Radiation measurement method

Abstract

To provide a radiation measurement method for enabling efficient and appropriate measurement of a spatial intensity distribution of a radiation source.SOLUTION: In a radiation measurement method, a radiation measurement unit 1 is used, which is constituted to allow a sensitivity for detecting a radiation to be made incident from a specified direction to be lower than a sensitivity for detecting a radiation to be made incident from a direction different from the specified direction. The method includes: a measurement step for measuring intensity of a radiation to be made incident to the radiation measurement unit 1 by the radiation measurement unit 1 while the radiation measurement unit 1 is arranged in each of a plurality of arrangement states where specified directions or positions mutually differ; and a ray source distribution estimation step for estimating an intensity distribution of a radiation source with the use of a measurement value of the intensity of the radiation obtained in the measurement step and also with the use of arithmetic processing of an inverse problem constructed based on a model which expresses relation between an intensity distribution of a radiation source corresponding to a relative direction or position with respect to the radiation measurement unit 1 and the intensity of the radiation to be made incident to the radiation measurement unit 1 by an integral equation in a form of Fredholm integral equation.SELECTED DRAWING: Figure 1

Description

[Technical Field]

[0001] The present invention relates to a method for measuring radiation. [Background technology]

[0002] A known conventional radiation measurement technique involves placing a shielding member that blocks radiation on the side of a radiation measurement device such as a scintillator, and measuring the radiation intensity at each predetermined rotation angle while rotating the shielding member around the radiation measurement device, as described in Patent Document 1. In this measurement technique, the direction of the radiation source is identified by identifying the angle at which the radiation intensity drops from data on measured radiation intensity corresponding to the rotation angle of the shielding member around the radiation measurement device.

[0003] Also, as seen in Patent Document 2, for example, there is known a technique for estimating a three-dimensional radiation distribution from information on the incident position and dose of radiation detected by three-dimensionally arranged scintillation fibers using a sampled pattern matching method. In this method, a radiation intensity distribution pattern is calculated on the assumption that a radiation source exists in each of a plurality of elements obtained by dividing an environmental space, and the position of the element is searched for so that the radiation intensity distribution pattern is closest to the actually measured radiation intensity distribution pattern, thereby estimating the distribution of the radiation sources. [Prior art documents] [Patent documents]

[0004] [Patent Document 1] Japanese Patent Application Laid-Open No. 2004-325417 [Patent Document 2] Japanese Patent Application Publication No. 9-15335 Summary of the Invention [Problem to be solved by the invention]

[0005] In an environment exposed to radiation, it is often desirable to measure the intensity distribution of a radiation source (a distribution indicating the intensity of a radiation source in a given direction or position) according to spatial direction, position, etc. However, the technology disclosed in Patent Document 1 is only capable of identifying the direction of a radiation source (the direction in which radiation travels), but is not capable of identifying the intensity of a radiation source in a given direction.

[0006] Furthermore, although the technology disclosed in Patent Document 2 can estimate the spatial distribution of radiation sources, it does so by assuming that the intensity of the radiation sources is constant, and therefore it is not possible to accurately estimate the intensity of the spatially distributed radiation sources. Furthermore, if the environmental space is divided into a large number of parts, it tends to take a long time to identify the distribution of radiation sources using the sampled pattern matching method.

[0007] The present invention has been made in view of the above background, and has an object to provide a radiation measurement method that makes it possible to efficiently and appropriately measure the spatial intensity distribution of a radiation source. [Means for solving the problem]

[0008] In order to achieve the above object, the radiation measurement method of the present invention includes a measuring step of using a radiation measurement device configured so that the sensitivity for detecting radiation incident from a specific direction is lower or higher than the sensitivity for detecting radiation incident from a direction different from the specific direction, and measuring the intensity of radiation incident on the radiation measurement device with the radiation measurement device in a plurality of arrangement states in which the specific direction of the radiation measurement device or the position of the radiation measurement device are different from each other; a radiation source distribution estimation step of estimating the intensity distribution of the radiation source using a calculation process for an inverse problem constructed based on a model that expresses the relationship between the intensity distribution of a radiation source according to a relative orientation or position with respect to the radiation measurement device and the intensity of radiation incident on the radiation measurement device according to the intensity distribution by an integral equation in the form of a Fredholm integral equation, and a radiation source distribution estimation step of estimating the intensity distribution of the radiation source using measured values ​​of the intensity of the radiation obtained in the measurement step. 、 the radiation measuring device is configured so that sensitivity for detecting radiation incident from a specific direction is lower than sensitivity for detecting radiation incident from a direction different from the specific direction, the measuring step includes a first measuring step of measuring, by the radiation measuring device, an intensity of radiation incident on the radiation measuring device in a plurality of arrangement states in which the specific direction of the radiation measuring device is different from one another around a predetermined axis of the radiation measuring device; The radiation source distribution estimation step includes a first radiation source distribution estimation step of estimating the first intensity distribution from a calculation process of an inverse problem constructed based on a model that expresses a relationship between a first intensity distribution, which is an intensity distribution of a radiation source according to a relative orientation around the predetermined axis with respect to the radiation measurement device, and the intensity of radiation incident on the radiation measurement device according to the first intensity distribution, by a first integral equation in the form of a Fredholm integral equation, and a measured value of the intensity of the radiation obtained in the first measurement step. (First invention).

[0009] Here, when the measurement step is carried out using a radiation measuring device configured so that its sensitivity in detecting radiation incident from a specific direction is lower or higher than its sensitivity in detecting radiation incident from a direction different from the specific direction, the relationship between the intensity distribution of the radiation source depending on the orientation or position relative to the radiation measuring device and the intensity of the radiation incident on the radiation measuring device depending on the intensity distribution can be modeled by an integral equation in the form of a Fredholm integral equation.

[0010] Then, by using the calculation process of the inverse problem that can be constructed based on this model and the measured values ​​of the radiation intensity obtained in the measurement step, it is possible to estimate the intensity distribution of the radiation source according to the relative orientation or position with respect to the radiation measurement device. In this case, the calculation process of the inverse problem based on the above model can be constructed as a relatively simple linear calculation process. Therefore, according to the first invention, it is possible to efficiently and appropriately measure the spatial intensity distribution of the radiation source.

[0011] The first invention above In the above The computational processing of the inverse problem in the first radiation source distribution estimation step may be configured to estimate the first intensity distribution using an inverse matrix of a circulant matrix obtained by discretizing a model represented by the first integral equation (second invention).

[0012] According to this, the calculation process of the inverse problem in the first radiation source distribution estimation step of estimating a first intensity distribution, which is an intensity distribution of a radiation source according to a relative orientation around a predetermined axis with respect to the radiation measurement device, can be configured to estimate the first intensity distribution using an inverse matrix of a circulant matrix obtained by discretizing a model represented by the first integral equation. The inverse matrix of the circulant matrix can be easily calculated at high speed using a computer or the like. Therefore, according to the second aspect of the present invention, the first intensity distribution can be estimated efficiently.

[0013] In the second aspect of the present invention, the measurement step is a step of performing the first measurement step for each of a plurality of positions in a state in which the radiation measurement device is disposed at each of the plurality of positions different from one another in the predetermined axis direction, and the radiation source distribution estimation step is a step of performing the first radiation source distribution estimation step for each of the plurality of positions, and estimating a relationship between a second intensity distribution, which is an intensity distribution of the radiation source according to a position in a predetermined direction arbitrarily selected from directions around the predetermined axis, and the intensity of the radiation incident on the radiation measurement device from the predetermined direction according to the second intensity distribution, by using a second integral equation in the form of a Fredholm integral equation. and a second radiation source distribution estimation step of estimating the second intensity distribution from an inverse problem calculation based on a model expressed by the second integral equation and the intensity of radiation in the predetermined direction defined by the first intensity distribution estimated in the first radiation source distribution estimation step, wherein the inverse problem calculation in the second radiation source distribution estimation step is calculation configured to estimate the second intensity distribution using an inverse matrix of a matrix obtained by discretizing the model expressed by the second integral equation, or a Moore-Penrose inverse matrix defined in accordance with the matrix (third invention).

[0014] According to this, the computational process of the inverse problem in the second radiation source distribution estimation step of estimating a second intensity distribution, which is an intensity distribution of a radiation source according to a position in a predetermined direction arbitrarily selected from directions around a predetermined axis of the radiation measurement device, can be configured to estimate the second intensity distribution using an inverse matrix of a matrix obtained by discretizing a model represented by the second integral equation, or a Moore-Penrose inverse matrix defined according to the matrix. The inverse matrix of the matrix or the Moore-Penrose inverse matrix can be easily calculated at high speed using a computer or the like. Therefore, according to the third aspect of the present invention, not only the first intensity distribution but also the second intensity distribution can be efficiently estimated.

