JP2006285667A - Fast interpolation of absorption coefficient in radiative transfer calculation - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a method for fast interpolation of an absorption coefficient in radiative transfer calculation. <P>SOLUTION: The function value interpolation method for interpolating values of a function mapping points in a three-dimensional space to points in a one-dimensional space by means of a computer has the steps of rearranging lattice point data in a two-dimensional lattice space as a part of the three-dimensional space in a one-dimensional column direction by a predetermined method, and generating a two-dimensional matrix for calculation having point sequence data on the other dimension in the three-dimensional space as the other dimension of the two-dimensional matrix for calculation. By the singular value decomposition of the two-dimensional matrix for calculation, 10% of larger singular values of a diagonal matrix of singular values approximate the lattice point data in the two-dimensional lattice space to calculate interpolation coefficients. <P>COPYRIGHT: (C)2007,JPO&INPIT

Description

  The present invention relates to a function interpolation method, and more particularly to an interpolation method and apparatus for performing interpolation by mapping from a three-dimensional space to a one-dimensional space.

  As a function interpolation method, for example, linear interpolation, spline interpolation, and many other interpolation methods are known. Interpolation is a method of calculating points between function values representing discretized numerical values, and the amount of calculation increases in proportion to the amount of data to be interpolated. In particular, in calculations that perform interpolation of a large amount of data, the amount of calculation becomes enormous, and it is difficult to perform calculations within a practical time. An example of such a calculation is atmospheric radiative transfer calculation.

  The atmospheric radiative transfer calculation is a calculation for observing the spectrum of light transmitted through the atmosphere to determine the concentration of the components in the atmosphere, and it is necessary to calculate the processes of absorption and scattering received by the light transmitted through the atmosphere. The calculation of the absorption coefficient uses three-dimensional data defined by air temperature × atmospheric pressure × wave number.

Conventionally, the absorption coefficient necessary for the radiative transfer calculation is converted into a numerical table of the spline interpolation coefficient at atmospheric pressure × temperature, and for example, this numerical table is input to a database or the like and is interpolated as necessary.
JP 2003-248664 A

  However, in such a method, if the wavelength region to be calculated and the number of points to be interpolated increase, the amount of calculation becomes enormous, and even if a workstation or the like is used, it takes several days to calculate. This is the same when applying a radiative transfer calculation, for example, calculating a measurement result and specifying a greenhouse gas, and it cannot be said that it is a practical calculation time. Therefore, a method for performing such interpolation calculation as fast as possible is desired.

  In view of the above problems, the present invention is a function value interpolation method for interpolating, using a computer, the value of a function that maps from a point in a three-dimensional space defined by temperature × atmospheric pressure × wave number to a point in a one-dimensional space, Rearranging lattice point data of a two-dimensional lattice space, which is a part of the three-dimensional space, in a one-dimensional column direction by a predetermined method, and calculating point sequence data of another dimension of the tertiary space A step of generating a two-dimensional matrix for calculation as another dimension of the two-dimensional matrix, performing singular value decomposition of the two-dimensional matrix for calculation, and 10% from the top in descending order of singular values in the singular value diagonal matrix Providing a method for calculating the interpolation coefficient by applying it to the lattice point data in the two-dimensional lattice space using the singular values before and after and the column part or row part of the other two matrices around 10%. To do. As a result, the amount of data required for the calculation can be greatly reduced, and high-speed interpolation calculation is possible.

に対して要素の平均値

に最も近い要素ａ_ｉ，ｊ又は該要素の中央値又は該要素の最大値を標準要素とし、
（２）該標準要素を用いて、

を、ｌを１，２，３，・・と順次求め、求める要求精度（有効桁数）ｄに対してｄ＜ｄ_ｌとなるｌが暫定の近似範囲であり、特異値行列の成分において降順に先頭からｌ個の成分が近似範囲となる。
（３）上記（１）、（２）を計算用２次元行列の全列に対して行い、最大のｌを近似範囲とする。 Further, the predetermined ratio can be obtained as follows.
A calculation for a two-dimensional matrix represents a U orthogonal matrices between columns are orthogonal, sigma singular value matrix, the components of said specific value matrix Σ σ _1> σ _2> σ in ₃ ···> _{σ n,} V ^In a singular value decomposition A = U · Σ · V ^t , where ^t is a rotation matrix,
(1) Column vector of the calculation two-dimensional matrix A

