Preamble

On the Spatial Distribution of Radiation Dose Using Dose Gradient and Least Squares Method Lin-Jiang Jiang Jun Liu and Yue Sun Xiao-Feng Yang

1 Chengdu University of Technology, Chengdu 610059, China 2 Chengdu University of Technology, Chengdu 610059, China.

In nuclear decommissioning scenarios, the distribution of radiation doses is usually unknown. To under- stand the current radiation situation in the monitoring of nuclear decommissioning facilities, it is important to generate continuously visualized radiation dose distribution images. The conventional method is to perform interpolation between measurement points to obtain continuous radiation dose distributions and predict high- dose areas. However, the results of conventional interpolation methods have low accuracy and poor prediction performance. Therefore, this paper proposes a method that combines gradients with least squares, aiming to improve the accuracy of radiation dose interpolation and the prediction of high-dose areas. In this study, we utilize the characteristics of radiation dose gradients to determine the direction of high-dose areas, combine this information with the least squares optimization method to predict high-dose regions, and obtain the distribution results through the radiation dose attenuation formula. Comparative experiments with conventional interpolation methods show that the proposed method reduces the mean squared error (MSE) of dose calculation by 82.1% compared to Kriging interpolation and by 47.9% compared to inverse distance weighting, significantly improv- ing interpolation accuracy. This method is suitable for radiation scenarios composed of gamma-ray ( -ray) detection point data and can provide technical support for radiation risk assessment and emergency pollution response, thereby enhancing the nuclear safety monitoring system.

Keywords

Dose gradient, Least squares method, Radiation scenario

INTRODUCTION

In the context of nuclear decommissioning scenarios, col- 2

lecting dose rate data at various locations within the scene and performing spatial interpolation on these measurement points can enable observation of the dose distribution in the radia- tion environment and predict high-dose areas. The resulting data can further be used to assess the potential impacts of the current scenario and provide core data support for developing the next disposal plan. Therefore, researching spatial interpo- lation calculation methods suitable for radiation scenarios has become an important research direction in the field of nuclear

safety monitoring. 12

Current research in academia and engineering on calculat- ing radiation dose distribution mainly focuses on two types of interpolation methods: Kriging interpolation and inverse distance weighting (IDW). Li Hua and others were the first to use the Kriging method to reconstruct and visualize radia- tion dose fields, achieving good reconstruction results, though with considerable accuracy deviations[ ]. Jin Guodong and colleagues presented the principles of the inverse distance weighted interpolation and Kriging interpolation methods, and then compared the two methods. Their conclusion was that Kriging interpolation performs better than inverse dis- tance weighting in most application scenarios[ ]. Building on this, Hu Jifeng and colleagues introduced the Radial Basis Function (RBF) interpolation method. Compared to Kriging, RBF generally performs less well overall, but its reconstruc- tion accuracy at the radiation source position exceeds that of Lin-Jiang Jiang, No. 1, East Third Road, Erx- ianqiao, Chenghua District, Chengdu, Sichuan Province, 610051, +86 Kriging[ ]. Xie Xingwen and others incorporated network function interpolation into traditional dose rate interpolation methods, using sparsely sampled nodes to reconstruct three- dimensional radiation fields[ ]. Chi Mingwen improved the Kriging method and conducted a comparative study with in- verse distance squared surface interpolation, finding that in- verse distance squared interpolation is efficient but less ac- curate, while the improved Kriging method provides better fitted data but with relatively lower efficiency; overall, Krig- ing interpolation is more suitable for practical engineering applications[ Zhang Biao and others found that radia- tion fields often contain mixtures of multiple radioactive nu- clides, making inversion extremely difficult, and therefore proposed an optimized inverse distance weighting algorithm to reconstruct gamma radiation fields. The reconstruction re- sults obtained using the optimized inverse distance weighting

algorithm had a significantly lower mean absolute percent- 45

age error compared to the results obtained using the Kriging method[ The aforementioned research methods demonstrate ad- vantages in computational efficiency in specific scenarios and can quickly complete radiation dose distribution predic- tions.