[0015] The radiation measurement method of the present invention includes a measurement step of using a radiation measurement device configured so that its sensitivity for detecting radiation incident from a specific direction is lower or higher than its sensitivity for detecting radiation incident from a direction different from the specific direction, and measuring the intensity of radiation incident on the radiation measurement device with the radiation measurement device in a plurality of arrangement states in which the specific direction or the position of the radiation measurement device differs from one another; a radiation source distribution estimation step of estimating the intensity distribution of the radiation source using a calculation process for an inverse problem constructed based on a model that expresses a relationship between an intensity distribution of a radiation source according to a relative orientation or position with respect to the radiation measurement device and the intensity of radiation incident on the radiation measurement device according to the intensity distribution by an integral equation in the form of a Fredholm integral equation, and a radiation source distribution estimation step of estimating the intensity distribution of the radiation source using measured values ​​of the radiation intensity obtained in the measurement step, the radiation measuring device is configured so that sensitivity for detecting radiation incident from a specific direction is lower than sensitivity for detecting radiation incident from a direction different from the specific direction, the measurement step includes a first measurement step of measuring the intensity of radiation incident on the radiation measurement device with the radiation measurement device in a state where the radiation measurement device is arranged in each of a plurality of arrangement states where the specific direction of the radiation measurement device is different from one another around a predetermined axis of the radiation measurement device, and is a step of performing the first measurement step for each of a plurality of positions in a state where the radiation measurement device is arranged in each of a plurality of positions which are different from one another in the predetermined axial direction, The radiation source distribution estimation step includes a first radiation source distribution estimation step of estimating a first intensity distribution, which is an intensity distribution of the radiation source according to a relative orientation around the predetermined axis with respect to the radiation measuring device, from the measured values ​​of the radiation intensity obtained in the first measurement step, for each of the plurality of positions, and a second intensity distribution, which is an intensity distribution of the radiation source according to a position in a predetermined direction arbitrarily selected from directions around the predetermined axis, and a first intensity distribution estimation step of estimating a relationship between the second intensity distribution, which is an intensity distribution of the radiation source according to a position in a predetermined direction arbitrarily selected from directions around the predetermined axis, and the intensity of the radiation incident on the radiation measuring device from the predetermined direction. and a second radiation source distribution estimation step of estimating the second intensity distribution from the radiation intensity in the predetermined direction defined by the first intensity distribution estimated in the first radiation source distribution estimation step, wherein the calculation of the inverse problem in the second radiation source distribution estimation step is carried out by using an inverse matrix of a matrix obtained by discretizing the model represented by the second integral equation or a Moore-Penrose inverse matrix defined in accordance with the matrix. The second intensity distribution is estimated by the calculation process. It is characterized by ( 4th invention).

[0016] According to this, in the first radiation source distribution estimation step, the first intensity distribution may be estimated by another method, not limited to the same method as in the second invention. Then, in the second radiation source distribution estimation step, the second intensity distribution can be estimated by the same method as in the third invention. Therefore, according to the fourth invention, the second intensity distribution can be estimated efficiently.

[0017] The radiation measurement method of the present invention includes a measurement step of using a radiation measurement device configured so that its sensitivity for detecting radiation incident from a specific direction is lower or higher than its sensitivity for detecting radiation incident from a direction different from the specific direction, and measuring the intensity of radiation incident on the radiation measurement device with the radiation measurement device in a plurality of arrangement states in which the specific direction or the position of the radiation measurement device differs from one another; a radiation source distribution estimation step of estimating the intensity distribution of the radiation source using a calculation process for an inverse problem constructed based on a model that expresses a relationship between an intensity distribution of a radiation source according to a relative orientation or position with respect to the radiation measurement device and the intensity of radiation incident on the radiation measurement device according to the intensity distribution by an integral equation in the form of a Fredholm integral equation, and a radiation source distribution estimation step of estimating the intensity distribution of the radiation source using measured values ​​of the radiation intensity obtained in the measurement step, the radiation measuring device is configured so that sensitivity for detecting radiation incident from a specific direction is higher than sensitivity for detecting radiation incident from a direction different from the specific direction, the measurement step includes a measurement step A of measuring the intensity of radiation incident on the radiation measurement device with the radiation measurement device in a state in which the radiation measurement device is disposed so that the specific direction coincides with an arbitrarily selected predetermined direction among directions around a predetermined axis of the radiation measurement device, The radiation source distribution estimation step includes an A-th radiation source distribution estimation step of estimating the A-th intensity distribution from an inverse problem calculation process constructed based on a model that expresses, by an A-th integral equation in the form of a Fredholm integral equation, a relationship between an A-th intensity distribution, which is an intensity distribution of a radiation source depending on a position in the predetermined direction among directions around the predetermined axis, and the intensity of radiation incident on the radiation measuring device from the predetermined direction depending on the A-th intensity distribution, and measured values ​​of the radiation intensity obtained in the A-th measurement step, and the A-th radiation source distribution estimation step is a calculation process constructed to estimate the A-th intensity distribution using an inverse matrix of a matrix obtained by discretizing the model expressed by the A-th integral equation, or a Moore-Penrose inverse matrix defined depending on the matrix. Characterized by (Fifth invention).

[0018] According to this, the measurement in the A measurement step can directly measure the intensity of radiation incident on the radiation measuring device from a predetermined direction around the predetermined axis. Then, by using the measured radiation intensity value, the A radiation source distribution estimation step can estimate the A intensity distribution, which is the intensity distribution of the radiation source according to the position in the predetermined direction, by the same method as the second radiation source distribution estimation step in the third or fourth invention. Therefore, according to the fifth invention, the A intensity distribution can be efficiently estimated. [Brief explanation of the drawings]

[0019] [Figure 1] FIG. 1A is a perspective view showing a schematic configuration of a radiation measuring device according to a first embodiment of the present invention, and FIG. 1B is a view showing the configuration of the radiation measuring device as viewed in the direction of the central axis of a measuring device main body included in the radiation measuring device according to the first embodiment. [Figure 2] 4 is a flowchart showing a measurement procedure in the first embodiment. [Figure 3] 4A and 4B are explanatory diagrams relating to the arrangement of shielding members around the central axis of the radiation measurement device body and the first radiation source distribution estimation process in the first embodiment. [Figure 4]FIG. 4 is an explanatory diagram relating to a plurality of measurement points where radiation measurement devices are arranged and the second radiation source distribution estimation process in the first embodiment. [Figure 5] 6 is a graph showing an example of a simulation result regarding the first radiation source distribution estimation process in the first embodiment. [Figure 6] FIG. 6A is a graph showing a first example of a simulation result related to the second radiation source distribution estimation process in the first embodiment, and FIG. 6B is a graph showing a second example of a simulation result related to the second radiation source distribution estimation process in the first embodiment. [Figure 7] 10 is a graph showing a third example of a simulation result regarding the second radiation source distribution estimation process in the first embodiment. [Figure 8] FIG. 10 is a diagram showing the configuration of a radiation measurement device according to a second embodiment of the present invention, as viewed in the direction of the central axis of a measurement device main body included in the radiation measurement device. [Figure 9] FIG. 10 is a diagram showing the configuration of a radiation measuring device according to a third embodiment of the present invention, as viewed in the direction of the central axis of a measuring device main body included in the radiation measuring device. [Figure 10] FIG. 11 is an explanatory diagram relating to a plurality of measurement points where radiation measurement devices are arranged and the second radiation source distribution estimation process according to the third embodiment. [Figure 11] 10 is a graph showing an example of a simulation result regarding the second radiation source distribution estimation process in the third embodiment. DETAILED DESCRIPTION OF THE INVENTION

[0020] [First embodiment] A first embodiment of the radiation measurement method of the present invention will be described with reference to Figures 1A to 7. With reference to Figures 1A and 1B, a radiation measurement device 1 in this embodiment is a radiation measurement device configured so that its sensitivity for detecting radiation incident from a specific direction is lower than its sensitivity for detecting radiation incident from a direction different from the specific direction. This radiation measurement device 1 includes a measurement device main body 1a that generates an output according to the intensity of the incident radiation, and a shielding member 1b that blocks radiation incident on the measurement device main body 1a from the specific direction.

[0021] The measuring device main body 1a is configured, for example, by a GM tube (Geiger-Mueller tube) type radiation measuring device such as a Geiger counter. However, the measuring device main body 1a is not limited to a GM tube type, and may be configured, for example, by a scintillator type radiation measuring device or an ionization chamber type radiation measuring device. Note that although the outer shape of the measuring device main body 1a is depicted as being cylindrical in FIGS. 1A and 1B, the outer shape of the measuring device main body 1a may be any shape. This also applies to other embodiments described below.

[0022] The shielding member 1b is made of a known material capable of shielding radiation, such as lead. The shielding member 1b is configured to cover an area around the measurement device main body 1a with a predetermined angular width θw1 around the central axis C of the measurement device main body 1a. Specifically, the shielding member 1b is disposed to the side of the measurement device main body 1a, for example, so as to be spaced from the measurement device main body 1a in a direction perpendicular to the central axis C and to extend in the same direction as the central axis C.

[0023] In this case, when viewed in the direction of the central axis C of the measuring device main body 1a, as shown in FIG. 1B, the inner surface (the surface facing the measuring device main body 1a) and the outer surface (the surface opposite the inner surface) of the shielding member 1b are formed to curve in an arc shape with an arc length of a predetermined angular width θw1 around the central axis C. The shielding member 1b is also formed to have a constant thickness in a direction perpendicular to the central axis C. The central axis C of the measuring device main body 1a is set (defined) in advance with respect to the measuring device main body 1a. For example, the axis of a GM tube provided in the measuring device main body 1a can be set as the central axis C of the measuring device main body 1a.

[0024] Since the shielding member 1b is configured as described above, when the shielding member 1b is positioned at an angle θ' around the central axis C as shown in FIG. 1B, radiation coming toward the measuring device main body 1a from a direction in the angular range of θ' to θ' + θw1 (for example, the radiation illustrated by arrow Y1 in FIG. 1B) is blocked by the shielding member 1b from entering the measuring device main body 1a, and is hardly detected by the measuring device main body 1a.

