within 5% in point and body source validation experiments with a CeBr3 detector \cite{26}.

In summary, this paper introduces a novel sourceless efficiency calibration method based on Boolean operations of the ray deposition process to calibrate the static airborne gamma-ray survey model. Based on this foundation, multiple static measurement models are superimposed to approximate the dynamic measurement model, upon which the dynamic terrain response matrix is constructed. This approach provides a solid theoretical foundation for the accurate inversion of surface radionuclide concentrations in aerial surveys.

2.1 Sourceless Efficiency Calibration Method

This paper employs a sourceless efficiency calibration method based on Boolean operations of the ray deposition process to calculate the detection efficiency of the airborne gamma-ray spectrometer \cite{26}. Analyzing the interaction between rays and matter, the process from radioactive source emission to photopeak recording by the detector can be divided into two stages: (1) attenuation of rays by each shielding layer (including the detector envelope) before entering the detector sensitive volume, designated as the AE component; and (2) the probability of complete energy deposition (intrinsic detection efficiency) due to Compton scattering, electron-positron pair production, and the photoelectric effect when rays enter the detector sensitive volume, designated as the εE,θ,ϕ,S component. Therefore, the detection efficiency formula for a radioactive source can be expressed by Equation (1):

$$
\eta_{E,V} = \int \int A_E \cdot \varepsilon_{E,\theta,\phi,S} \, d\Omega \, dV
$$

where ηE,V is the photopeak detection efficiency of the radiation source, Ω is the angular range of ray emission, and V is the spatial range of the radiation source.

The parameters involved in the shielding process include: point source location distribution V, shield geometry and material information (including air, aircraft floor, detector mounting box, detector housing, vibration damping layers, etc.), and ray angular distribution Ω (elevation and azimuth angles). To solve the AE component quickly and flexibly, continuous variables in the measurement environment can be discretized (Fig. 1 [FIGURE:1] shows a schematic diagram of the discretization of the detector sensitivities of the airborne gamma-ray spectrometer GR820, which consists of 15 NaI(Tl) crystals measuring 10.16 cm × 10.16 cm × 40.64 cm).

After similar discretization of each shielding component, the angle and distance relationship between the coordinates of each scattering point and the ray equation in a specific emission direction is used to obtain the ray penetration distance dl in each shielding body. Thus, Equation (1) can be converted into:

$$
\eta_{E,V} = \sum \sum e^{-\sum_l \mu_l dl} \cdot \varepsilon_{E,\theta,\phi,S} \Delta\Omega \Delta V
$$

where μE,l is the linear absorption coefficient of shielding body l for rays of energy E, obtained through winXcom software calculations.

For the εE,θ,ϕ,S component, Geant4 can be used to obtain and record the deposition process of NE particles with energy E in an infinitely large sensitive volume at a specific incidence angle and position as original coordinates Dn. These coordinates are then spatially transformed according to the ray incidence direction (ϕin, θin) and incidence point coordinates Sin, as shown in Equation (3):

$$
D'n = D_n \cdot R_z \cdot R_x + S
$$

where D′n are the deposited coordinates after coordinate system transformation, and Rz, Rx are the rotation matrices of the coordinate system around the z-axis and x-axis, respectively.

The transformed D′n coordinates are then intersected with the sensitive volume through graphical Boolean operations, as illustrated in Fig. 2 [FIGURE:2] (using a rectangular detector from the GR820 as an example). Points located inside the sensitive volume are assigned a value of 1, while those outside are assigned 0, stored in column vector Np in subscript order. The intrinsic detection efficiency εE,θ,ϕ,S, representing cases where the entire deposition process occurs within the sensitive volume, can then be computed using Equation (4):

$$
\varepsilon_{E,\theta,\phi,S} = \frac{N_{in,E,\theta,\phi,S}}{N_E} = \frac{(\bigotimes_n P_n)^T \times \text{ones}}{N_E}
$$

where Nin,E,θ,ϕ,S represents the number of rays fully deposited within the detector, Pn is the medium density (e.g., 2.2 g/cm³ with an SiO₂ substrate), ones is a unit column vector with the same length as Pn, and NE is the total number of incident particles. Finally, detection efficiency is obtained by combining Equation (4) with Equation (2). During efficiency calibration for narrow-beam gamma rays in the energy range of 59.54–2620 keV, this method's computational efficiency is approximately 900–7200 times higher than conventional Monte Carlo simulations, with further improvements for longer ray path lengths (CPU @ R5800H).