However, they generally exhibit limitations in sce- nario adaptability: methods such as Kriging and RBF inter- polation perform well in dose inversion within the coverage

area of measurement points, but tend to produce significant 54

deviations when extrapolating to areas outside the coverage extrapolation predictions within a certain range under simple scenarios or a single radiation source, in complex situations such as multiple overlapping radiation sources and obstacle

shielding, not only does the interpolation accuracy signifi- 60

cantly decrease, but there may also be incorrect predictions of dose distribution. When the radiation source is located at

the edge of the monitoring area, or in areas that are difficult 63

for detection equipment to reach, existing methods struggle

to achieve radiation source localization and dose inversion.

Therefore, this paper conducts further research on these is- sues.

This paper proposes a radiation dose distribution calcula- tion method that integrates dose gradient analysis with least squares optimization. It aims to address the accuracy of radi- ation dose interpolation in complex scenarios and to achieve three-dimensional visualization of predicted high-dose distri- bution areas. The principle and implementation process of up comparative experiments between the method proposed in this paper and the Kriging and inverse distance weighting methods. The effectiveness of our method is verified through error comparison and image scene comparison, with the re- sults analyzed in Section 4. Finally, Section 5 summarizes the main conclusions of this paper.

METHOD DESIGN As shown in Fig. . , this section focuses on gradient meth- ods and least squares methods, and explains in detail their the- oretical basis, computational logic, and specific design ideas through three steps: regional division, gradient method re- gional constraints, and least squares method simulation cal- culations.

Gradient method The radiation dose distribution of a single radiation source in space has distinctive characteristics: it exhibits a mono-

tonic decreasing gradient outward in all directions from the 91

center of the source (as shown in Fig. ). This figure includes three measurement locations, labeled 1, 2, and 3, as well as the extended lines of rays between each pair of locations.

By observing the distribution characteristics, key patterns can be extracted:Positions 1 and 2 fall within different gradient ranges, and the spatial distance between the two points is rel- atively short. The direction of radiation extending from low to high dose is closer to the center of the radiation source. Po- sitions 2 and 3 belong to the same gradient range, with con- sistent dose attenuation characteristics. Although positions 1 and 3 are in different gradient ranges, the distance between them is relatively large, and the direction of radiation extend- ing from low to high dose along the line connecting the two points deviates from the center of the radiation source.

Based on the above rules, the approximate direction of the

radiation source can be determined by the magnitude and di- 107

rection of the gradient. The core logic is: the more significant 108

the gradient change, the region where radiation shifts from low to high dose in a more concentrated direction is closer to the radiation source. The calculation of a single gradient

magnitude is centered on the "dose difference between two 112

points" and the "spatial distance between two points," with the formula defined as follows:

T = | ∆ D | (∆ r ) 2 (1) 115

Here, represents the absolute value of the dose difference between two points, and is the distance between the two points along the branch line.

However, the gradient directions of radiation from low to high doses do not necessarily point directly at the center of the radiation source, and the occurrence of 1,3-line extension cable deviations is more common. Therefore, in most cases, the prediction area can only be narrowed down by calculating multiple gradient rays. The specific formula is as follows:

T ( D i ) = max � | D i − D j | ( P i − P j ) 2 | i ̸ = j � (2) 125

Among them, is the set of gradient calculations, is the dose rate ( Gy/h) at the i-th measurement point,

the dose rate at the j-th measurement point (j ̸ = i), and P i and 128

are the spatial coordinates of the i-th and j-th measurement points, respectively. The gradient calculation result of the cur- rent measurement point i with all other measurement points is taken as the maximum value to serve as the core gradient fea- ture of this point, and the direction vector is calculated using the following formula:

⃗α i = ⃗P i − ⃗P j (3) 135

Here, is the ray direction vector corresponding to the max- imum gradient at the i-th measurement point (direction from the low dose point to the high dose point), and the spatial position vectors of the i-th and j-th measurement points, respectively. The extension direction of this vector is the candidate orientation of the radiation source.