[0025] On the other hand, radiation coming toward the measuring device main body 1a from directions other than the angular range of θ' to θ' + θw1 around the central axis C (for example, the radiation illustrated by the arrow Y2 in Figure 1B) is not blocked by the shielding member 1b, but enters the measuring device main body 1a and is detected by the measuring device main body 1a.

[0026] As a result, the radiation measuring instrument 1 is configured so that its sensitivity in detecting radiation incident from a specific direction (a direction in the angular range of θ' to θ' + θw1 around the central axis C) is lower than its sensitivity in detecting radiation incident from a direction different from the specific direction.

[0027] Furthermore, the shielding member 1b is connected to a rotary actuator 2 configured by an electric motor or the like, and is provided so as to be rotatable relative to the measuring device main body 1a about the central axis C by the rotational driving force applied from the rotary actuator 2. In this case, the rotary actuator 2 has an output shaft 2a coaxial with the central axis C, and the shielding member 1b is connected to this output shaft 2a via a connecting mechanism 3. As a result, the rotational driving force output to the output shaft 2a of the rotary actuator 2 causes the shielding member 1b to rotate about the central axis C.

[0028] Furthermore, by stopping the rotation of the output shaft 2a of the rotary actuator 2 at any rotation angle, it is possible to stop the shielding member 1b at any angular position around the central axis C relative to the measurement device main body 1a. This makes it possible to change the specific direction in which the radiation detection sensitivity of the measurement device main body 1a becomes low to any direction around the central axis C.

[0029] 1A shows a simplified view of the coupling mechanism 3 between the shielding member 1b and the output shaft 2a of the rotary actuator 2. However, the coupling mechanism 3 may have any structure as long as it can transmit a rotational driving force from the rotary actuator 2 to the shielding member 1b, causing the shielding member 1b to rotate relatively around the central axis C with respect to the measurement device main body 1a. The power transmission path between the rotary actuator 2 and the shielding member 1b may be provided with a reducer, a clutch mechanism, a brake mechanism, or the like. The shielding member 1b is not limited to an arc-shaped shape and may have other shapes. Furthermore, the radiation measurement device 1 may be configured, for example, so that the shielding member 1b can be manually rotated around the measurement device main body 1a (it need not include an actuator that rotationally drives the shielding member 1b).

[0030] Next, a measurement process for measuring the intensity (dose or dose rate) of radiation using the radiation measurement device 1 configured as described above will be described. In this embodiment, the radiation measurement device 1 is sequentially installed at each of a plurality of measurement points that are different in vertical position (i.e., height) with the central axis C of the measurement device main body 1a facing up and down (vertical or nearly vertical). Then, at each measurement point, the shielding member 1b is rotated to a plurality of angular positions that are different from each other around the central axis C (in other words, with the orientation of the shielding member 1b around the central axis C as viewed from the measurement device main body 1a facing each of a plurality of orientations), and the intensity (dose or dose rate) of the radiation incident on the measurement device main body 1a is measured by the measurement device main body 1a.

[0031] More specifically, the radiation intensity measurement process is performed, for example, according to the procedure shown in the flowchart in Fig. 2. In STEP 1, the measurement point number m is set to "1," which is the number of the first measurement point. Then, in STEP 2, the radiation measuring device 1 is installed at the mth measurement point. In this case, the radiation measuring device 1 is installed at the mth measurement point with the central axis C of the measuring device main body 1a facing up and down.

[0032] Next, in STEP3, as the number n of the angular position of the shielding member 1b of the radiation measuring device 1, the initial number "1" is set. Then, in STEP4, the rotary actuator 2 rotates the shielding member 1b around the central axis C so that it is disposed at the angular position of the n-th angle θn (the n-th angular position).

[0033] Next, in STEP5, with the shielding member 1b disposed at the n-th angular position, the intensity of the radiation incident on the measuring device main body 1a is measured by the measuring device main body 1a. This measured value is stored and held in a storage medium such as a memory (not shown) in association with the measurement location and the angular position of the shielding member 1b.

[0034] Next, in STEP6, it is determined whether or not the number n of the angular position of the shielding member 1b has reached a predetermined total number N. If the determination result is negative (when n < N), then in STEP7, after increasing the value of the number n by only "1", the process from STEP4 is repeated.

[0035] Also, when the determination result in STEP6 becomes affirmative by the number n of the angular position of the shielding member 1b reaching the total number N (when n ≥ N), then next, in STEP8, it is determined whether or not the number m of the measurement location has reached a predetermined total number M. And if the determination result in STEP8 is negative (when m < M), then in STEP9, after increasing the value of the number m by only "1", the process from STEP2 is repeated. Also, when the determination result in STEP8 becomes affirmative by the number m of the measurement location reaching the total number M (when m ≥ M), the measurement process of the radiation intensity is terminated.

[0036] As described above, by performing the measurement process of the radiation intensity, with the radiation measuring device 1 installed at each of the M measurement locations from the first measurement location to the M-th measurement location, the shielding member 1b is sequentially rotated around the central axis C of the measuring device main body 1a to the N angular positions from the first angular position to the N-th angular position, and at each angular position, the intensity of the radiation incident on the measuring device main body 1a is measured and stored and held.

[0037] Here, the multiple (N) angular positions from the first angular position to the Nth angular position can be set, for example, as shown in Fig. 3. In this case, for example, a reference angular position, which is an angular position of a reference orientation predetermined with respect to the measuring device main body 1a, is set as the first angular position. Then, each angular position from the second angular position to the Nth angular position is set as an angular position shifted around the central axis C by a predetermined increment of angle δθ from the reference angular position (= first angular position).

[0038] Therefore, the nth angular position (n=1, 2, ..., N) is set as an angular position rotated around the central axis C by an angle θn that corresponds to (n-1) δθ from the reference angular position. Note that the interval angle δθ between adjacent angular positions is set as the angle (=2π / N) obtained by dividing 2π [rad] (=360°) by the total number N of angular positions. Additionally, the reference angular position and the total number N of angular positions can be set arbitrarily in the design.

[0039] Furthermore, the multiple (M) measurement points from the first measurement point to the Mth measurement point can be set, for example, as shown in Fig. 4. In Fig. 4, the Z axis represents the coordinate axis in the up-down direction (the same direction as the central axis C of the measuring device main body 1a), and the X axis represents the coordinate axis in the direction perpendicular to the Z axis direction (the direction of the central axis C of the measuring device main body 1a installed at each measurement point).

[0040] 4, a reference height point, which is a point at a predetermined reference height (vertical position) at the location where the radiation measuring device 1 is installed, is set as the first measurement point. In this case, the height (vertical position) of the reference height point can be set in advance to be, for example, approximately the same height as the height of a radiation source that actually exists in the vicinity (e.g., ground level). However, it is more preferable to set the height of the reference height point to a height different from the height of an actually existing radiation source so that radiation can be easily measured even if the radiation source is located far away. Then, each of the measurement points from the second measurement point to the Mth measurement point is set as a point shifted in the Z-axis direction from the first measurement point as the reference height point by a predetermined increment height δz.

[0041] Therefore, the mth measurement point (m = 1, 2, ..., M) is set as a point displaced in the Z-axis direction from the reference height point by a distance zm, which is equal to (m-1) · δz. Note that the interval height δz and the total number M of measurement points can be set arbitrarily in design. Also, in FIG. 4, the measurement points other than the first measurement point are arranged above the first measurement point, but some of the measurement points other than the first measurement point may be set below the first measurement point, which is the reference height point.

[0042] 2, the rotation of the shielding member 1b by the rotary actuator 2 at each measurement point and the measurement of the radiation intensity may be performed manually by an operator, but the operation control of the rotary actuator 2 and the acquisition of the measurement data of the radiation intensity may be performed automatically using a control device or a computer. Also, the installation of the radiation measuring device 1 at each measurement point may be performed manually by an operator, but may also be performed automatically using a robot or the like.

[0043] Next, a process for estimating the spatial distribution of radiation sources using the measurement data of radiation intensity obtained by the measurement process described above will be described. In this embodiment, first, a process (hereinafter referred to as a first radiation source distribution estimation process) is performed to estimate, for each measurement point, a first intensity distribution, which is the intensity distribution of the radiation sources according to the orientation around the central axis C of the measurement device body 1a of the radiation measurement device 1 installed at each of the multiple measurement points. In this embodiment, the step of performing this first radiation source distribution estimation process corresponds to the first radiation source distribution estimation process in the present invention. The first radiation source distribution estimation process will be described below.

[0044] The first intensity distribution for each measurement point is expressed as a function f(θ) of angle θ around the central axis C of the measurement device body 1a of the radiation measurement device 1 installed at each measurement point. f(θ) is the intensity (intensity per unit angle) of radiation coming from a radiation source existing within a plane in the direction of angle θ toward the measurement device body 1a. Hereinafter, f(θ) will be referred to as the first intensity distribution function.

[0045] The relationship between this first intensity distribution function f(θ) and the intensity g(θ') of the radiation incident on the measurement device body 1a in accordance with the first intensity distribution function f(θ) when the shielding member 1b of the radiation measurement device 1 is disposed at an angle θ' around the central axis C with respect to the measurement device body 1a can be modeled by an integral equation in the form of a Fredholm integral equation, for example, as shown in the following equations (1) and (1)'. This integral equation corresponds to the first integral equation in this invention.