2.2 Simplification of the Calibration Model

The ground gamma-ray injection rate model typically simplifies the ground gamma-ray source as an infinite cylindrical disc source and the detector as a point detector for injection rate calculations \cite{27}. However, as a 3D model, the volumetric source model increases detection efficiency computation time. Therefore, based on the concept of radioactivity spectrum equilibrium, the volumetric source model can be simplified to a surface source model to improve computational efficiency.

In quantitative inversion of airborne gamma-ray spectra, the primary focus is on counts within the characteristic peak regions of the U-series, Th-series, and 40K. Consequently, the simplification process only analyzes the equilibrium process for the characteristic gamma rays of each nuclide.

Gamma rays are emitted to the surface from underground rock with uniformly distributed radionuclides and density ρ. The characteristic gamma-ray injection at point O can be considered as the superposition of gamma rays emitted by spherical shell rock with radius r = 0 ∼ R, expressed by Equation (5). In reality, the natural gamma-ray surface injection rate reaches 95% of the infinite body source value before R = 50 cm. Since the range of lithological variations and aerial survey coverage are generally much larger than 50 cm, this paper simplifies the aerial gamma-ray body source model to a surface source model and performs dynamic terrain correction based on surface sourceless efficiency calibration. The relationship between natural radionuclide content on the surface and the surface injection rate can be expressed by Equation (6):

$$
\psi_O = \frac{\lambda \cdot P \cdot q \cdot \rho}{2\mu} \cdot \int_0^{2\pi} \int_0^{\pi/2} \int_0^R (1 - e^{-\mu R}) e^{-\mu r} \sin\theta \, d\theta \, d\phi \, dr
$$

where the rock is considered a homogeneous medium with density ρ (g/cm³), radionuclide content q (g/g), gamma-ray absorption coefficient μ (cm⁻¹), gamma-ray emission probability P, and decay constant λ.

From Equation (5), when R approaches infinity, the injection rate at point O equals that of an infinite body source and becomes isotropic in spatial distribution. Thus, the surface injection under an infinite radioactive body source is numerically equal to the injection at point O, and theoretically each point conforms to the 2π steradian angular distribution of a point source. In Equation (5), all parameters are constants except R. Using the surface injection rate under an infinite body source as a benchmark, the saturation of natural gamma-ray surface injection rate versus R for 1.46, 1.76, and 2.62 MeV is shown in Fig. 3 [FIGURE:3] (with a medium density of 2.2 g/cm³ and SiO₂ substrate).

2.3 Dynamic Terrain Correction Method

2.3.1 Dynamic Terrain Aerial Survey Forward Model

During aerial surveys, the distribution of surface radioactivity can be divided into finite elements according to influencing factors such as measurement point location, time, and measurement distance (as shown in Fig. 4 [FIGURE:4], with nj divisions in latitude and nk divisions in longitude). When the spectrometer flies from recording point ti−1 to the next point ti, its measurement range moves dynamically with the spectrometer. Consequently, all radioactive finite elements within the total measurement range of a single flight segment contribute to the spectrometer counts, and the detection efficiency of each finite element changes with the spectrometer coordinates.

Combining the simplified model from Section 2.2, the counts in a single spectrum can be considered as the superposition of contributions from all radiation field injection rates within the measurement range during that flight segment. Moreover, since each geological body has different radioactivity content, the spectrum count rate Ci of the i-th measurement can be expressed by Equation (7):

$$
C(i) = \iint A(x, y) \eta(x, y, t) \, dx \, dy
$$

where t is the measurement time, [x, y] is the measurement range of the i-th measurement, A is the ray injection rate of a single radioactive finite element, and η is the detection efficiency at the corresponding measurement coordinates at time t.