Mean of elements for

The element a _{i, j} closest to or the median value of the element or the maximum value of the element as a standard element,
(2) Using the standard element,

L is sequentially obtained as 1, 2, 3,..., And l is a provisional approximation range where d < _dl with respect to the required accuracy (number of significant digits) d to be obtained, and descending order in the components of the singular value matrix L components from the top are approximate ranges.
(3) The above (1) and (2) are performed on all the columns of the calculation two-dimensional matrix, and the maximum l is set as the approximation range.

  The two-dimensional lattice space is an oblique lattice, and a method is provided in which the lattice space is divided into two or more regions and calculation is performed for each region. Furthermore, a computer program capable of performing the above-described method and a computing device having the program are provided.

  The present invention will be described below with reference to the drawings. First, the data of the absorption coefficient is data 1 in a three-dimensional space in which data of atmospheric pressure × temperature is arranged for each wave number as shown in FIG. FIG. 1B shows the temperature × barometric pressure data extracted for a certain wave number. As can be seen from FIG. 1B, an oblique grid is used, which is divided into three sections. In the present embodiment, it is divided into three sections, but the present invention is not limited to this, and those skilled in the art can divide into appropriate sections as necessary.

  Next, two-dimensional data of temperature × atmospheric pressure is mapped to a one-dimensional space for each section. As shown in FIG. 2, one oblique section is extracted, and the first vertical data string 3 of the lattice points in the section is arranged in the column direction of the matrix. Similarly, the adjacent second vertical data row 4 is continuously arranged in the row direction. That is, the vertical data string is sequentially extracted as one unit and arranged in the column direction. This process is repeated, and the vertical data strings are sequentially arranged as a third vertical data string 5, a fourth vertical data string 6, and so on.

  In this way, the two-dimensional data of the lattice points in one oblique section can be mapped as one-dimensional data in the column direction of the matrix. A two-dimensional number table is created by performing this for all data arranged in the wave number direction. When data is rearranged in one oblique space in this way, the absorption coefficient number table 7 is generated. It is obvious to those skilled in the art that the number of elements in the column direction is a matrix in which the number of elements in the row direction is the number of grid points in the oblique section and the number of elements in the row direction is the number of data in the wavenumber axis direction. . That is, the absorption coefficient number table 7 can be restated as an absorption coefficient matrix.

但し、Ａは吸収係数行列、Ｕは列間は直交する直交行列、Σは特異値行列（σ₁＞σ₂＞・・＞σ_n）、Ｖ^ｔは回転行列である。
ここで特異値対角行列Σに注目すると、Σの対角要素を特異値の降順に並べて見ると図３のようなグラフになる。グラフから明らかなように特異値は急激に減少している。該グラフによると第２の領域９では特異値の値はほぼ一定となっており、さらに第１の領域８の値と比較すると１桁以上小さい値となっている。
従来のＰＴ（気圧：Ｐ，気温：Ｔ）テーブルを用いた計算（例えば特開２００３−２４８６７４号公報参照）では、波数点断面（（気圧×気温）格子点）の格子点値の４次スプライン補間係数を例えばデータベース等に記憶しておき、任意の気圧×気温点に対し全波数点について補間係数を計算していた。
補間のベースになる気圧×気温格子点の断面が、上記特異値分解結果の直交行列Ｖ^ｔの列方向に対応する。Ｖ^ｔの列を元の２次元気圧×気温格子に戻し、２次元スプライン補間係数を計算しておき、任意の気圧×気温に対応する列ベクトルＶ（Ｐ，Ｔ）を補間すれば、格子点値を高速で計算出来ることになる。 Next, singular value decomposition is performed on the absorption coefficient number table 7. Although known to those skilled in the art, the singular value decomposition is expressed by the following equation.