[0046]

number

[0047] Then, by dividing the integral interval on the right side of equation (1) into N parts and discretizing it, the following equation (2) is obtained. [gj]=(2π / N)·[Cjk]·[fk]……(2) j=1,2,…,N, k=1,2,…,N

[0048] Here, gj in formula (2) is the intensity of radiation incident on the measuring device main body 1a when the shielding member 1b is disposed at an angle θj around the central axis C of the measuring device main body 1a, and [gj] is the vertical vector (=[g1, g2, ..., gN]) that lists the radiation intensities g1 to gN corresponding to N different angles θ1 to θN of the arrangement position of the shielding member 1b. T ), fk is the function value of the first intensity distribution function f(θ) at angle θk (i.e., the intensity of the radiation traveling from the radiation source in the direction of angle θk toward the measurement device main body 1a), [fk] is a vertical vector (=[f1, f2, ..., fN]) that lists the function values ​​f1 to fN of the first intensity distribution function f(θ) corresponding to N angles θ1 to θN of the direction in which the radiation travels toward the measurement device main body 1a. T )

[0049] Furthermore, Cjk is a constant value (=0 or 1) defined by the proviso in formula (1) corresponding to the pair of angle θj of the arrangement position of shielding member 1b and angle θk of the incident direction of radiation to measuring device main body 1a, and [Cjk] is a matrix in which constant values ​​Cjk corresponding to each pair of N types of angles of the arrangement position of shielding member 1b and N types of angles of the incident direction of radiation are arranged vertically and horizontally (a matrix in which the value of the component in the jth row and kth column is Cjk). In this case, the matrix [Cjk] is a circulant matrix, and when the number of angles among the N types of angles where the constant value Cjk is 1 has no common divisor with N, its inverse matrix [Cjk] -1 Therefore, the following equation (3) is obtained from equation (2). [fk]=(N / 2π)·[Cjk] -1 ·[gj] …(3)

[0050] This formula (3) represents the calculation process of the inverse problem for estimating the first intensity distribution function f(θ). In the first radiation source distribution estimation process of this embodiment, for each measurement point (for each of the 1st to Mth measurement points), a plurality of function values ​​fk (function values ​​corresponding to a plurality of angles around the central axis C of the measurement device body 1a) of the first intensity distribution function f(θ) are calculated based on formula (3) from the measured values ​​of radiation intensities measured in STEP 3 corresponding to each of N types of angular positions (1st to Nth angular positions) of the shielding member 1b.

[0051] Specifically, for each measurement point (each of the first to Mth measurement points), the shielding member 1b is positioned at the first angular position, the second angular position, ..., the Nth angular position, and the radiation intensity measured by the measuring device main body 1a is used as the values ​​of g1, g2, ..., gN to calculate the right side of equation (3), thereby estimating the intensities f1, f2, ..., fN of the radiation coming toward the measuring device main body 1a from the directions of the first angle θ1, the second angle θ2, ..., the Nth angle θN. In this case, the inverse matrix [Cjk] of the circulant matrix [Cjk] -1is calculated in advance. As a result, the function values ​​of the first intensity distribution function f(θ) corresponding to each of the multiple angles θ1 to θN around the central axis C of the measuring device main body 1a are estimated. Note that the inverse matrix [Cjk] -1 The calculation of and the operation of equation (3) are performed using, for example, a computer, etc. In this embodiment, the first radiation source intensity estimation process is performed as described above.

[0052] As described above, after the first radiation source intensity distribution estimation process is performed, a process (hereinafter referred to as second radiation source distribution estimation process) is performed to estimate a second intensity distribution, which is the intensity distribution of the radiation source according to the position in each direction (more specifically, the distance from the measurement point in each direction), for each of a plurality of angles around the central axis C of the measurement device main body 1a. In this embodiment, the step of performing this second radiation source distribution estimation process corresponds to the second radiation source distribution estimation process in the present invention. The second radiation source distribution estimation process will be described below.

[0053] In this embodiment, as described above, the height (vertical position) of the reference height point serving as the first measurement point is set in advance to be approximately the same height as the height of the radiation sources actually present in the vicinity. Therefore, when attention is paid to radiation arriving toward the measurement device main body 1a from a direction at any one angle around the central axis C (hereinafter, this angle will be referred to as angle θa), the radiation sources emitting the radiation can be considered to be distributed on a straight line that passes through the reference height point (first measurement point) and extends in the direction of angle θa, which is the direction in which the radiation of interest arrives. For example, in Figure 4, the radiation sources can be considered to be distributed on the X-axis that passes through the reference height point (first measurement point).

[0054] In this case, the second intensity distribution in the direction of angle θa is expressed as a function ρ(θa,x) of the angle θa and the position x (distance x from the reference height) in the X-axis direction extending from the reference height in the direction of angle θa. The function ρ(θa,x) is the intensity (intensity per unit length in the X-axis direction) of the radiation output from a radiation source located at a distance x from the reference height in the direction of angle θa. Hereinafter, ρ(θa,x) will be referred to as the second intensity distribution function. In the following description, unless otherwise specified, the X-axis will be the coordinate axis extending from the reference height in the direction of angle θa around the central axis C at the same height as the reference height. In addition, ρ(θa,x) may also be simply referred to as ρ(x).

[0055] Here, the first intensity distribution function as seen from the mth measurement point, whose height from the reference height point (first measurement point) is zm (=(m-1)·δz), is expressed as f(θ,zm) as a function of the angle θ around the central axis C and the height zm. In other words, the intensity of radiation coming from a direction of an arbitrary angle θ relative to the radiation measuring device 1 at the mth measurement point at height zm is expressed as f(θ,zm).

[0056] In this case, the relationship between the second intensity distribution ρ(θa,x) in the direction of the one angle θa and the intensity f(θa,zm) of the radiation incident from a radiation source distributed in the direction of angle θa at the height (z1) of the reference height point (a radiation source distributed on the X-axis in the direction of angle θa) toward the m-th measurement point at height zm (in other words, the function value at angle θa of the first intensity distribution function f(θ,zm) seen from the m-th measurement point at height zm) can be modeled by an integral equation in the form of a Fredholm integral equation, as shown in the following equation (4): In this embodiment, this integral equation corresponds to the second integral equation in the present invention.

number

[0057] Then, when the integral interval on the right side of equation (4) is divided into M parts, the same number as the number of measurement points, and discretized, the following equation (5) is obtained. [fr]=[Hrs]·[ρs] ……(5) r=1,2,…,M, s=1,2,…,M

[0058] Here, fr in equation (5) is the function value of the first intensity distribution function f(θ, zr) at the angle θa seen at the rth measurement point (the intensity of radiation coming from the direction of angle θa at the rth measurement point), and [fr] is the vertical vector (=[f1, f2, ..., fM]) that lists the radiation intensities f1 to fM corresponding to each of the M measurement points (1st to Mth measurement points). T ), ρs is the function value of the second intensity distribution function ρ(θa, x) at a position (see Figure 4) that is a distance xs in the X-axis direction from the reference height point (first measurement point) (in other words, the intensity of the radiation output from the radiation source at a distance xs toward the radiation measuring device 1), [ρs] is a vertical vector (=[ρ1, ρ2, ..., ρM]) that lists the function values ​​ρ1 to ρM of the second intensity distribution function ρ(θa, x) that correspond to M distances x1 to xM from the reference height point (first measurement point), respectively. T )

[0059] Furthermore, [Hrs] is a matrix given by the following equation (6), and Hrs is the component of its rth row and sth column (=α·zr / (xs 2 +zr 2 ) 3 / 2 )

number

[0060] In this case, the matrix [Hrs] is a regular matrix, and its inverse matrix [Hrs] -1 Therefore, the following equation (7) is obtained from equation (5). [ρs] = [Hrs] -1 [fr] ……(7)

[0061] Equation (7) represents the calculation process of the inverse problem for estimating the second intensity distribution function ρ(θa, x) in the direction of an arbitrary angle θa around the central axis of the measuring device body 1a. In the second radiation source distribution estimation process of this embodiment, multiple function values ​​(function values ​​corresponding to multiple distances in the X-axis direction) of the second radiation source distribution function ρ(θa, x) in the direction of an arbitrarily selected angle θa around the central axis C of the measuring device body 1a are calculated based on Equation (7) from the function value f(θa, zm) at the angle θa of the first intensity distribution function f(θ, zm) (m = 1, 2, ..., M) estimated by the first radiation source distribution process for each of multiple measurement points (1st to Mth measurement points).

[0062] Specifically, for example, angles θ1 to θN corresponding to the first to Nth angular positions in the first radiation source estimation process are selected as angles θa for which the second intensity distribution is to be estimated. Furthermore, for each of angles θ1 to θN as angle θa, a plurality of (M) distances x1 to xM are set in the X-axis direction, which is the direction of angle θa, from the reference height point (first measurement point). In this case, distances x1 to xM are set, for example, to distances shifted at predetermined intervals from the reference height point in the X-axis direction. Note that the intervals of distances x1 to xM can be set arbitrarily by design.

[0063] Then, for each of the angles θ1 to θN as the angle θa, the function values ​​ρ(θa,x1) to ρ(θa,xM) of the second source distribution function ρ(θa,x) at each of the distances x1 to xM are calculated based on equation (7) from the function values ​​f(θa,z1) to f(θ,zM) at the angle θa of the first intensity distribution functions f(θ,z1) to f(θ,zM) estimated by the first source distribution processing corresponding to each of the first to Mth measurement points.