As shown in Fig. , gradient methods can efficiently com- pute the gradient direction in scenarios with a single radia- tion source. However, in the multi-radiation source scenario shown in Fig. , multiple sources create overlapping high- gradient regions, causing confusion in the ray direction and interfering with the determination of the radiation source ori- entation.

Therefore, before performing gradient evaluation, the data needs to be processed. On one hand, high-dose points are seg- mented, and on the other hand, points in low-dose regions are

excluded as initial points for gradient rays. Using dose thresh- 152

olds and spatial distance thresholds as criteria, the measure- ment points are divided into multiple independent high-dose regions, defined by the following formula:

� D j ≥ D Threshold r j ≤ r Threshold (4) 156

Here, represents the i-th high-dose region, is the po- sition of the j-th measurement point, Threshold is the preset dose determination threshold, and only high-dose points are retained for subsequent calculations.

Threshold is the preset distance threshold that controls the spatial extent of a sin- gle region. When the j-th measurement point satisfies both

“dose ≥ D Threshold ” and “distance to region A i ≤ r Threshold ”, 163

the point is assigned to region To ensure the accuracy of regional division, it is neces-

sary to update the minimum distance from each measurement 166

point to its corresponding region in real time. The formula is as follows:

r j = min � ( P j − P i ) 2 | P i ∈ A i � (5) 169

Here, r j is the minimum distance from the j-th measurement 170

point to its corresponding region , and is any measure- ment point within region Through the above preprocessing, the multi-source mixed regions in Fig. can be divided into two separate high-dose areas, red and blue, as shown in Fig. , eliminating the gradi- ent interference from low to medium doses. Subsequently, it is only necessary to calculate the gradient ray direction within each independent area to constrain the radiation source local- ization region.

Least Squares Method The core idea of the least squares method is to quantify the difference between the “model-predicted values” and the “ac-

tual observed values” by defining a “sum of squared errors 183

function”, and then to find the model parameters that mini- 184

mize the error. Specifically, this is done by defining an error 185

function that measures the degree of difference between the observed value y and the function f(x) at the corresponding x, typically using the sum of squared errors as this error func- tion. That is, for a given set of n data points (x1, y1), (x2, y2),

..., (xn, yn), if the fitting function is y = f(x), then the sum of 190

squared errors S is:

i =1 ( y i − f ( x i )) 2 (6) 192

The ultimate goal is to find a set of parameters that mini- 193

mizes the sum of squared errors S. In this process, this method is used to optimize the simulated position of the radiation source. By constructing an error function between the sim- ulated dose values and the actual values, the simulation posi- tion that corresponds to the smallest error is identified as the optimal simulated position of the radiation source.

Under realistic conditions, radioactive materials are gener- ally stored in containers and pipelines, or attached to other objects. Therefore, we introduced scene modeling data as the basis for the attachment of radioactive materials, limiting the simulation range of radiation source locations to the mesh- modeled scenes. In the gradient method of part A, we have obtained a series of gradient rays. After extending these rays from low dose to high dose to the scene-modeled locations, we determined the specific areas of the radiation sources.

Within each area, source data is simulated:

D 1 × i = | D 1 D 2 · · · D i | (7) 210

· · · · · ·  

L i × j =

Here, represents the simulated source dose matrix for each region, represents the distance from the simulated source position to the measurement point, and represents the squared distance matrix from all simulated sources to all measurement points.

