CN110727914B - Vector operation-based plumb line deviation single-point calculation method - Google Patents

Vector operation-based plumb line deviation single-point calculation method Download PDF

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CN110727914B
CN110727914B CN201910944229.3A CN201910944229A CN110727914B CN 110727914 B CN110727914 B CN 110727914B CN 201910944229 A CN201910944229 A CN 201910944229A CN 110727914 B CN110727914 B CN 110727914B
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黄炎
王庆宾
冯进凯
赵东明
谭勖立
黄子炎
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Information Engineering University of PLA Strategic Support Force
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Abstract

The invention relates to a vector operation-based vertical deviation single-point calculation method, which changes the storage mode of each parameter involved in vertical deviation prime-unitary component and meridian component spherical harmonic series calculation into a mode of storing all the parameters in a one-dimensional array, converts the existing serial logic-based vertical deviation prime-unitary component and meridian component spherical harmonic series calculation formula, converts the cyclic calculation and accumulation summation processes in the original calculation formula into the calculation formulas of corresponding element operation and array inner product in the corresponding array, effectively converts scalar calculation into vector calculation, and realizes the parallel calculation of components with different truncation orders and orders through multi-thread parallel calculation logic, thereby greatly reducing the time of vertical deviation single-point calculation and improving the calculation efficiency.

Description

Vector operation-based plumb line deviation single-point calculation method
Technical Field
The invention belongs to the technical field of vertical deviation calculation, and particularly relates to a vector operation-based vertical deviation single-point calculation method.
Background
The vertical deviation is one of the gravity field elements of the earth, and has important research significance. The main methods for calculating the vertical line deviation at home and abroad comprise: a bit coefficient model method, a Stokes integral method, a point mass model method, a single layer density method, a function approximation method, an earth gravity field bit coefficient model method and the like. Each of these methods has advantages, but still suffers from some disadvantages. Some methods require a large amount of measured data, and the gravity measured data often cannot meet the requirements due to various reasons. Some methods can only calculate the vertical line deviation of a local area, and cannot calculate the vertical line deviation of irregular discrete points.
The earth gravity field model is to approximate the earth gravity field by a gravitational potential spherical harmonic series truncated to a finite order. The process of calculating the vertical deviation by using the method can be divided into two parts: the recursion operation of the Legendre function and the operation of the spherical harmonic series are associated. These two sections will be described in detail below.
The associated Legendre function calculation part selects a cross-order recursion method and a Belikov recursion method for calculation, and the ultra-high-order associated Legendre function can be calculated quickly and efficiently without divergence.
According to the cross-order recursion principle, a fully normalized associated Legendre function of order n and order m
Figure BDA0002223731700000011
The expression of (a) is:
Figure BDA0002223731700000012
in the formula:
Figure BDA0002223731700000013
Figure BDA0002223731700000014
Figure BDA0002223731700000015
Figure BDA0002223731700000021
Figure BDA0002223731700000022
according to the principle of Belikov recursion method, a complete normalized associated Legendre function with the order of n and the order of m
Figure BDA0002223731700000023
The expression of (a) is:
Figure BDA0002223731700000024
will be provided with
Figure BDA0002223731700000025
Conversion to fully normalized associative legendre function
Figure BDA0002223731700000026
Figure BDA0002223731700000027
In the formula:
Figure BDA0002223731700000028
the earth gravity field model is to approximate the earth gravity field by a gravitational potential spherical harmonic series truncated to a finite order, and the disturbance potential series is represented by the following formula:
Figure BDA0002223731700000029
wherein rho is the radial distance from the point to be solved to the earth center, R is the earth radius, G is the universal gravitation constant, M is the earth mass,
Figure BDA00022237317000000210
is geocentric latitude, and λ is geocentric longitude.
Under the condition of sphere approximation, the relationship between the vertical deviation xi and eta disturbance positions is as follows:
Figure BDA00022237317000000211
further, the calculation formula of the vertical line deviation prime-unitary component and the meridian component spherical harmonic series can be deduced in the local northeast coordinate system as follows:
Figure BDA0002223731700000031
in the formula, xi and eta are a prime component and a meridian component of vertical deviation respectively,
Figure BDA0002223731700000032
and
Figure BDA0002223731700000033
and (4) completely normalizing the gravity field model bit coefficient for an n-order m-order.