[0064] In this case, the vertical vector of function values ​​f(θa,z1) to f(θa,zM) corresponding to the first to Mth measurement points is used as [fr] on the right side of equation (7), and the vertical vector of function values ​​ρ(θa,x1) to ρ(θa,xM) of the second source distribution function ρ(θa,x) at distances x1 to xM is used as [ρs] on the left side of equation (7). Also, the inverse matrix [Hrs] of the matrix [Hrs] is -1 is calculated in advance. As a result, function values ​​ρ(θa,x1) to ρ(θa,xM) of the second intensity distribution function ρ(θa,x) at each of a plurality of distances x1 to xM from the reference height point are estimated for each of a plurality of angles θ1 to θN around the central axis C of the measuring device main body 1a. Note that the inverse matrix [Hrs] -1 The calculation of and the operation of equation (7) are performed using, for example, a computer, etc. In this embodiment, the second radiation source intensity estimation process is performed as described above.

[0065] Supplementally, in the above-described second radiation source estimation process, multiple function values ​​ρ(θa,x1) to ρ(θa,xM) of the second intensity distribution function ρ(θa,x) are estimated for each of multiple angles θ1 to θN around the central axis C of the measurement device main body 1a, i.e., for each angle changed by a predetermined angle increment δθ around the central axis C. However, the angle θa for estimating the function value of the second intensity distribution function ρ(θa,x) can be selected arbitrarily. For example, the function value of the second intensity distribution function ρ(θa,x) may be estimated only for one or more angles at which the function value of the first intensity distribution function f(θ) is relatively large at each measurement point.

[0066] In the second radiation source intensity estimation process, if the value of the determinant of the matrix [Hrs] is close to zero, the inverse matrix [Hrs] -1 In this case, it is difficult to accurately calculate the inverse matrix [Hrs] -1 If the calculation of the above formula (7) is performed using this as is, an error is likely to occur in the function value of the estimated second intensity distribution function ρ(θa,x).

[0067] In such a case, it is preferable to estimate the function value of the second intensity distribution function ρ(θa,x) using, for example, the Moore-Penrose inverse matrix (pseudo-inverse matrix). In this case, the calculation process for estimating the function value of the second intensity distribution function ρ(θa,x) using the Moore-Penrose inverse matrix instead of equation (5) is given by the following equation (8): [ρs]=(a·[I]+[Hrs] T [Hrs]) -1 [Hrs] T [fr] ……(8)

[0068] Here, in equation (8), a is a constant, [I] is the identity matrix, and (a·[I]+[Hrs] T [Hrs]) -1 [Hrs] T is the Moore-Penrose inverse matrix (pseudo-inverse matrix). In this case, by appropriately setting the value of the constant a taking into account the error in [fr], etc., it is possible to calculate multiple function values ​​ρ(θa,x1) to ρ(θa,xM) of the second intensity distribution function ρ(θa,x) so that the norm of [ρs] is minimized using equation (8). This makes it possible to estimate multiple function values ​​ρ(θa,x1) to ρ(θa,xM) with high reliability as function values ​​of the second intensity distribution function ρ(θa,x).

[0069] In this embodiment, as described above, a first radiation source distribution estimation process and a second radiation source distribution estimation process are executed. As a result, a first intensity distribution and a second intensity distribution of a radiation source are estimated. In this case, the first radiation source distribution estimation process can calculate the function value of the first intensity distribution function f(θ) using the inverse matrix of a circulant matrix, thereby enabling efficient estimation of the first intensity distribution. Furthermore, the second radiation source distribution estimation process can calculate the function value of the second intensity distribution function ρ(θa,x) in the direction of an arbitrary angle θa using the inverse matrix of the matrix [Hrs] or a Moore-Penrose inverse matrix corresponding to the matrix [Hrs], thereby enabling efficient estimation of the second intensity distribution.

[0070] Next, a simulation for verifying the accuracy of the first intensity distribution estimated by the first radiation source distribution estimation process and the accuracy of the second intensity distribution estimated by the second radiation source distribution estimation process will be described.

[0071] [Simulation 1] At any measurement point where the radiation measuring device 1 is installed, the actual (true) first intensity distribution function f(θ)in according to the angle (orientation) around the central axis C of the measuring device main body 1a is assumed to be a function expressed, for example, by the following equation (9), and at each of multiple angular positions of the shielding member 1b, the virtual measurement values ​​of the radiation intensity gn (n=1, 2, ..., N) measured by the measuring device main body 1a are set to include an error.

[0072] In this case, when the shielding member 1b is placed at the angular position of angle θn (nth angular position), the virtual measurement value of the radiation intensity gn measured by the measuring device main body 1a is calculated by multiplying the true value gn0 of the radiation intensity by the mean value μ and variance σ of predetermined values, as shown in the following equation (10): 2 The error N(μ,σ) is normally distributed 2 ) is added to the true value gn0 of the radiation intensity. The true value gn0 of the radiation intensity is calculated from the true first intensity distribution function f(θ)in by the following equation (11) based on equation (1).

number

[0073] Then, from the hypothetical measured values ​​of radiation intensities gn (n = 1, 2, ..., N), function values ​​of the first intensity distribution function f(θ) at multiple angles θn (n = 1, 2, ..., N) were calculated using equation (3), and the function values ​​were compared with the true value defined by the true first intensity distribution function f(θ)in. Hereinafter, the first intensity distribution function f(θ) estimated from the hypothetical measured values ​​will be referred to as f(θ)out.

[0074] 5 shows a graph of the true first intensity distribution function f(θ)in defined as above, and an example of function values ​​at multiple angles θm of the first intensity distribution function f(θ)out estimated by the first radiation source distribution estimation process from the radiation intensity measurement value gm including an error. In this example, the number of angle divisions (=N) in the first radiation source distribution estimation process is 50, the angle width θw1 of the shielding member 1b is an angle width (=3 δθ) three times the interval angle δθ (=2π / 50 [rad]), and the error N(μ, σ 2 ) mean value μ and variance σ 2 are 0.001, respectively.

[0075] 5, even if the measured value of the radiation intensity gn contains an error, the multiple function values ​​(function values ​​at multiple angles) of the first intensity distribution function f(θ)out obtained by the first radiation source distribution estimation process are calculated as values ​​that are roughly in line with the true first intensity distribution function f(θ)in. Therefore, it can be seen that the first intensity distribution function f(θ)out can be estimated with high accuracy by the first radiation source distribution estimation process.

[0076] [Simulation 2] Assume that the actual (true) second intensity distribution function ρ(x)in in the direction of an arbitrary angle θa is a function expressed by, for example, the following equation (12). From the multiple function values ​​of the second intensity distribution function ρ(x)in (function values ​​at each of multiple distances x1 to xM), the function values ​​fm (m = 1, 2, ..., M) of the first intensity distribution function f(θ) seen at each of multiple measurement points (measurement points 1 to M) in the direction of angle θa are calculated as true values ​​using equation (5). Note that exp( ) on the right-hand side of equation (12) represents an exponential function with the base e of the natural logarithm. Furthermore, μ and σ in equation (12) are constant values, e.g., μ = 10 and σ = 2. ρ(x)in=exp(-(x-μ) 2 / (2·σ 2 )) …(12)

[0077] Then, the function values ​​fm (m=1, 2, ..., M) calculated by the above-mentioned formula (5) are each truncated to a predetermined number of digits to obtain values ​​f'm (m=1, 2, ..., M), which are set as virtual estimated values ​​obtained by the first radiation source distribution estimation process, and the inverse matrix [Hrs] of the matrix [Hrs] is calculated from the virtual estimated values ​​f'm (m=1, 2, ..., M) by the above-mentioned formula (7). -1 (by a second radiation source distribution estimation process using (a)), a plurality of function values ​​of the second intensity distribution function ρ(x) (function values ​​at each of a plurality of distances x1 to xM) were calculated, and the function values ​​were compared with the true value defined by the true second intensity distribution function ρ(x)in. Hereinafter, the second intensity distribution function ρ(x) estimated from the virtual estimated value f'm of the function value of the first intensity distribution function will be denoted as ρ(x)out.

[0078] 6A and 6B are graphs of the true second intensity distribution function ρ(x)in defined as above, and the second source distribution estimation process (the inverse matrix [Hrs] of the matrix [Hrs]) from the virtual estimated value f'm of the first intensity distribution function. -1 6A and 6B show an example of function values ​​at a plurality of distances xm of the second intensity distribution function ρ(x)out estimated by the second radiation source distribution estimation process using the function (second radiation source distribution estimation process using the function (x)out). In this case, in Fig. 6A, the virtual estimated value f'm is a value obtained by rounding down the true value fm to the 13th decimal place, and in Fig. 6B, the virtual estimated value f'm is a value obtained by rounding down the true value fm to the 14th decimal place. In addition, the distance x on the horizontal axis of the graphs in Figs. 6A and 6B is a normalized distance obtained by dividing the distance in length units by a predetermined interval, and the number of divisions of the distance x (=M) is 30.

[0079] When the hypothetical estimated value f'm is the true value fm rounded down to the 13th decimal place (in other words, when the accuracy of f'm is low), the accuracy of the function value of the second intensity distribution function ρ(x)out tends to be low, as shown in Fig. 6A. However, when the hypothetical estimated value f'm is the true value fm rounded down to the 14th decimal place (in other words, when the accuracy of f'm is high), the function value of the second intensity distribution function ρ(x)out can be calculated with high accuracy, as shown in Fig. 6B. Therefore, it can be seen that by measuring the first intensity distribution function f(θ) with high accuracy, the second intensity distribution function ρ(x) can be estimated by the second source distribution processing using the matrix [Hrs].