Classify and process the measurement point data, setting high, medium, and low dose ranges. In each range, use 10%

of the data as interpolated unknown points, and the remaining 219

data as the set of interpolated known points. ������� �������

D Known j × 1 =

������� �������

D Unknown k × 1 =

Establish a matrix equation based on the radiation attenuation function and solve it using simulated source dose data:

D 1 × i · L i × j = D Known j × 1 (11) 225

Obtain the simulated point dose data for each region in the matrix , and then perform multi-source joint superposi- tion calculations of the dose at the interpolated unknown point positions.

n =1 D n · L n · e − µ n · d n ( k ) (12) 230

D Total ( k ) =

Here, Total represents the dose rate at the k-th interpo- lation unknown point calculated from the simulated data, N represents the number of simulated sources, is the refer- ence dose rate of the n-th simulated source, is the distance from the position of the n-th simulated source to the k-th in- terpolation unknown point, is the linear attenuation coeffi- cient of the medium, which is related to the type of radiation and the material of the medium, and is the thickness of the medium that the radiation passes through.

Finally, the error is calculated using the least squares method:

k =1 ( D Total ( k ) − D Unknown ( k )) 2 (13) 242

Here, represents the current error calculation value, the number of measurement points, and Unknown repre- sents the value of the k-th interpolated unknown point.

Within the range of the gradient-limited area, find the next

scene grid unit, update the simulation point, and recompute. 247

After traversing all situations, find the minimum value from 248

the final array

S Best = min( S ) (14) 250

Among them, each regional simulated source position cor- responding to is the optimal predicted position of the resulting radiation source.

The best simulated sources obtained from each region are combined and superimposed, with their dose rate attenuation strictly following the inverse square law and medium absorp- tion corrections, to calculate the radiation dose rate at an un- known point, as shown in the following formula:

i =1 D i · r 2 i r 2 x · e − µ · d (15) 259

D x =

Here, represents the dose rate at any unknown point be- ing measured, represents the dose rate of the i-th simulated radiation source point, represents the number of simulated radiation sources, represents the distance from the simu- lated source point to the actual source, represents the dis- tance from the position of the unknown point to the actual source, represents the “linear attenuation coefficient” of the medium, and represents the thickness of the medium tra- versed.

EXPERIMENTAL DESIGN The experimental data are sourced from actual measure- ments at a nuclear waste treatment site, including dose data at measurement points and site point cloud reconstruction data.

The experimental environment was built using self-developed 3D radiation field imaging analysis software. Among them, the dose rate at measurement points ranges from 0.02 to 0.9 Gy/h. The experiment grouped the data according to the rule of “dividing one dose interval for every 0.1 Gy/h.” Within

each interval, 10% of the measurement points were selected as interpolation unknown points for comparison and verifica-

tion, while the remaining 90% were used as the training set 280

for subsequent model calculations and performance valida- tion.

To verify the effectiveness of the proposed method, this pa- per conducts an analysis through comparative experiments.

The comparison objects include the Kriging interpolation method and the inverse distance weighting method. evaluation criterion is designed using the cross-validation method, with mean absolute error (MAE) and mean squared error (MSE) as the core evaluation indicators. The perfor- mance of the Kriging method, the inverse distance weighting method, and the method designed in this paper is quantita- tively compared. The calculations are repeated five times, and the results are retained to four decimal places to avoid the im- pact of sampling randomness. The MSE is calculated based on the average MAE of the five repetitions as the final evalua- tion criterion. At the same time, to explore the error distribu- tion of each method in different dose intervals, the unknown interpolation points are ranked from high to low according to their true values. Then, the computed value of each unknown interpolation point is compared with the actual value, and fi- nally, the absolute errors of the three methods are compared.