Equation (6) is a theoretical basis generated for computer serial calculation, which is based on scalar operation in the calculation process, and the calculation steps are divided into front and back parts and must be performed in sequence. The calculation of the model vertical deviation is performed by taking the increment of the truncation order N and the order m as a calculation sequence, and (N + 2) × (N + 1) times of cyclic calculation are required for calculating the vertical deviation of each point, which consumes a lot of time. For example, the calculation of the model vertical deviation with the truncation order of 2160 requires 466 ten thousand cycles to complete the calculation.
In addition, when the computer calculates the formula (6), each required parameter is stored in a matrix form, and the matrix is a lower triangular matrix, and the storage mode is as shown in fig. 1,
Figure BDA0002223731700000034
and N is the truncation order of the point location model to be solved. For example, for the gravitational field model potential coefficient therein
Figure BDA0002223731700000035
Figure BDA0002223731700000036
The storage mode is shown in fig. 2. Other parameters
Figure BDA0002223731700000037
Etc. are stored in this manner. If stored in this manner, it is difficult to index a single location when performing parallel computations. In addition, in the process of performing parallel computation in a matrix form, the spare part of the lower triangular matrix needs to be filled with zero elements, so that a large number of threads do useless work, and the computation efficiency is seriously influenced.
From the overall analysis, in the process of calculating the single point of the vertical deviation, due to the front and rear correlations of the recursion process of the calculation of the associated legendre function, the parallel processing is difficult to perform, the calculation time is about 10% of the total time, when the single point calculation is performed on the spherical harmonic series part, the latitude and the longitude are already determined, the calculation process is only related to the truncation order n and the order m, the calculation time is more than 90% of the total time of the single point calculation, a large amount of time is consumed, and the calculation efficiency is low. For example, the vertical line deviation value with a truncation order of 2160 at a certain point is calculated, the original data is stored as a lower triangular matrix with 2161 rows, and a (2161 + 1) × 2161=4672082 bit coefficient model needs to be calculated and read
Figure BDA0002223731700000038
Recalculated (2161 + 1) + 2161/2=2336041 associated legendre functions, followed by another 10 6 The magnitude spherical harmonic series addition, subtraction, multiplication, division and trigonometric function operation has low calculation efficiency.
Disclosure of Invention
The invention provides a vector operation-based vertical line deviation single-point calculation method, which is used for solving the problems of more calculation time consumption and low efficiency in the prior art.
In order to solve the technical problems, the technical scheme and the beneficial effects of the invention are as follows:
the invention relates to a vector operation-based vertical line deviation single-point calculation method, which comprises the following steps of:
1) Completely normalizing gravity field model bit coefficients of each n-order m-order
Figure BDA0002223731700000041
Fully normalized associative legendre function of order m, order n and order m
Figure BDA0002223731700000042
And an n-th order m-level fully normalized associative Legendre function
Figure BDA0002223731700000043
To pair
Figure BDA0002223731700000044
Derivative of (2)
Figure BDA0002223731700000045
All are stored in a one-dimensional array form, and an array [ C ] is correspondingly obtained]And array [ S ]]Array [ M ]]Array [ P ]]And array [ dP](ii) a Storing cosine angles cosm lambda of the geocentric longitudes of different levels and sine angles sinm lambda of the geocentric longitudes of different levels in a one-dimensional array form to correspondingly obtain an array [ cos lambda ]]And array [ sin ]](ii) a Wherein, the order n and the order m corresponding to the elements at the same position in each array are the same,
Figure BDA0002223731700000046
is geocentric latitude, lambda is geocentric longitude, N is more than or equal to 0 and less than or equal to N, m is more than or equal to 0 and less than or equal to N, and N is truncation order;
2) Converting a calculation formula of spherical harmonic series of prime component and meridian component of vertical deviation into a calculation formula of corresponding elements in corresponding arrays and a calculation formula of corresponding inner products of the arrays;
3) Calculating the converted calculation formula by adopting multiple threads to obtain each t ξ (i) And each t η (i) (ii) a Each t obtained ξ (i) Adding to obtain the Mao-unitary component of the vertical deviation; each t obtained η (i) Adding to obtain the meridional component of the vertical deviation; wherein:
t ξ (i)＝-(C i ×cos i +S i ×sin i )×dP i
Figure BDA0002223731700000047
wherein i is more than or equal to 1 and less than or equal to J,
Figure BDA0002223731700000048
C i 、cos i 、S i 、sin i 、dP i 、M i 、Pcos i are respectively array [ C]Group of [ cos ]]Array [ S ]]Array [ sin ]]Array [ dP ]]Array [ M ]]Array [ P ]]The ith element in (1).