To ensure inversion accuracy in aerial surveys, the range of radioactive finite elements is often larger than the map elevation accuracy, resulting in terrain relief within each radioactive finite element. Additionally, the intersection region between the measurement range and the corresponding radioactive finite element changes dynamically with time. Therefore, after discretizing the single measurement time t and converting the inversion object into a radioactive finite element A(j, k) containing a height distribution, the contribution of the radioactive finite element to the spectrometer counts at the i-th measurement can be expressed by Equation (8):

$$
C(i, j, k) = A(j, k) \cdot \sum S(t_i, j, k) \cdot \eta(t_i, j, k)
$$

where S(ti, j, k) is the intersection area of the (j, k) radioactive finite element with the measurement range during the i-th measurement, and η(ti, j, k) is the average detection efficiency of this intersection area. Further, the spectrometer count C(i) at the i-th measurement can be converted from Equation (7) to Equation (9):

$$
C(i) = \sum_j \sum_k A(j, k) \cdot \sum S(t_i, j, k) \cdot \eta(t_i, j, k)
$$

Thus, the entire measurement process from the first measurement to the n-th measurement can be expressed in matrix form as Equation (10):

$$
\begin{bmatrix}
C(1) \
\vdots \
C(n)
\end{bmatrix}
=
\begin{bmatrix}
\sum S(t_1, 1, 1)\eta(t_1, 1, 1) & \cdots & \sum S(t_1, n_j, n_k)\eta(t_1, n_j, n_k) \
\vdots & \ddots & \vdots \
\sum S(t_n, 1, 1)\eta(t_n, 1, 1) & \cdots & \sum S(t_n, n_j, n_k)\eta(t_n, n_j, n_k)
\end{bmatrix}
\begin{bmatrix}
A(1, 1) \
\vdots \
A(n_j, n_k)
\end{bmatrix}
$$

where B is the response of the radioactive finite element to the spectrometer count rate.

In radioactive measurements, radionuclide decay is a random process. As a result, even under constant source and measurement conditions, the spectrometer count rate is not fixed but exhibits statistical fluctuations. If a nucleus decays with probability rate τ, then the probabilities of decay and non-decay within time interval t are (1 − e^{−τt}) and e^{−τt}, respectively. When the spectrometer has detection efficiency η, the probability p that a decay is both emitted and detected, and the probability q that it does not produce a count, can be defined as shown in Equation (11):

$$
p = (1 - e^{-\tau t}) \cdot \eta \
q = (1 - e^{-\tau t}) \cdot (1 - \eta) + e^{-\tau t}
$$

Thus, radionuclides obey a binomial distribution. Given measurements on N₀ radioactive nuclei, the probability that the spectrometer count ξ equals n at time t can be expressed by Equation (12):

$$
P(\xi = n) = C_n^{N_0} p^n q^{N_0-n} = \frac{N_0!}{(N_0 - n)! n!} p^n q^{N_0-n}
$$

During aerial surveys, N₀ is much greater than p. By applying a limiting approximation to Equation (12), it can be shown that P(ξ = n) follows a Poisson distribution with mean λ = N₀p, as given in Equation (13):

$$
P(\xi = n) = \frac{e^{-N_0 p} (N_0 p)^n}{n!}
$$

In summary, after incorporating background count rate δ and Poisson statistical fluctuations, the dynamic terrain aerial survey forward model can be expressed by Equation (14):

$$
C = \text{Poisson}(B \cdot A + \delta)
$$

Therefore, the PSO-MLEM algorithm \cite{28} based on Poisson statistical fluctuations is employed to solve the matrix equation.

2.3.2 Construction of the Dynamic Terrain Response Matrix

As derived in Section 2.3.1, the dynamic terrain response matrix B is constructed by superimposing the contributions of each radioactive finite element under static measurement conditions to obtain their total contributions under dynamic measurement conditions.