[0080] [Simulation 3] As in Simulation 2, function values ​​fm (m = 1, 2, ..., M) of the first intensity distribution function f(θ) seen at each of multiple measurement points (the first to Mth measurement points) in the direction of angle θa were calculated using Equation (5) from multiple function values ​​of the second intensity distribution function ρ(x)in defined by Equation (12). Then, from this function value fm, multiple function values ​​(function values ​​at each of multiple distances x1 to xM) of the second intensity distribution function ρ(x) were calculated using Equation (8) (by second source distribution determination processing using the Moore-Penrose inverse matrix), and the function values ​​were compared with the true value defined by the true second intensity distribution function ρ(x)in.

[0081] Figure 7 shows a graph of the true second intensity distribution function ρ(x)in, defined in the same way as in Simulation 2, and an example of function values ​​at multiple distances xm of the second intensity distribution function ρ(x)out, estimated from the function value fm of the first intensity distribution function by a second source distribution determination process using a Moore-Penrose inverse matrix. In this case, the distance x on the horizontal axis of the graph in Figure 7 is a normalized distance, as in Figures 6A and 6B. The number of divisions of the distance x (= M) is 20.

[0082] As shown in Figure 7, the inverse matrix [Hrs] of the matrix [Hrs] -1It can be seen that the second intensity distribution function ρ(θ) can be estimated with high accuracy by the second source distribution estimation process using the Moore-Penrose inverse matrix, even without using the Moore-Penrose inverse matrix. It was also confirmed that when the Moore-Penrose inverse matrix is ​​used, the second intensity distribution function ρ(θ) can be estimated with high accuracy even if the number of divisions of the distance x is changed or even if an error is included in the first intensity distribution function f(θ).

[0083] [Second embodiment] Next, a second embodiment of the present invention will be described with reference to Fig. 8. A radiation measuring device 11 of this embodiment is a radiation measuring device configured so that its sensitivity for detecting radiation incident from a specific direction is higher than its sensitivity for detecting radiation incident from directions other than the specific direction. This radiation measuring device 11 includes a measuring device main body 11a that generates an output according to the intensity of the incident radiation, and a shielding member 11b that blocks radiation incident on the measuring device main body 11a from directions other than the specific direction. The measuring device main body 11a is the same as the measuring device main body 1a of the first embodiment.

[0084] In this embodiment, shielding member 11b is configured to cover the entire periphery of measuring device body 11a except for an area having a predetermined acute angle width θw2 around the central axis C of measuring device body 11a. Specifically, shielding member 11b is formed, for example, in a cylindrical shape having kerf 11b1, and is arranged around measuring device body 11a coaxially with central axis C. Kerf 11b1 extends between both ends of shielding member 11b in the same direction as central axis C, and has a constant acute angle width θw2 around central axis C.

[0085] Since the shielding member 11b is configured as described above, when the shielding member 11b is arranged around the measuring device main body 11a so that the cutting groove 11b1 is at an angle θ' around the central axis C, radiation coming toward the measuring device main body 11a from a direction in the angular range of θ' to θ' + θw2 (for example, the radiation illustrated by arrow Y3 in Figure 8) is not blocked by the shielding member 11b, but enters the measuring device main body 11a and is detected by the measuring device main body 11a.

[0086] On the other hand, radiation coming toward the measuring device main body 11a from directions other than the angular range of θ' to θ' + θw2 around the central axis C (for example, the radiation illustrated by arrow Y4 in Figure 8) is blocked from entering the measuring device main body 11a by the shielding member 11b and is hardly detected by the measuring device main body 11a.

[0087] As a result, the radiation measuring instrument 11 is configured so that the sensitivity for detecting radiation incident from a specific direction (a direction within an angular range of θ' to θ' + θw2 around the central axis C) is higher than the sensitivity for detecting radiation incident from a direction different from the specific direction. Note that, like the shielding member 1b of the first embodiment, the shielding member 11b can be driven to rotate about the central axis C by a rotary actuator such as an electric motor (not shown). Alternatively, the shielding member 11 may be designed to be manually rotatable. The shape of the shielding member 11b is not limited to a cylindrical shape and may be another shape.

[0088] In this embodiment, the measurement process for measuring the radiation intensity (dose or dose rate) by the radiation measurement instrument 11 is performed in the same manner as in the first embodiment (for example, according to the procedure shown in the flowchart in FIG. 2). Note that the interval angle δθ (the angle between adjacent angle positions) of the rotation angle of the shielding member 11b is set to, for example, the same angle width as the angle width θw2 of the kerf 11b1 of the shielding member 11b.

[0089] In this case, if the angle of kerf 11b1 at each angular position (1st to Nth angular positions) of shielding member 11b at each measurement point is defined as θn (n = 1, 2, ..., N), the measured value of the radiation intensity measured by measuring device main body 11a at each angle θn position, converted into a radiation intensity per unit angle around central axis C, corresponds to the function value of first intensity distribution function f(θ) at angle θn. Therefore, by measuring the radiation intensity at multiple angular positions (1st to Nth angular positions) of shielding member 11b at each measurement point, function values ​​f(θ1), f(θ2), ..., f(θN) of first intensity distribution function f(θ) corresponding to multiple angles θn (n = 1, 2, ..., N) of kerf 11b1 can be obtained.

[0090] In this embodiment, next, a second radiation source distribution estimation process is executed to estimate the second intensity distribution for each of a plurality of angles θn (n = 1, 2, ..., N) of the kerf 11b1 of the shielding member 11b (for each of a plurality of angular positions of the shielding member 11b). In this case, the method of the second radiation source distribution estimation process in this embodiment is the same as the method of the second radiation source distribution estimation process in the first embodiment. As a result, for each of a plurality of angles θ1 to θN around the central axis C of the measuring device body 11a, function values ​​ρ(θa,x1) to ρ(θa,xM) of the second intensity distribution function ρ(θa,x) (θa is each of θ1 to θN) at each of a plurality of distances x1 to xM in the X-axis direction from the reference height point are estimated.

[0091] As in the first embodiment, the angle θa at which the function value of the second intensity distribution function ρ(θa,x) is estimated can be selected arbitrarily. For example, the function value of the second intensity distribution function ρ(θa,x) may be estimated only for one or more angles at which the function value of the first intensity distribution function f(θ) is relatively large at each measurement point.

[0092] Supplementally, in this embodiment, the step of measuring the radiation intensity at a plurality of angular positions (1st to Nth angular positions) of the shielding member 11b at each measurement point corresponds to the Ath measurement step in the present invention. Also, in this embodiment, the second intensity distribution corresponds to the Ath intensity distribution in the present invention, and the step of executing the second radiation source distribution estimation process corresponds to the Ath radiation source distribution estimation step in the present invention. Also, the above-mentioned formula (4) related to the second radiation source distribution estimation process corresponds to the Ath integral equation in the present invention.

[0093] In this embodiment, as described above, the first intensity distribution is estimated from the measured values ​​of the radiation intensity for each of the multiple angles θ1 to θN at each of the multiple measurement points, and then the second intensity distribution is estimated by the second radiation source distribution estimation process. In this case, the second radiation source distribution estimation process can use the inverse matrix of the matrix [Hrs] or the Moore-Penrose inverse matrix corresponding to the matrix [Hrs] to find the function value of the second intensity distribution function ρ(θa,x) in the direction of an arbitrary angle θa, so that the second intensity distribution can be estimated efficiently.

[0094] [Third embodiment] Next, a third embodiment of the present invention will be described with reference to FIGS. 9 to 11 as a variation of the second intensity distribution. Referring to FIG. 9, a radiation measuring device 111 of this embodiment is configured so that the sensitivity for detecting radiation incident from a specific direction is higher than the sensitivity for detecting radiation incident from directions other than the specific direction. This radiation measuring device 111 includes a measuring device main body 111a that generates an output according to the intensity of the incident radiation, and a shielding member 111b that blocks radiation incident on the measuring device main body 111a from directions other than the specific direction. The configuration of the measuring device main body 111a is the same as that of the measuring device main body 1a of the first embodiment. Note that FIG. 9 shows the radiation measuring device 111 as viewed from a direction perpendicular to the central axis C of the measuring device main body 1a of the first embodiment.

[0095] In this embodiment, the shielding member 111b is configured to cover the entire periphery of the measuring device body 111a except for an area having a predetermined angular width θw3 around the central axis C2 of the measuring device body 111a. Specifically, the shielding member 111b is formed, for example, in a cylindrical shape having a kerf 111b1 and is arranged coaxially around the central axis C2 around the measuring device body 111a. The kerf 111b1 extends between both ends of the shielding member 111b in the same direction as the central axis C2 and has a certain angular width θw3 around the central axis C2. Note that the central axis C2 of the measuring device body 111a in this embodiment is an axis perpendicular to the central axis C in the first embodiment.

[0096] Since the shielding member 111b is configured as described above, when the shielding member 111b is arranged around the measuring device main body 111a so that the cutting groove 111b1 is at an angle θ'' around the central axis C2, radiation coming toward the measuring device main body 111a from a direction in the angular range of θ'' to θ''+θw3 (for example, the radiation illustrated by arrow Y5 in Figure 9) is not blocked by the shielding member 111b, but enters the measuring device main body 111a and is detected by the measuring device main body 111a.

[0097] On the other hand, radiation coming toward the measuring device main body 111a from directions other than the angular range of θ'' to θ''+θw3 around the central axis C2 (for example, the radiation illustrated by arrow Y6 in Figure 9) is blocked from entering the measuring device main body 111a by the shielding member 111b and is hardly detected by the measuring device main body 111a.