The beneficial effects are as follows: the method changes the storage mode of each parameter involved in the calculation of the vertical line deviation Mao-unitary component and the meridian component spherical harmonic series, wherein the parameters comprise
Figure BDA0002223731700000049
Figure BDA00022237317000000410
cosm lambda and sinm lambda are changed into a mode that all the cosm lambda and the sinm lambda are stored in a one-dimensional array form, and then the existing calculation formula of the spherical harmonic series of the vertical line deviation prime and the meridian component based on the serial logic is converted, namely: the original calculation formula which needs to be subjected to cyclic calculation and accumulation summation is converted into the calculation formula of the operation of corresponding elements in the corresponding array and the inner product of the array, scalar calculation is effectively converted into vector calculation, and t of different truncation orders and different orders is realized through multi-thread parallel calculation logic ξ (i) And each t η (i) The time of the single-point calculation of the vertical line deviation is greatly reduced, the logic of the parallel calculation equipment is fully matched, and the calculation efficiency is improved.
Further, in step 3), in order to further reduce the calculation time, multiple threads are adopted to obtain each t ξ (i) An addition operation is performed.
Further, in step 3), to further reduce the computation time, multiple threads are used to obtain each t η (i) An addition operation is performed.
Further, the converted meterThe calculation formula is as follows:
Figure BDA0002223731700000051
in the formula (I), the compound is shown in the specification,
Figure BDA0002223731700000055
representing the multiplication of corresponding elements in an array of equal length, "·" represents the inner product of the array,
Figure BDA0002223731700000052
are respectively array [ C]Group of [ cos ]]Array [ S ]]Array [ sin ]]Array [ M ]]The corresponding row vector is set to the corresponding row vector,
Figure BDA0002223731700000053
are respectively an array [ dP]Array [ P ]]The corresponding column vector.
Further, in step 3), multithreading is implemented in the GPU.
Drawings
FIG. 1 is a diagram illustrating a parameter storage method in the prior art;
FIG. 2 is a prior art
Figure BDA0002223731700000054
A schematic diagram of a storage mode;
FIG. 3 is a diagram illustrating a modified parameter storage method according to the present invention;
FIG. 4 is a diagram of a GPU-side group summing scheme of the present invention.
Detailed Description
When the vertical deviation calculation is carried out, because the association legendre function recursion process is difficult to carry out parallelization processing and the total consumption time is short, excessive processing is not carried out on the association legendre function recursion process, and the association legendre function recursion process is carried out in a serial mode on the CPU end. Aiming at the problem that the calculation of the spherical harmonic series part consumes more time, the embodiment provides the vector operation-based vertical line deviation single-point calculation method, and the vector operation formula suitable for parallel calculation is derived based on the principle of vector operation so as to fully fit the logic of parallel calculation equipment and effectively improve the calculation efficiency.