As illustrated in Fig. 5 [FIGURE:5], during static measurements, the intersection area between each radioactive finite element and the measurement range varies with the detector position. Moreover, terrain variations within each finite element also significantly impact detection efficiency. Therefore, the main challenges in constructing the response matrix include: (1) determining the static measurement range and obtaining spatial information for all radioactive finite elements within it; (2) rapidly computing the detection efficiency of surface sources after acquiring the spatial distribution of finite elements; and (3) determining the minimum number of convergent segments required to approximate dynamic measurements through static measurement superposition.

Among naturally occurring radionuclides, the 2.62 MeV gamma ray from the Th-series nuclide 208Tl undergoes less attenuation in matter compared to the 1.46 MeV gamma ray from 40K and the 1.76 MeV gamma ray from the U-series nuclide 214Bi. Consequently, its detection efficiency is more significantly affected by the measurement range and incident angle within the detector sensitive volume. Since the response matrix construction process is largely consistent across different gamma-ray energies, this section focuses on constructing the response matrix using the 2.62 MeV gamma ray as an example, with the GR820 airborne gamma spectrometer at a flight altitude of 100 meters.

Step 1: Spatial information of radioactive finite elements within the static measurement range. First, the static measurement range of the detector is determined. Based on the previously described sourceless efficiency calibration method, a model is constructed assuming a measurement altitude of 100 meters. Surface detection intensity for 2.62 MeV gamma rays is calculated over radial distances ranging from 100 m to 1000 m in 100 m increments. The resulting trend of detection efficiency versus measurement radius is shown in Fig. 6 [FIGURE:6], with the 98% saturation point indicated by a red asterisk.

After determining the measurement range at a given flight altitude, it is necessary to identify the area and spatial altitude distribution of the radioactive finite elements within this range that contribute to the spectrometer count. As shown in Fig. 8 [FIGURE:8], each radioactive finite element exhibits an altitude distribution. To account for altitude variation within each finite element, a vertical correction is applied by further subdividing the element into altitude sub-elements—regions with different elevations but the same radioactive activity. The detection efficiency of each altitude sub-element is calculated individually based on its intersection with the measurement range, and the contributions from all altitude sub-elements belonging to the same radioactive finite element are aggregated to determine the total contribution of that element to the spectrometer, as illustrated in Fig. 7 [FIGURE:7].

To efficiently record the intersection information between each altitude sub-element and the measurement range (including intersection area and geographic coordinates), the measurement range is simplified as a square with side length 2R and center coordinates (Ox, Oy). Using Equation (15), the vertex coordinates of each altitude sub-element rectangle (xm,n,l, ym,n,l) are processed to compute their intersection with the square measurement range. The resulting intersection regions are all rectangles (where l = 1 or 2 indicates the minimum and maximum values of x or y in the rectangle). Consequently, for each altitude sub-element within the measurement range, the intersection area Sm,n and the coordinates of the four rectangle vertices (Vxm,n,l, Vym,n,l) are recorded. These serve as essential input parameters for subsequent detection efficiency calculations.

$$
V_{xm,n,l} = \text{sort}[Ox + R, Ox - R, x_{m,n,1}, x_{m,n,2}]{(2,3)} \
V]} = \text{sort}[Oy + R, Oy - R, y_{m,n,1}, y_{m,n,2{(2,3)} \
S)} = (V_{xm,n,1} - V_{xm,n,2}) \cdot (V_{ym,n,1} - V_{ym,n,2
$$

In the equation, "sort" refers to arranging elements in ascending order, with the two middle values from the sorted set assigned as the minimum and maximum x and y coordinates of the intersection rectangle's vertices.

In addition to obtaining intersection information for altitude sub-elements within the measurement range, it is necessary to identify obstructed elements. Although elements with smaller angles can still be hidden, this requires very rugged terrain, as illustrated in Fig. 8 [FIGURE:8]. Since gamma-ray attenuation in rock and soil is significantly greater than in air, contributions from obstructed altitude sub-elements during each measurement are set to zero. To determine whether an altitude sub-element is obstructed, the angle θ between the sub-element and spectrometer is first calculated. Then all other altitude sub-elements along the line of sight between the sub-element and spectrometer are evaluated. If any intermediate sub-element has an angle to the spectrometer greater than θ, the original sub-element is considered obstructed and its contribution is set to zero. If all intermediate angles are smaller than θ, the sub-element is considered unobstructed.