To intuitively display the experimental scene, measurement

data, and positioning results, image rendering is performed 303

using 3D visualization technology. The specific parameter settings are as follows: the scene adopts a meshing method to construct a 3D model of the experimental site, with a mesh radius set to 0.005 m to balance point cloud density and ren- dering efficiency, ensuring clear presentation of the site struc- ture details; measurement points are marked in the form of point clouds, with a point size set to 1.0, making it easy to distinguish measurement points from the scene background and highlight the locations of the original data; within the spatial range covered by the measurement points and scene data, interpolation calculation points are generated at 0.1 m intervals. Continuous visualization of radiation dose distribu- tion is achieved through volume rendering, allowing intuitive presentation of continuous dose gradients; the transparency of the scene model and measurement points is set to 1.0 to ensure the original scene and data are clearly visible, while the transparency of the interpolation results is set to 0.5 to avoid obscuring the original data while highlighting the dose distribution trend.

Kriging interpolation method Kriging interpolation is based on spatial autocorrelation and achieves the optimal unbiased estimation of unknown points through linear weighting of known sample points. Its core calculation formula is as follows:

i =1 λ i · Z ( x i ) (16) 328

Z ( x 0 ) =

Here, represents the estimated value at the unknown point represents the weight of the i-th known point, represents the attribute value of the known point, and n represents the number of known sample points involved in the calculation.

By combining the spatial attenuation law of radiation dose 334

and quantifying spatial correlation, this experiment matches the weight to the exponential variogram model, with the specific form as follows:

λ i = Nuggest + Sill × � 1 − e ( − d Range ) � (17) 338

The nugget is set to 0 by default, and the sill, which is the “stable maximum value” of the variogram, is set to 1.0. The distance d represents the distance from an unknown point to a known point . The range is the “maximum distance” of spatial correlation. Based on the range between known and unknown points, the distance range of the test data in this study is within 8 meters, so the range is set to 8.0 here.

Inverse Distance Weighting Method The core logic of inverse distance weighting is that the closer a known sample point is to the unknown point, the greater its weight on the estimated value of the unknown point, which conforms to the attenuation characteristic of ra- diation dose being “larger when near and smaller when far”.

The dose estimation formula is as follows:

� n i =1 1 d p i 0 · Z ( x i )

Z ( x 0 ) =

� n i =1 1 d p i 0

Here, represents the estimated value at the unknown point represents the actual attribute value of the i- th known sample point, represents the direct distance be-

tween the known point x i and the unknown point x 0 ( d i 0 = 0 357

to avoid division by zero), p represents the distance decay co- efficient, and according to the rule of radiation decaying with the square of the distance, is set to 2 here. represents the number of known sample points involved in the calculation.

Gradient and Least Squares Method Regional Division Part: In this experiment, the strategy first calculates the maximum dose rate of all known interpo- lation points and ranks them. Starting from the first known interpolation point as the core, it extends outward by 1.0 m

(0.1 times the adaptive scenario range) to form the initial re- 367

gional boundary. Then, new known interpolation points are evaluated: if a new known interpolation point falls within the existing regional boundary, a boundary extending 1.0 m from the new point is combined with the original region to form a new regional range, ensuring coverage of all associated high- dose points; if a new known interpolation point falls outside all existing regional boundaries and its dose rate is greater than 0.25 times the maximum dose rate (with a dose attenu- ation factor of 0.25 at 1.0 m), a new independent calculation

region is created centered on this point according to the initial 377

regional rules. Gradient Calculation Section: Within each independent re- gion, the candidate range of radiation sources is screened based on gradient directions. Using the maximum dose rate of interpolated known points within the current region as a ref-

erence, a screening threshold is set at 0.8 times the maximum 383

dose rate of the current region (the amplification threshold is set for high dose intervals in this study). Only interpolated known points with dose rates greater than this threshold are retained for gradient calculation, thereby eliminating inter- ference from low-dose points. For each filtered interpolated known point, the gradient value with respect to other interpo- lated known points in the region is calculated, and the direc- tion with the largest gradient is selected as the core gradient direction. This direction is extended from low dose to high dose until it reaches the boundary of the experimental sce- nario. The “starting point coordinates” and “boundary end- point coordinates” of all core gradient rays are collected, and

the maximum and minimum values along the X, Y, and Z axes 396

are extracted. The rectangular space formed by these six ex- treme values then represents the candidate range for radiation sources in the region. Subsequent searches for the radiation source location are carried out only within this range, thereby reducing computational effort.