The method comprises the following specific steps:
step one, changing a parameter storage mode, and changing the original parameters stored in a lower triangular matrix mode into a storage mode as shown in fig. 2, that is: rearranging the data in the form of the traditional lower triangular matrix into one-dimensional array data with a row length of J as shown in the lower part of figure 3 according to the sequence of row numbers and the sequence of each row from left to right,
Figure BDA0002223731700000061
the specific parameters and the processing process thereof comprise:
1. completely normalizing gravity field model bit coefficients of each n-order m-order
Figure BDA0002223731700000062
The cosine angle cosm λ of the geocentric longitude of different levels and the sine angle sinm λ of the geocentric longitude of different levels are correspondingly obtained into an array [ C ] according to the array storage modification mode shown in figure 3]Array [ S ]]Group of [ cos ]]And array [ sin ]]. To accommodate equation (9) as described below, the array [ C ] may be used]Array [ S ]]Group of [ cos ]]And array [ sin ]]Are respectively regarded as
Figure BDA0002223731700000063
Figure BDA0002223731700000064
The four 1 xJ row vectors of (1) are respectively shown in formulas (7-1) to (7-4):
Figure BDA0002223731700000065
Figure BDA0002223731700000066
Figure BDA0002223731700000067
Figure BDA0002223731700000068
2. fully normalized associative Legendre function of order m, order n, order m
Figure BDA0002223731700000069
And
Figure BDA00022237317000000610
ratio of (A to B)
Figure BDA00022237317000000611
And an n-th order m-level fully normalized associative legendre function
Figure BDA00022237317000000612
To pair
Figure BDA00022237317000000613
Derivative of (2)
Figure BDA00022237317000000614
Also according to the mode of FIG. 3, the data are stored in the form of one-dimensional arrays, and the arrays [ M ] are obtained correspondingly]And the array [ Pcos]And array [ dP]. To accommodate equation (9) described below, the array [ M ] may be set]Array [ Pcos ]]And array [ dP]Are respectively regarded as
Figure BDA00022237317000000615
The three column vectors of J × 1 are respectively shown in formulas (8-1) to (8-3):
Figure BDA0002223731700000071
Figure BDA0002223731700000072
Figure BDA0002223731700000073
and the data storage is changed, and the data types stored in the memory of the computer are changed into array data forms so as to finish the derivation of the parallel calculation formula in the next step.
And step two, deducing a vector operation formula for calculating the vertical line deviation by using the earth gravity field model according to the formula (6), wherein the deduced formula is as follows:
Figure BDA0002223731700000074
in the formula (I), the compound is shown in the specification,
Figure BDA00022237317000000714
representing the multiplication of corresponding elements within vectors of the same length, "·" represents the inner product of the vectors.
Through the formula, derivation of a vectorized vertical line deviation calculation formula is completed, the calculation formulas of the circular calculation and accumulation summation form in the original calculation formula (6) are changed into vector operation among the groups, namely the calculation formulas are converted into operation forms of inner product of vector operation, addition, subtraction, multiplication, division, reduction, summation and the like of corresponding elements, scalar calculation is effectively converted into vector calculation, a theoretical basis is provided for further realizing parallel calculation, and the calculation logic of the parallel calculation and the processing mode of calculation equipment are effectively matched.
Analysis is performed on equation (9), which can be developed as:
Figure BDA0002223731700000075
Figure BDA0002223731700000076
in the formula, C i 、cos i 、S i 、sin i 、dP i 、M i 、Pcos i Are respectively array [ C]Group of [ cos ]]Array [ S ]]Array [ sin ]]Array [ d ]P]Array [ M ]]And the array [ Pcos]The ith element in (1), i.e. the row vector
Figure BDA0002223731700000077
Line vector
Figure BDA0002223731700000078
Line vector
Figure BDA0002223731700000079
Line vector
Figure BDA00022237317000000710
Column vector
Figure BDA00022237317000000711
Column vector
Figure BDA00022237317000000712
Column vector
Figure BDA00022237317000000713
The ith element in (1).
And step three, because the calculation results of the addition, subtraction, multiplication and division operations among the position elements corresponding to the arrays with the same length cannot be changed due to the calculation sequence, the operations of each position element are independent and do not influence each other, and the magnitude is large, different threads of the GPU can be used for processing respectively. And (4) finishing the operation of addition, subtraction, multiplication and division on the corresponding elements of the logarithm by utilizing the GPU on the basis of the formula (9). In order to better distribute tasks to the CPU thread, the method refines the deduced parallel calculation formula, deduces the calculation task of splitting to a specific thread so as to accurately control and distribute the calculated amount, and the formula of the thread calculation unit after splitting is as follows:
Figure BDA0002223731700000081
in the formula, t ξ (i)、t η (i) The value of the ith element of the new array obtained after calculation.