However, A is an absorption coefficient matrix, U is an orthogonal matrix in which columns are orthogonal, Σ is a singular value matrix (σ ₁ > σ ₂ >...> Σ _n ), and V ^t is a rotation matrix.
If attention is paid to the singular value diagonal matrix Σ, the diagonal elements of Σ are arranged in descending order of singular values, and the graph is as shown in FIG. As is apparent from the graph, the singular value decreases rapidly. According to the graph, the value of the singular value is almost constant in the second region 9 and is smaller by one digit or more than the value of the first region 8.
In a calculation using a conventional PT (atmospheric pressure: P, temperature: T) table (see, for example, Japanese Patent Application Laid-Open No. 2003-248664), a quadratic spline of lattice point values of a wavenumber point cross section ((atmosphere × temperature) lattice point). The interpolation coefficient is stored in, for example, a database, and the interpolation coefficient is calculated for all wave numbers for an arbitrary atmospheric pressure × temperature point.
Cross-section of pressure × temperature lattice point as the base of the interpolation corresponds to the column direction of the orthogonal matrix V ^t of the singular value decomposition results. If the column of V ^t is returned to the original two-dimensional atmospheric pressure × temperature grid, a two-dimensional spline interpolation coefficient is calculated, and a column vector V (P, T) corresponding to an arbitrary atmospheric pressure × temperature is interpolated, the grid point The value can be calculated at high speed.

The inventor of the present application has found that the data of the absorption coefficient number table 7 can be approximated with sufficient accuracy when the calculation is performed using only the data of the first region 8. That is, since the absorption coefficient number table 7 can be calculated with sufficient accuracy by calculating only the data in the first region 8, the amount of calculation can be greatly reduced, and interpolation calculation can be performed at high speed. In addition, when the absorption coefficient is calculated by a table lookup method, the amount of data stored in the database can be reduced. Here, the data in the first area 8 is about 10% of the entire data. The data corresponding to the first region 8 corresponds to a range surrounded by a dotted line in the following expression when viewed in terms of an expression.

Here, it will be described using data that the absorption coefficient can be approximated with sufficient accuracy with 10% data as described above. The upper graph in FIG. 4 is an absorption coefficient, and is a graph showing the difference from the original data when this is approximated by 5% data from the top in descending order of singular values by the above method. As is apparent from the graph, an error occurs on the order of 10 ⁻⁷ .

  Next, FIG. 5 is a graph when the same calculation is performed with 10% data. As can be seen from the lower graph of FIG. 5, the error is almost zero. In other words, it can be seen that if the absorption coefficient is calculated using 10% of data from the top in descending order of the singular value as described above, it can be approximated with a sufficiently small error.

Thus, since it can calculate with sufficient precision with a part of data, when storing an absorption coefficient in a database by prior calculation, the amount of data to be stored can be reduced. Alternatively, all the data including the area 9 can be calculated and stored in the database, and 10% of the data can be extracted and used from the beginning during operation.
When the number of rows of the absorption coefficient matrix is large, calculation time is required for the householder transformation of the decomposition process in the singular value decomposition. Further, the order of the lattice point value greatly changes in the wave number region, and better approximation accuracy can be obtained by dividing the large wave number region than the decomposition target. Therefore, a unit obtained by further dividing the row direction of the absorption coefficient matrix can be used as a processing unit for singular value decomposition.