Radiation Source Localization: Within the candidate range of each area, the simulated radiation source position is iter- atively optimized using the least squares method to achieve

precise localization. To balance positioning accuracy and 405

computational efficiency, a three-dimensional grid is gener- ated at 0.1 m intervals, with each grid point serving as a can- didate simulated radiation source. Combinations of candidate simulated sources from all areas are then considered, and the least squares method is used to calculate the sum of squared errors between the simulated radiation doses and the actual doses at interpolated unknown points. The position parame- ters of candidate simulated sources in each area are iteratively updated, repeating the error calculation process. Finally, the

simulated source position with the minimum sum of squared 415

errors is selected as the final localization result of the radia- tion source for that area.

Dose Rate Calculation: Using the radiation dose rate at- tenuation model (Equation 15), dose rate simulation calcula- tions are performed at interpolated unknown points. In this experiment, since the material and thickness data of the scene medium are unknown, medium attenuation corrections are not considered, and the linear absorption attenuation coef- ficient is set to 0 (the calculation results may be overes- timated); the simulated radiation source is treated as a real source, with set to 1.0 and being the distance from the interpolated unknown point to the simulated radiation source.

EXPERIMENTAL RESULTS The Kriging method calculation results are shown in ta- and Fig. , with the calculated mean square error being 0.0715 ( Gy/h) Number Average calculated Actual value Mean Absolute value Error Performance of the Kriging method in different dosage ranges The computed results using the inverse distance weighting method are shown in table and Fig. , and the calculated mean square error is 0.0206 ( Gy/h) The calculation results of the method designed in this paper are shown in table and Fig. , and the calculated mean square error result is 0.0128 ( Gy/h) The absolute errors of the three methods at different dosage ranges are shown in Fig.

Based on the MSE quantification results, there are signif- 440

icant differences in the accuracy performance of the three methods. The MSE of the method designed in this study is ap- proximately 82.1% lower than that of the Kriging method and about 47.9% lower than that of the inverse distance weighting method. In terms of performance across different dose ranges, the method designed in this study shows higher accuracy than both the Kriging and inverse distance weighting methods in the low-dose range; in the medium-dose range, it outperforms

Number Average calculated Actual value Mean Absolute value Error Number Average calculated Actual value Mean Absolute value Error the Kriging method and has similar accuracy to the inverse distance weighting method; in the high-dose range, although the calculated results are slightly higher, the accuracy still sur- passes that of the other two methods. This indicates that the method proposed in this study effectively improves the ac- curacy of dose estimation, with the inverse distance weight- ing method being second, and the Kriging method having the lowest accuracy.

Next, the measured point dose data will be superimposed in 3D with the simulated dose distribution data using exper- imental software for visualization, applying a color mapping

rule where the dose ranges from high to low are represented by red yellow green blue. Additionally, all visualiza- tion results will be ensured to use the same viewing angle and observation area to eliminate the interference of perspective differences on the analysis results.

The initial scene is shown in Fig. 10 [FIGURE:10] , with the image 465

containing measurement points marked in color and the site 466

background model. The red high-dose measurement points are clustered around the cylindrical barrel-like objects in the scene, the yellow-green medium-dose points spread around the high-dose points, and the blue low-dose points are dis- tributed along the periphery of the site, clearly reflecting the spatial dose gradient characteristics of the measurement data.

After dividing the high-dose regions, the gradient rays from low dose to high dose are drawn as shown in Fig. : the gradient rays consist of continuous red dots, starting from the high-dose measurement points and extending along the direction of maximum gradient to the boundary of the site.