That is, one lineThe program calculates a t ξ (i) Multiple threads computing multiple t simultaneously ξ (i) One thread calculates one t η (i) With multiple threads computing multiple t simultaneously η (i) In that respect After the above calculation is completed, two new arrays are obtained
Figure BDA0002223731700000082
Figure BDA0002223731700000083
Step four, carrying out each t ξ (i) And each t η (i) The addition operation of (1). In order to further shorten the computation time, a summation computation is performed at the GPU side using multiple threads, and the specific computation is shown in fig. 4. The elements in the array to be summed are first grouped, the grouping principle is that starting from the first element, two adjacent elements are grouped into one group, and if the total elements of the array are odd, the last remaining element is singly grouped. At the moment, different threads in the GPU are utilized to calculate the sum of two elements in each group at the same time, so that a new array which is half shorter than the original array can be obtained through one calculation. And then repeating the calculation process by taking the new array as a target array until the newly generated array is a unit array with only one element, and finishing the calculation. The sum of the specifications performed in this way effectively reduces the number of computations, via int [ (J + 1)/2%]And obtaining a result after secondary calculation.
In the whole view, the method enables data to be orderly and effectively used by parallel computing threads when the data are read, written and computed through a certain regular arrangement.
In this embodiment, when the parameter is stored in the first step, the parameter is directly stored
Figure BDA0002223731700000084
The corresponding formula after conversion is shown as formula (9). Due to the fact that
Figure BDA0002223731700000085
Is constant, so it is otherEmbodiments may be stored therein
Figure BDA0002223731700000086
So as to obtain a one-dimensional array [ P]The corresponding column vector is
Figure BDA0002223731700000091
Corresponding to the need of adding pairs in subsequent calculation
Figure BDA0002223731700000092
The process of (1) is such that the formula (9) is transformed into the formula (12), and the formula (11) is modified into the formula (13).
Figure BDA0002223731700000093
Figure BDA0002223731700000094
In this embodiment, the parameter stores are stored in the order of increasing sequence numbers n and m, and as another embodiment, the parameter stores may be stored in the order of decreasing sequence or even in a random sequence, but the order n and the order m corresponding to the elements at the same position in each array (including array [ C ], array [ cos ], array [ S ], array [ sin ], array [ dP ], array and array [ Pcos ]) must be the same.
In this embodiment, the converted formula corresponds to formula (9), but it should be noted that, for the computer, only a string of sequential numbers is known to be processed, i.e. the calculation is performed according to the formula corresponding to formula (11), but the calculation of formula (11) is realized, and the corresponding converted formula may not be shown in formula (11), for example, as shown in formula (14), but at this time, the corresponding formula is corresponding to formula (9)
Figure BDA0002223731700000095
Can be regarded as a column vector, corresponding
Figure BDA0002223731700000096
It is considered as a row vector, but the processing result is consistent for the computer regardless of the formula (9) or the formula (14).
Figure BDA0002223731700000097
The effectiveness of the method is illustrated below by way of a specific example.
The experimental environment is as follows: a desk-top high-performance workstation with a processor
Figure BDA0002223731700000099
Core TM i9-9900K CPU@3.60GHz; GPU NVIDIA TITAN V; the memory is 32.00GB.
Based on the experiment platform, a simulation point location is used for carrying out an experiment, a cross-order recursion method and a Belikov recursion method are respectively used for calculating an association Legendre function, and the serial single-point calculation time consumption, the parallel single-point time consumption and the acceleration ratio of the deviation of the vertical line of different orders are calculated under different experiment environments and are respectively counted to the table 1 and the table 2.
TABLE 1 calculation of time consumption for single point calculation of vertical line deviations of different orders using cross-order recursion in an experimental environment
Figure BDA0002223731700000098
Figure BDA0002223731700000101
Table 2 calculation of time consumption of single-point calculation of vertical line deviation of different orders by means of Belikov recursion times in experimental environment
Figure BDA0002223731700000102
As can be seen from tables 1 and 2, as the order increases, the time consumption of the serial algorithm for calculating the vertical line deviation by using the cross-order recursion method increases rapidly, when the truncation order reaches 2160 order, the time consumption of single-point calculation in the experimental environment is 450ms, and the time consumption of calculation by using the Belikov recursion method is more than 540 ms. After a formula is improved by using vector operation, the calculation efficiency is obviously improved, and both a cross-order recursion method and a Belikov recursion method reach over 6 times of acceleration ratio and reach 9.69 times at most. The invention improves the traditional serial computing mode into the parallel computing mode, and effectively improves the computing efficiency of the model vertical deviation.