に対して要素の平均値

に最も近い要素ａ_ｉ，ｊ又は該要素の中央値又は該要素の最大値を標準要素とし、
（２）該標準要素を用いて、

を、ｌを１，２，３，・・と順次求め、求める要求精度（有効桁数）ｄに対してｄ＜ｄ_ｌとなるｌが暫定的な近似範囲であり、特異値行列の成分において降順に先頭からｌ個の成分が近似範囲となる。
（３）上記（１）、（２）を計算用２次元行列の全列に対して行い、最大のｌを近似範囲とする。
例えば吸収線のピークが顕著な場合には最大値を選ぶようにすることで精度を高めることができる。 In this embodiment, 10% data from the top is used in descending order of singular values, but this is not limited to 10%. For example, a predetermined ratio may be obtained from the graph shown in FIG. Moreover, it is preferable to obtain | require by the following methods.
The predetermined ratio is obtained as follows.
A is a calculation two-dimensional matrix, that is, an absorption coefficient matrix, U is an orthogonal matrix in which columns are orthogonal, Σ is a singular value matrix, and components of the singular value matrix Σ are σ ₁ > σ ₂ > σ _{3. In} the singular value decomposition A = U · Σ · V ^t , where V ^t is a rotation matrix,
(1) Column vector of the calculation two-dimensional matrix A

Mean of elements for

The element a _{i, j} closest to or the median value of the element or the maximum value of the element as a standard element,
(2) Using the standard element,

And the l 1, 2, 3, sequentially determined and · ·, l which is a d <d _l respect required accuracy (number of significant digits) d required by a preliminary approximation range, the components of the singular value matrix L components from the top in descending order are approximate ranges.
(3) The above (1) and (2) are performed on all columns of the calculation two-dimensional matrix, and the maximum l is set as the approximation range.
For example, when the peak of the absorption line is remarkable, the accuracy can be increased by selecting the maximum value.

In the present invention, when the singular value decomposition is performed on the absorption coefficient matrix 7 as described above, the numerical value is drastically reduced when the singular value matrix is arranged in descending order. In particular, the flat portion of the graph (see FIG. 3), that is, Since the numerical value of the area 9 of 2 is smaller by one digit or more than the numerical value of the first area 8, the absorption coefficient matrix 7 can be approximated by only the data of the first area 8. In the present embodiment, the data in the first region 8 is around 10% of the entire data, so that the absorption coefficient matrix 7 can be approximated in this portion is extremely advantageous in speeding up the calculation.
In addition, in the calculation of the absorption coefficient, the U / Σ of U is calculated in advance and stored in the database, so it is only necessary to store the portion U surrounded by the dotted line in Equation 8 above. Can be reduced. As a result, the data amount is reduced to 1/5 of the conventional amount, and the calculation amount is reduced accordingly.

FIG. 1A shows three-dimensional data of atmospheric pressure × temperature × wave number, and FIG. 1B shows a certain atmospheric pressure × temperature data. It is a figure of the process which rearranges the two-dimensional data of an oblique section into one-dimensional data. It is a graph when the components of a singular value matrix are arranged in descending order. It is the graph which showed the data and error when approximating the data of an absorption coefficient by the data of 5% from the head in descending order of a singular value. It is the graph which showed the data and error when approximating the data of an absorption coefficient by the data of 10% from the head in the descending order of a singular value.

Explanation of symbols

DESCRIPTION OF SYMBOLS 1 3D space data 3 1st vertical direction data sequence 4 2nd vertical direction data sequence 5 3rd vertical direction data sequence 6 4th vertical direction data sequence 7 Absorption coefficient number table 8 1st area | region 9 1st 2 areas

Claims (12)

A function value interpolation method for interpolating a function value mapping from a point in a three-dimensional space to a point in a one-dimensional space using a computer,
Rearranging grid point data in a two-dimensional grid space, which is a part of the three-dimensional space, in the direction of a one-dimensional column by a predetermined method, and calculating point sequence data in another dimension of the tertiary space Generating a computational two-dimensional matrix as another dimension of the two-dimensional matrix;
Perform a singular value decomposition of the two-dimensional matrix for calculation, and approximate it using lattice point data in the two-dimensional lattice space by using a predetermined ratio of singular values from the top of the singular value diagonal matrix in descending order. To calculate the interpolation factor.   The method of claim 1, wherein the two-dimensional lattice space is an oblique lattice.   The method according to claim 1, wherein the two-dimensional lattice space is divided into two or more regions, and calculation is performed for each region.   The method according to any one of claims 1 to 3, wherein the two-dimensional lattice space is stretched by an axis of temperature and pressure.   The method according to claim 4, wherein the two-dimensional lattice space is located in a three-dimensional space having a wave number as an axis. The method according to claim 1, wherein the predetermined ratio is obtained as follows.
A calculation for a two-dimensional matrix represents a U orthogonal matrices between columns are orthogonal, sigma singular value matrix, the components of said specific value matrix Σ σ _1> σ _2> σ in ₃ ···> _{σ n,} V ^In a singular value decomposition A = U · Σ · V ^t , where ^t is a rotation matrix,
(1) Column vector of the calculation two-dimensional matrix A