The concentrated directionality of the rays provides an intu-

itive basis for determining the candidate range of radiation 479

sources. Within the candidate range defined by the gradient rays, the simulated radiation source points obtained through itera- tive optimization using the least squares method are shown in optimal simulated radiation source location, which happens to be inside a cylindrical barrel. This barrel is a known con- tainer for radioactive materials in the experimental site. The simulation results highly correspond to the actual scenario,

validating the rationality of this positioning method. 489

The final dose distribution visualization results are shown in Fig. : the high-dose red region not only fully encom- passes the high-dose measurement points but also extends into the interior of the cylindrical container, consistent with the actual location of the radioactive material; the yellow- green medium-dose region only covers the area correspond- ing to the yellow measurement points, and the blue low-dose region is highly consistent with the low-dose measurement points, with dose boundaries closely matching the actual gra- dient pattern.

The visualization results of the dose distribution using or- dinary Kriging interpolation are shown in Fig. : the high-

dose red areas ( ≥ 0.8 µ Gy/h) do not align with the high-dose 502

measurement points, instead concentrating in the blank areas between medium-high dose measurement points; the yellow- green medium-dose areas (0.4–0.6 Gy/h) are excessively spread, encompassing many green and blue low-dose mea- surement points, which does not correspond to the actual dose gradient pattern, reflecting the failure of this method’s spatial correlation assumption in multi-source scenarios.

The dose distribution results of the inverse distance weight- ing method are shown in Fig. : the high-dose red region accurately covers the high-dose measurement points without

obvious deviation; the yellow-green medium-dose region in- cludes only a few blue low-dose measurement points, and the dose boundaries are relatively clear; however, the high-dose region spreads outward from the measurement points, cover- ing only the upper surface and top of the cylindrical bucket.

In summary, evaluating from the perspectives of the match- ing degree at different dose measurement points and the pre- dictive performance in high-dose regions, the gradient and least squares method designed in this study yielded the best results, followed by the inverse distance weighting method, Yuan, Liye, Yunshi, radiation field interpola- reconstruction visualization based theory[J].Radiation Protection,2019,39(6):475-482.

Jin Guo dong, Liu Yancong, Niu Wenjie. Comparison of distance-weighted inverse interpolation method and Kri-

gin interpolation method[J]. Journal of Changchun Univer- 569

sity of Technology(Natural Science Edition),2003,24(3):53-57. with the kriging method performing the worst. This result further validates the effectiveness of the method proposed in this study, and the visualization of the radiation dose distri- bution better reflects the characteristics of the actual radiation field.

CONCLUSION

To address the issues of low accuracy and large prediction deviations in traditional interpolation algorithms for radiation dose distribution calculations, this paper proposes a radiation dose distribution calculation method that integrates dose gra- dient analysis with least squares optimization. Experimental results show that the MSE of the proposed method is 82.1% lower than that of the ordinary kriging method and 47.9% lower than that of the inverse distance weighting method, fully demonstrating the advantage of this method in dose in- terpolation accuracy. Furthermore, three-dimensional visu- alization results indicate that the simulated radiation source points generated by the proposed method can be precisely located inside the storage barrels of radioactive materials, consistent with the physical laws of actual radiation fields,

whereas the kriging method produces offset positioning and 543

the inverse distance weighting method only covers the bar- rel surfaces, both failing to reflect the actual spatial positions of radiation sources.

This demonstrates the superiority of the proposed method in radiation source prediction accuracy.

Considering both error accuracy and practical features, the above analysis confirms the feasibility and practicality of the proposed method in complex radiation scenarios.

Although the method presented in this paper demonstrates advantages in accuracy and practicality, there are still two areas that need optimization: compared with traditional in- terpolation algorithms, the method designed in this paper in- volves more computational steps (excluding error calculation, the computation time is 200 ms); when the number of mea-

surement points in high-dose regions is small (< 2), insuffi- 557

cient input data can easily lead to large deviations in radiation source prediction. These issues need to be further optimized in the future.

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