While the present invention has been described in detail with reference to the preferred embodiments, it should be understood that the above description should not be taken as limiting the invention. Various modifications and alterations to this invention will become apparent to those skilled in the art upon reading the foregoing description. Accordingly, the scope of the invention should be determined from the following claims.

Claims (5)

1. A vertical line deviation single-point calculation method based on vector operation is characterized by comprising the following steps:
1) Completely normalizing gravity field model bit coefficients of each n-order m-order
Figure FDA0002223731690000011
And
Figure FDA0002223731690000012
fully normalized associative legendre function of order m, order n and order m
Figure FDA0002223731690000013
And an n-th order m-level fully normalized associative legendre function
Figure FDA0002223731690000014
For is to
Figure FDA0002223731690000015
Derivative of (2)
Figure FDA0002223731690000016
All are stored in a one-dimensional array form, and an array [ C ] is correspondingly obtained]And array [ S ]]Array [ M ]]Array [ P ]]And array [ dP](ii) a Storing cosine angles cosm lambda of the geocentric longitudes of different levels and sine angles sinm lambda of the geocentric longitudes of different levels in a one-dimensional array form to correspondingly obtain an array [ cos lambda ]]And array [ sin ]](ii) a Wherein, the order n and the order m corresponding to the elements at the same position in each array are the same,
Figure FDA0002223731690000017
is geocentric latitude, lambda is geocentric longitude, N is more than or equal to 0 and less than or equal to N, m is more than or equal to 0 and less than or equal to N, and N is truncation order;
2) Converting a calculation formula of spherical harmonic series of prime and unitary components of vertical deviation into a calculation formula of operation of corresponding elements in a corresponding array and a calculation formula of an inner product of the array;
3) Calculating the converted calculation formula by adopting multiple threads to obtain each t ξ (i) And each t η (i) (ii) a Each t obtained ξ (i) Adding to obtain the Mao-unitary component of the vertical deviation; each t obtained η (i) Adding to obtain the meridional component of the vertical deviation; wherein:
t ξ (i)＝-(C i ×cos i +S i ×sin i )×dP i
Figure FDA0002223731690000018
wherein i is more than or equal to 1 and less than or equal to J,
Figure FDA0002223731690000019
C i 、cos i 、S i 、sin i 、dP i 、M i 、Pcos i are respectively array [ C]Group of [ cos ]]Array [ S ]]Array [ sin ]]Array [ dP ]]Array [ M ]]Array [ P ]]The ith element in (1).
2. The orientation-based of claim 1The single-point calculation method for the vertical line deviation of the quantity operation is characterized in that in the step 3), each t obtained by adopting multithreading ξ (i) An addition operation is performed.
3. The method for computing the single point of vertical deviation based on vector operation as claimed in claim 1, wherein in step 3), each t obtained by multithreading is used η (i) An addition operation is performed.
4. The method for calculating a single point of vertical deviation based on vector operation of claim 1, wherein the converted calculation formula is:
Figure FDA0002223731690000021
in the formula (I), the compound is shown in the specification,
Figure FDA0002223731690000025
representing the multiplication of corresponding elements in an array of equal length, "·" represents the inner product of the array,
Figure FDA0002223731690000022
Figure FDA0002223731690000023
are respectively array [ C]Array [ cos ]]Array [ S ]]Array [ sin ]]Array [ M ]]The corresponding row vector is set to the corresponding row vector,
Figure FDA0002223731690000024
are respectively an array [ dP]Array [ P ]]The corresponding column vector.
5. The method for computing the single point of vertical deviation based on vector operation according to any one of claims 1 to 4, wherein in step 3), multiple threads are implemented in the GPU.
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