[0098] As a result, the radiation measuring instrument 111 is configured so that the sensitivity for detecting radiation incident from a specific direction (a direction within an angular range of θ'' to θ''+θw3 around the central axis C2) is higher than the sensitivity for detecting radiation incident from a direction different from the specific direction. Note that, like the shielding member 111b of the first embodiment, the shielding member 111b can be driven to rotate about the central axis C2 by a rotary actuator such as an electric motor (not shown). Alternatively, the shielding member 111 may be designed to be manually rotatable. The shape of the shielding member 111b is not limited to a cylindrical shape and may be another shape.

[0099] Supplementally, the angular width θw3 of the kerfs 111b1 of the shielding member 111b may be the same as the angular width θw2 of the kerfs 11b1 of the shielding member 11b of the second embodiment. Furthermore, the central axis C2 of the measuring body 111a of the present embodiment may be set to the same axis as the central axis C of the measuring body 1a of the first embodiment (or the measuring body 11a of the second embodiment). Therefore, the radiation measuring device 111 of the present embodiment may be a measuring device with the same configuration as the radiation measuring device 11 of the second embodiment.

[0100] Next, a measurement process for measuring radiation intensity (dose or dose rate) using the radiation measurement device 111 configured as described above will be described. In this embodiment, the radiation measurement devices 111 are sequentially installed at a plurality of measurement points that have a constant height (vertical position) and different horizontal positions (positions in the same direction as the central axis C2) with the central axis C2 of the measurement device body 111a facing laterally (horizontal or approximately horizontally). For example, as illustrated in FIG. 10, a plurality of measurement points (first to Mth measurement points) are set to be aligned at a constant height z0 in a line at equal intervals in the same direction as the central axis C2 (the X-axis direction in FIG. 10), and the radiation measurement devices 111 are sequentially installed at each of the measurement points.

[0101] 10, the arrangement direction of the multiple measurement points (the direction of the central axis C2 at each measurement point) is defined as the X-axis direction, the up-down direction (height direction) as the Z-axis direction, and the horizontal direction perpendicular to the X-axis and Z-axis as the Y-axis direction, and the position (height) of each measurement point in the Z-axis direction is defined as z0, and its position in the Y-axis direction is defined as zero. In this case, the plane where the Z-axis position is zero (the plane where z=0) is the distribution plane of the radiation source (for example, the ground).

[0102] Then, at each measurement point, the shielding member 111b is rotated to a plurality of angular positions with different angles around the central axis C2 (in other words, the orientation of the shielding member 111b around the central axis C2 as seen from the measuring device main body 111a is oriented in each of a plurality of orientations), and the intensity (dose or dose rate) of the radiation incident on the measuring device main body 111a through the cutting groove 111b1 of the shielding member 111b is measured by the measuring device main body 111a.

[0103] As in the first embodiment, this measurement process is performed, for example, according to the procedure shown in the flowchart of Fig. 2. Note that the interval angle δθ (the angle between adjacent angle positions) of the rotation angle of the shielding member 111b at each measurement point is set to, for example, the same angle width as the angle width θw3 of the kerf 111b1 of the shielding member 111b.

[0104] Next, the spatial distribution of the radiation source is estimated using the measurement data of the radiation intensity obtained by the above measurement process. In this case, in this embodiment, by measuring the radiation intensity at each measurement point, function values ​​f(θ1), f(θ2), ..., f(θN) of the first intensity distribution function f(θ) corresponding to multiple angles θn (n = 1, 2, ..., N) of the kerf 111b1 are obtained, as in the second embodiment. However, in this embodiment, the first intensity distribution function f(θ) is a function that represents the intensity distribution (first intensity distribution) of the radiation source according to the orientation around the lateral central axis C2 at each measurement point. In the following description, as shown in FIG. 10, the X-axis position of each measurement point (position in the direction of the central axis C2) will be denoted as xm (m = 1, 2, ..., M), and the first intensity distribution function f(θ) measured at the measurement point at the X-axis position xm will be denoted as f(θ, xm).

[0105] In the present embodiment, next, for each of one or more angles (hereinafter referred to as target angles θb) arbitrarily selected from angles around the central axis C2, a process (second radiation source distribution estimation process) is executed to estimate a second intensity distribution, which is the intensity distribution of the radiation source according to the position in the direction of each target angle θb. In this case, as the target angle θb, for example, each of a plurality of angles at which the kerfs 111b1 of the shielding member 111b face the radiation source distribution plane (the plane at z=0 in FIG. 10) may be selected, or an angle at which the function value of the first intensity distribution function f(θ) is relatively large may be selected from the plurality of angles.

[0106] More specifically, the second intensity distribution in the direction of each target angle θb is an intensity distribution of the radiation source according to the position on the intersection between the plane (plane including the central axis C2) whose angle around the central axis C2 at each measurement point coincides with the target angle θb and the distribution surface of the radiation source (in FIG. 10, for example, the position in the X-axis direction on line L).

[0107] 10, the distribution surface of the radiation source is assumed to be a plane at z=0, and the intensity distribution of the radiation source on this distribution surface is expressed as ρ(x, y). In this case, the relationship between the intensity of the radiation incident on measurement device main body 111a from the direction of target angle θb at the m-th measurement point (m=1, 2, ..., M) at a height of z0 (in other words, the function value f(θb, xm) of the first intensity distribution f(θ, xm)) and the intensity distribution ρ(x, y) of the radiation source on the distribution surface (on the plane at z=0) can be approximated by the integral equation of the following formula (13) (an integral equation in the form of a Fredholm integral equation):

number

[0108] In this embodiment, the integral equation of the above formula (13) is an integral equation that models the relationship between the second intensity distribution ρ(x, z0 tan θ) of the radiation source distributed on the distribution plane (plane where z=0) in the direction of the target angle θb and the intensity f(θb, xm) of the radiation traveling from the radiation source toward the radiation measuring device 111 installed at the m-th measurement point. This integral equation corresponds to the second integral equation in the present invention.

[0109] If the range in the X-axis direction where the radiation source exists is bounded, the integral equation of the following equation (14) can be obtained from equation (13). In equation (14), the range in the X-axis direction where the radiation source exists is set to the range [-R, R] (R is a constant).

number

[0110] 10, the range of positions in the X-axis direction of the measurement points where radiation measuring device 111 is installed (range xm), in other words, the range in which radiation measuring device 111 is moved, is set to the range [xmid-Rag / 2, xmid+Rag / 2], and xmid and Rag are set so that this range [xmid-Rag / 2, xmid+Rag / 2] is included in the range [-R, R] in the X-axis direction where the radiation source is located. In this case, the integral intervals of Equation (14) for each of the first to Mth measurement points are discretized into M equations, and the resulting equations are combined for all measurement points into M simultaneous equations, resulting in the following Equation (15), which has the same form as Equation (5) above.

number

[0111] In this embodiment, the second intensity distribution function ρ(x, z0 · tan θb) in the direction of the target angle θb is estimated by calculating an inverse problem based on the above equation (15) from the function value f(θb, xm) at the target angle θb of the first intensity distribution function f(θ, xm) (m = 1, 2, ..., M) measured at each of the first to M measurement points. This calculation of the inverse problem is performed in the same manner as in the first embodiment, using an inverse matrix or pseudo-inverse matrix (Moore-Penrose inverse matrix) of the matrix [Hrs']. In this embodiment, the second radiation source distribution estimation process is performed as described above. In this case, the second radiation source distribution estimation process can obtain the function value of the second intensity distribution function ρ(x, z0 · tan θb) in the direction of the required target angle θb using the inverse matrix or pseudo-inverse matrix (Moore-Penrose inverse matrix) of the matrix [Hrs'], so that the second intensity distribution can be estimated efficiently.

[0112] Next, a simulation for verifying the accuracy of the second intensity distribution estimated by the second radiation source distribution estimation process in this embodiment will be described. [Simulation 4] It is assumed that the actual (true) second intensity distribution function ρ(x)in (=ρ(x, z0 tan θb)) in the direction of an arbitrary target angle θa is a function expressed by a composite function of multiple normal distribution functions, and from the multiple function values ​​of the second intensity distribution function ρ(x)in (function values ​​at each of the X-axis positions x1 to xM), the function values ​​fr' (r=1, 2, ..., M) of the first intensity distribution function f(θ, xm) seen at each of the multiple measurement points (1st to Mth measurement points) in the direction of the target angle θb are calculated as true values ​​using equation (15).

[0113] Then, the function values ​​fr' (r = 1, 2, ..., M) calculated by the above formula (15) are each truncated to a predetermined number of digits to obtain values ​​fr'' (r = 1, 2, ..., M), which are set as hypothetical estimated values ​​obtained at each measurement point. From these hypothetical estimated values ​​fr'' (r = 1, 2, ..., M), the inverse matrix [Hrs'] of the matrix [Hrs'] in formula (15) is calculated. -1 By the second radiation source distribution estimation process using the above, multiple function values ​​of the second intensity distribution function ρ(x, z0 tan θb) (function values ​​at multiple positions x1 to xM in the x-axis direction) were calculated, and the function values ​​were compared with the true value defined by the true second intensity distribution function ρ(x)in. Hereinafter, the second intensity distribution function ρ(x, z0 tan θb) estimated from the virtual estimated value fr'' of the function value of the first intensity distribution function will be denoted as ρ(x)out.