Step 2: Fast computation of surface source detection efficiency during aerial surveys. After obtaining the spatial information of each altitude sub-element that contributes to the spectrometer count, it is necessary to calculate the surface source detection efficiency for each valid altitude sub-element. In practical aerial surveys, thousands of measurement records are typically generated, with each record decomposed into multiple static measurement superpositions, and each static measurement involving dozens of altitude sub-elements. This results in an extremely large computational workload when constructing the dynamic terrain response matrix.

To address this issue, the characteristics of detection efficiency distribution within the measurement range during aerial surveys are analyzed to optimize the efficiency computation process. The distribution within the first quadrant of a 500 × 500 m area is sufficient to represent the detection efficiency pattern within a 500 m radius at a flight altitude of 100 meters. With a grid resolution of 1 meter, a total of 250,000 points are calculated, and the resulting detection efficiency distribution is shown in Fig. 9 [FIGURE:9].

Due to spatial symmetry, the geometric center of each segment can be used to approximate the average detection efficiency within that region. From Fig. 9, it is observed that although the overall detection efficiency distribution resembles a two-dimensional Gaussian distribution, closer inspection of local regions (e.g., the area within [100–160, 60–140] in Fig. 9) reveals an approximately planar distribution. Within such regions, the detection efficiency at the geometric center provides an effective approximation of the average detection efficiency for the entire planar area. To identify sub-regions that can be approximated as planar segments, a profile line is drawn outward from the center and piecewise linear fitting is performed, as shown in Fig. 10 FIGURE:10. Segments with a coefficient of determination R² > 0.99 are retained. Using this method, the 500 m-radius measurement range is divided into 12 linearly fitted segments, each corresponding to a region where the detection efficiency exhibits approximately linear variation, as illustrated in Fig. 10 FIGURE:10.

Step 3: Minimum number of discrete segments for dynamic measurement. According to Equation (14), the count rate within a characteristic energy window statistically follows a Gaussian distribution determined by the product of radioactive intensity and detection efficiency. Even when flying over the same area at different speeds (i.e., with different measurement times but the same flight distance), the detection efficiency for a given ground region remains constant as it depends only on geometry. Therefore, discretizing a dynamic measurement into multiple static measurements is primarily related to flight altitude and distance rather than measurement duration. For example, if a flight distance of 10 m requires n discrete segments, a flight distance of 20 m would require 2n segments by linear superposition.

As shown in Figs. 6 and 9, the spectrometer count rate is mainly contributed by a small region directly beneath the detector, and this region is most sensitive to variations in flight distance. Therefore, as illustrated in Fig. 11 [FIGURE:11], this section analyzes the influence of the number of discrete segments n on detection efficiency within the left and right 50 × 50 m zones for a GR820 spectrometer flying at 100 m altitude.

Based on the static measurement parameters derived in the previous sections, the number of discrete segments is varied from 1 to 20. The corresponding variations in detection efficiency within the left and right side regions under different discretization numbers are shown in Fig. 12 [FIGURE:12]. When the discretization number is 1, the detection efficiency corresponds to the static measurement value at the detector position (25, 25, 100), resulting in lower initial detection efficiency on the left side compared to the right side. As the discretization number increases, the detection efficiencies of both regions gradually converge. Using a ±2% deviation as the convergence criterion, the minimum discretization number required for convergence at a flight altitude of 100 m is determined to be 4.

In summary, after obtaining the spatial distribution of radioactive finite elements within the measurement range (Step 1) and developing a fast surface source detection efficiency calibration method based on linear segment midpoint approximation (Step 2), the static response matrix construction method is established. Then, using the minimum discretization number for dynamic measurement obtained in Step 3, the static response matrices are superimposed to construct the dynamic terrain response matrix. The complete workflow is illustrated in Fig. 13 [FIGURE:13], with the relative error of each step remaining within 2%. According to the principle of error propagation, the overall error is expected to remain within 3.5%.