Mean of elements for

The element a _{i, j} closest to or the median value of the element or the maximum value of the element as a standard element,
(2) Using the standard element,

Are sequentially obtained in descending order as 1, 2, 3,..., And l that satisfies d <d ₁ with respect to the required accuracy d to be obtained is defined as a provisional approximation range.
(3) The above (1) and (2) are performed on all columns of the calculation two-dimensional matrix, and the maximum l is set as the approximation range. A function value interpolation program for interpolating using a computer the value of a function that maps from a point in 3D space to a point in 1D space,
Rearranging grid point data in a two-dimensional grid space, which is a part of the three-dimensional space, in the direction of a one-dimensional column by a predetermined method, and calculating point sequence data in another dimension of the tertiary space Generating a computational two-dimensional matrix as another dimension of the two-dimensional matrix;
Perform a singular value decomposition of the two-dimensional matrix for calculation, and approximate it using lattice point data in the two-dimensional lattice space by using a predetermined ratio of singular values from the top of the singular value diagonal matrix in descending order. Program that calculates interpolation coefficients.   The program according to claim 7, wherein a predetermined ratio is obtained using the method of claim 6. A function value interpolation device for interpolating a value of a function for mapping from a point in a three-dimensional space to a point in a one-dimensional space using a computer,
A means for rearranging lattice point data in a two-dimensional lattice space, which is a part of the three-dimensional space, in a one-dimensional column direction by a predetermined method, and for calculating point sequence data in another dimension of the tertiary space Means for generating a computational two-dimensional matrix as another dimension of the two-dimensional matrix;
Perform a singular value decomposition of the two-dimensional matrix for calculation, and approximate it using lattice point data in the two-dimensional lattice space by using a predetermined ratio of singular values from the top of the singular value diagonal matrix in descending order. A device that calculates interpolation coefficients. A function value interpolation device for interpolating a value of a function for mapping from a point in a three-dimensional space to a point in a one-dimensional space using a computer,
Means for rearranging lattice point data of a two-dimensional lattice space, which is a part of the three-dimensional space, in the direction of a one-dimensional column by a predetermined method; and 2 for calculating point sequence data of another dimension of the tertiary space Means for generating a two-dimensional matrix for calculation as another dimension of the dimensional matrix;
The calculation 2D matrix A, as the singular value diagonal matrix sigma, A = an apparatus for calculating the singular value decomposition by U · Σ · V ^t, the proportion from the beginning of the predetermined calculation values in descending order of U · sigma That stores the data in the PT table. A function interpolation device for interpolating a value of a function for mapping from a point in a three-dimensional space to a point in a one-dimensional space using a computer,
Means for rearranging lattice point data of a two-dimensional lattice space, which is a part of the three-dimensional space, in the direction of a one-dimensional column by a predetermined method; and 2 for calculating point sequence data of another dimension of the tertiary space Means for generating a two-dimensional matrix for calculation as another dimension of the dimensional matrix;
The calculations for two-dimensional matrix A, as the singular value diagonal matrix sigma, the apparatus for calculating the singular value decomposition by ^{A = U · Σ · V t} , to save the calculated value of U · sigma in PT table, the A device that performs function interpolation using a predetermined ratio of data from the top of the PT table data in descending order during operation of the device.   The method according to any one of claims 1 to 6, wherein an interpolation coefficient is calculated within a specified accuracy by designating a ratio of the singular values of the singular value diagonal matrix to be adopted from the top in descending order.

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