[0114] Figure 11 shows the graph of the true second intensity distribution function ρ(x)in defined above and the virtual estimated value fr'' of the first intensity distribution function, which is used to estimate the second source distribution (the inverse matrix [Hrs'] of the matrix [Hrs]). -1 11 illustrates an example of function values ​​(function values ​​at multiple positions in the X-axis direction) of the second intensity distribution function ρ(x)out estimated by the second radiation source distribution estimation process using (second radiation source distribution estimation process using). In this case, the height z0 of each measurement point is 20, the target angle θb is π / 3 [rad], the number of measurement points is 100, and the variable range Rag of the measurement points in the X-axis direction and its central position xmid are 80 and 50, respectively. As can be seen from FIG. 11, in this embodiment as well, it is possible to accurately calculate the function values ​​of the second intensity distribution function ρ(x)out.

[0115] The present invention is not limited to the first to third embodiments described above, and other embodiments may also be employed. For example, the first intensity distribution may be estimated using a method different from that of the first or second embodiment. In this case, for example, if the measuring device main body 1a, 11a, or 111a is configured to have high directivity in a specific direction, it is also possible to estimate the first intensity distribution by measuring the intensities of radiation coming from each of a plurality of angles around the central axis C or C2 without using the shielding member 1b, 11b, or 111b. Furthermore, if necessary, the first radiation source distribution estimation process may be performed while the second radiation source distribution estimation process is omitted.

[0116] In the third embodiment, the radiation measurement instrument 111 is a measurement instrument having the same shielding member 111b as in the second embodiment, but may be provided with a shielding member having a structure similar to that of the oblique side member 1b in the first embodiment instead of the shielding member 111b. Then, by executing the same first radiation source distribution estimation process as in the first embodiment, the first intensity distribution f(θ, xm) at each measurement point at height z0 may be estimated. [Explanation of symbols]

[0117] 1,11,111...radiation detector.

Claims

1. a measuring step of using a radiation measuring device configured so that its sensitivity in detecting radiation incident from a specific direction is lower or higher than its sensitivity in detecting radiation incident from a direction different from the specific direction, and measuring the intensity of radiation incident on the radiation measuring device with the radiation measuring device in each of a plurality of arrangement states in which the specific direction of the radiation measuring device or the position of the radiation measuring device differs from one another; a radiation source distribution estimation step of estimating the intensity distribution of the radiation source using a calculation process for an inverse problem constructed based on a model that expresses a relationship between an intensity distribution of a radiation source according to a relative orientation or position with respect to the radiation measurement device and the intensity of radiation incident on the radiation measurement device according to the intensity distribution by an integral equation in the form of a Fredholm integral equation, and a radiation source distribution estimation step of estimating the intensity distribution of the radiation source using measured values ​​of the radiation intensity obtained in the measurement step, the radiation measuring device is configured so that sensitivity for detecting radiation incident from a specific direction is lower than sensitivity for detecting radiation incident from a direction different from the specific direction, the measuring step includes a first measuring step of measuring, by the radiation measuring device, an intensity of radiation incident on the radiation measuring device in a plurality of arrangement states in which the specific direction of the radiation measuring device is different from one another around a predetermined axis of the radiation measuring device; the radiation source distribution estimation step includes: a first radiation source distribution estimation step of estimating the first intensity distribution from: an inverse problem calculation process constructed based on a model that expresses, by a first integral equation in the form of a Fredholm integral equation, a relationship between a first intensity distribution, which is an intensity distribution of a radiation source according to a relative orientation around the predetermined axis with respect to the radiation measurement device, and an intensity of radiation incident on the radiation measurement device according to the first intensity distribution; and measurement values ​​of the radiation intensity obtained in the first measurement step.

2. 2. The radiation measurement method according to claim 1, a calculation process for solving the inverse problem in the first radiation source distribution estimation step, the calculation process being constructed to estimate the first intensity distribution using an inverse matrix of a circulant matrix obtained by discretizing a model represented by the first integral equation.

3. 3. The radiation measurement method according to claim 2, the measuring step is a step of placing the radiation measuring device at each of a plurality of positions different from one another in the predetermined axial direction, and performing the first measuring step for each of the plurality of positions; the radiation source distribution estimation step, after performing the first radiation source distribution estimation step for each of the plurality of positions, further includes: a computational process of an inverse problem constructed based on a model expressing, by a second integral equation in the form of a Fredholm integral equation, a relationship between a second intensity distribution, which is an intensity distribution of a radiation source according to a position in a predetermined direction arbitrarily selected from directions around the predetermined axis, and an intensity of radiation incident on the radiation measurement device from the predetermined direction according to the second intensity distribution; and a second radiation source distribution estimation step of estimating the second intensity distribution from the intensity of radiation in the predetermined direction defined by the first intensity distribution estimated in the first radiation source distribution estimation step, wherein the computational process of the inverse problem in the second radiation source distribution estimation step is a computational process constructed to estimate the second intensity distribution using an inverse matrix of a matrix obtained by discretizing the model expressed by the second integral equation, or a Moore-Penrose inverse matrix defined in accordance with the matrix.

4. A measurement process using a radiation measuring device configured so that its sensitivity in detecting radiation incident from a specific direction is lower or higher than its sensitivity in detecting radiation incident from a direction different from the specific direction, and measuring the intensity of radiation incident on the radiation measuring device with the radiation measuring device while the radiation measuring device is positioned in each of a plurality of positions in which the specific direction of the radiation measuring device or the position of the radiation measuring device are different from each other; a radiation source distribution estimation step of estimating the intensity distribution of the radiation source using a calculation process for an inverse problem constructed based on a model that expresses a relationship between an intensity distribution of a radiation source according to a relative orientation or position with respect to the radiation measurement device and the intensity of radiation incident on the radiation measurement device according to the intensity distribution by an integral equation in the form of a Fredholm integral equation, and a radiation source distribution estimation step of estimating the intensity distribution of the radiation source using measured values ​​of the radiation intensity obtained in the measurement step, the radiation measuring device is configured so that sensitivity for detecting radiation incident from a specific direction is lower than sensitivity for detecting radiation incident from a direction different from the specific direction, the measurement step includes a first measurement step of measuring, by the radiation measurement device, the intensity of radiation incident on the radiation measurement device in a state where the radiation measurement device is arranged in each of a plurality of arrangement states where the specific direction of the radiation measurement device is different from one another around a predetermined axis of the radiation measurement device, and a step of performing the first measurement step for each of a plurality of positions in a state where the radiation measurement device is arranged in each of a plurality of positions which are different from one another in the predetermined axial direction, The radiation source distribution estimation step includes a first radiation source distribution estimation step of estimating a first intensity distribution, which is an intensity distribution of the radiation source according to a relative orientation around the predetermined axis with respect to the radiation measuring device, from the measured values ​​of the radiation intensity obtained in the first measurement step, for each of the plurality of positions, and a second intensity distribution, which is an intensity distribution of the radiation source according to a position in a predetermined direction arbitrarily selected from directions around the predetermined axis, and a first intensity distribution estimation step of estimating a relationship between the second intensity distribution, which is an intensity distribution of the radiation source according to a position in a predetermined direction arbitrarily selected from directions around the predetermined axis, and the intensity of the radiation incident on the radiation measuring device from the predetermined direction. and a second radiation source distribution estimation step of estimating the second intensity distribution from the radiation intensity in the predetermined direction defined by the first intensity distribution estimated in the first radiation source distribution estimation step, wherein the calculation of the inverse problem in the second radiation source distribution estimation step is carried out by using an inverse matrix of a matrix obtained by discretizing the model represented by the second integral equation or a Moore-Penrose inverse matrix defined in accordance with the matrix. A radiation measurement method, characterized in that the calculation process is constructed to estimate the second intensity distribution.

5. A measurement process using a radiation measuring device configured so that its sensitivity in detecting radiation incident from a specific direction is lower or higher than its sensitivity in detecting radiation incident from a direction different from the specific direction, and measuring the intensity of radiation incident on the radiation measuring device with the radiation measuring device while the radiation measuring device is positioned in each of a plurality of positions where the specific direction of the radiation measuring device or the position of the radiation measuring device are different from each other; a radiation source distribution estimation step of estimating the intensity distribution of the radiation source using a calculation process for an inverse problem constructed based on a model that expresses a relationship between an intensity distribution of a radiation source according to a relative orientation or position with respect to the radiation measurement device and the intensity of radiation incident on the radiation measurement device according to the intensity distribution by an integral equation in the form of a Fredholm integral equation, and a radiation source distribution estimation step of estimating the intensity distribution of the radiation source using measured values ​​of the radiation intensity obtained in the measurement step, the radiation measuring device is configured so that sensitivity for detecting radiation incident from a specific direction is higher than sensitivity for detecting radiation incident from a direction different from the specific direction, the measurement step includes a measurement step A of measuring the intensity of radiation incident on the radiation measurement device with the radiation measurement device in a state where the radiation measurement device is disposed so that the specific direction coincides with an arbitrarily selected predetermined direction among directions around a predetermined axis of the radiation measurement device, the radiation source distribution estimation step includes an A-th radiation source distribution estimation step of estimating the A-th intensity distribution from an inverse problem calculation based on a model that expresses, by an A-th integral equation in the form of a Fredholm integral equation, a relationship between an A-th intensity distribution, which is an intensity distribution of a radiation source depending on a position in the predetermined direction among directions around the predetermined axis, and the intensity of radiation that enters the radiation measuring device from the predetermined direction depending on the A-th intensity distribution, and measured values ​​of the radiation intensity obtained in the A-th measurement step, and the inverse problem calculation in the A-th radiation source distribution estimation step is a calculation process that is implemented to estimate the A-th intensity distribution using an inverse matrix of a matrix obtained by discretizing the model expressed by the A-th integral equation, or a Moore-Penrose inverse matrix defined in accordance with the matrix.

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