3.1 Airborne and Ground Gamma Spectrometry Measurements

The experimental area is located in the Baiyun'ebo iron ore mining district, Darhan Muminggan United Banner, Baotou City, Inner Mongolia Autonomous Region. The site features abundant mineral samples and low vegetation cover, making it suitable for subsequent ground gamma-ray measurements and geological mapping to validate the airborne survey data. The measurement instrument used is the GR820 airborne gamma-ray spectrometer, with all operations strictly following the "Specifications for Airborne Gamma-Ray Spectrometry" (EJ/T 1032-2005). The selected survey area spans from 110°00′11″E to 110°04′11″E and 41°37′29″N to 41°53′31″N, covering approximately 165 km².

The survey line layout is shown in Fig. 14 [FIGURE:14]. A total of 20 main survey lines are arranged longitudinally (east-west) with a spacing of approximately 350 meters, and 5 transverse tie lines are arranged latitudinally (north-south) with an average spacing of 7.4 kilometers, yielding 10,085 measurement points. To ensure flight safety, the flight altitude was slightly elevated, with approximately 81.40% of the survey conducted at altitudes between 80 and 120 meters and an average flight altitude of about 100 meters. The test zone includes 10 main survey lines and 1 tie line, comprising 987 measurement points.

To verify the effectiveness of the dynamic terrain correction algorithm, a test area was selected within the airborne survey region, as shown in the red box in Fig. 14(a). This area is located near the Baiyun'ebo iron ore deposit, covering coordinates 110°01′05″E to 110°03′04″E and 41°47′45″N to 41°50′50″N, with an approximate length of 5.7 km and width of 2.7 km. Satellite imagery indicates that the terrain within this region is relatively flat, with clearly visible strip-shaped ore belts in the central and upper portions. The lower left corner of the area is adjacent to the Baiyun'ebo mining zone. During the survey, the ground was dry and vegetation coverage was low, making the area well-suited for ground-based gamma spectrometry (sampling locations are marked by red lines in the figure). The ground investigation section includes 10 airborne gamma survey lines, comprising 2,083 measurement points. Measurements were conducted using the GR-320 ground gamma-ray spectrometer, strictly following the "Technical Specifications for Ground Gamma-Ray Spectrometry" (DZ/T 0205–1999).

3.2 Evaluation and Analysis of the Method

Based on the spatial information provided by the airborne survey system, the dynamic terrain response matrix (with detection efficiency corrected to sensitivity) is constructed using the method described in the previous chapter. The PSO-MLEM inversion algorithm is then applied to background-corrected airborne gamma-ray data for eU, eTh, and K to estimate their surface concentrations. The matrix construction parameters are summarized in Table 1 [TABLE:1], and the inversion results are presented as contour maps in Figs. 15, 16, and 17.

TABLE 1. Inversion parameters for dynamic terrain correction
| No. | Parameter | Indicator |
|-----|-----------|-----------|
| 5 | Number of Discretizations | 4 |

As shown in Fig. 15 [FIGURE:15], four anomalous zones were delineated based on eU content from ground measurements. After applying conventional altitude correction to the airborne survey results, anomalous contours were observed in all zones except Zone 2. Although the eU content values were relatively low, the spatial variation pattern indicated good accuracy in the airborne survey. After dynamic terrain correction, all anomalous zones became directly observable through contour maps. A comparison of average eU content in each anomalous zone (Table 2 [TABLE:2]) shows that values obtained after dynamic terrain correction were generally closer to ground measurements, with accuracy improvements ranging from 3.12% to 25.79% compared with altitude correction.

TABLE 2. Comparison of eU content and relative error in anomaly zones
| Anomalous Zones | Ground (ppm) | Altitude correction (ppm) | Err.(%) | Terrain correction (ppm) | Err.(%) |
|-----------------|--------------|---------------------------|---------|--------------------------|---------|

As shown in Fig. 16 [FIGURE:16], five anomalous zones were delineated based on eTh content from ground measurements. After altitude correction, anomalous contours were observed in all zones except Zones 1 and 4, though the boundaries were relatively indistinct. Following dynamic terrain correction, more distinct contours emerged in all zones except Zone 4. However, due to limited flight line density, Zone 1 exhibited some spatial deviation. A comparison of average Th content in each anomalous zone (Table 3 [TABLE:3]) indicates that dynamic terrain correction improved inversion accuracy by 7.50% to 48.82% compared with altitude correction. Moreover, the dynamically corrected results show that inversion accuracy for eTh content was significantly higher than for eU, attributable to generally higher Th concentrations in the survey area and the stronger penetration capability of characteristic gamma rays from the Th decay series, which reduces the impact of statistical fluctuations on inversion accuracy.

TABLE 3. Comparison of eTh content and relative error in anomaly zones
| Anomalous Zones | Ground (ppm) | Altitude correction (ppm) | Err.(%) | Terrain correction (ppm) | Err.(%) |
|-----------------|--------------|---------------------------|---------|--------------------------|---------|

As shown in Fig. 17 [FIGURE:17], five anomalous zones were delineated based on K content from ground measurements. After altitude correction, only Zones 3 and 5 exhibited clear anomalies, while no obvious anomalies were observed at other locations. After dynamic terrain correction, corresponding anomalies appeared at Zones 3, 4, and 5; however, positional deviations were observed at Zones 1 and 2, likely related to the sparser survey line layout of the aerial survey compared to ground measurements. A comparison of average K content in each anomalous zone (Table 4 [TABLE:4]) indicates that dynamic terrain correction improved inversion accuracy by 9.10% to 36.52% compared with altitude correction.

TABLE 4. Comparison of K content and relative error in anomaly zones
| Anomalous Zones | Ground (ppm) | Altitude correction (ppm) | Err.(%) | Terrain correction (ppm) | Err.(%) |
|-----------------|--------------|---------------------------|---------|--------------------------|---------|

Based on the comprehensive analysis of Figs. 15, 16, 17 and Tables 2–4, dynamic terrain correction produced U, Th, and K anomaly zone values closer to ground measurements than altitude correction, with accuracy improvements ranging from 3.12% to 48.82%, and yielded clearer anomaly contours. However, due to statistical fluctuations and the inherent nature of matrix-based inversion, each survey point is influenced by neighboring points within the measurement range, causing elevated measurement values to absorb lower ones and resulting in an apparent increase in the number of anomalous zones, as shown by the yellow blocks in Fig. 17(c). Nevertheless, the locations of high-anomaly zones (represented by red and white blocks) generally align well with ground survey results.

It should also be noted that apart from unavoidable measurement errors such as radar altimeter noise, GPS positioning inaccuracies, and flight attitude variations, the aerial survey used fewer and non-corresponding survey lines compared to ground measurements, resulting in slight positional deviations between anomalous points on the dynamically terrain-corrected contour maps and those derived from ground measurements.

4. Summary

Based on a novel sourceless efficiency calibration method, this study derives a static measurement model of airborne gamma-ray spectrometry into a dynamic measurement model and further improves the inversion accuracy of surface radionuclide concentrations by constructing and solving a dynamic terrain response matrix. This approach addresses the influence of detection efficiency calibration accuracy on inversion results in traditional terrain correction methods, as well as the low simulation efficiency of Monte Carlo programs for airborne survey modeling. Airborne and ground gamma-ray spectrometry experiments conducted in the Baiyun'ebo iron ore district revealed that: (1) after applying dynamic terrain correction, anomalous zones of eU, eTh, and K concentrations became more pronounced; using ground measurements as reference, the corrected radionuclide concentrations showed higher accuracy than conventional airborne inversion results, with a maximum relative error reduction of 30.58%, and anomaly boundaries became clearer and more consistent with ground survey distributions; (2) due to statistical fluctuations in measurement and inherent matrix inversion characteristics, elevated values tend to absorb nearby lower values, resulting in expansion of anomaly zones, although the locations of high-concentration anomalies remain largely consistent; and (3) due to slight deviations between airborne and ground survey line layouts, the positions of inversion anomalies show minor discrepancies compared to ground-measured anomaly zones.

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