CN102930176B - Nine-grid five-spline interpolation method for geoid model - Google Patents

Nine-grid five-spline interpolation method for geoid model Download PDF

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CN102930176B
CN102930176B CN201210478131.1A CN201210478131A CN102930176B CN 102930176 B CN102930176 B CN 102930176B CN 201210478131 A CN201210478131 A CN 201210478131A CN 102930176 B CN102930176 B CN 102930176B
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interpolation
point
node
height anomaly
height
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CN102930176A (en
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邓兴升
孙虹虹
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Changsha University of Science and Technology
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Changsha University of Science and Technology
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Abstract

A method utilizing (quasi) geoid grid interpolation for calculating height anomaly of any point belongs to the field of geodesy and measurement engineering technology. GPS (Global Positioning System) geoid height measurement has advantages of high-speed and high-precision, however, the normal height system in which the GPS ground height has to be converted into the normal height is adopted by China. The method provided by the invention, which is based on the (quasi) geoid grid, utilizes the nine-grid five-spline interpolation to calculate the height anomaly. According to the method, based on nine grids and sixteen joints closest to a point for interpolation, four joints on the same straight line are respectively adopted to form polishing spline curves interpolation which are used for calculating the height anomaly of the middle point, and then spline curves formed by the middle points are adopted to calculate the height anomaly of the point for interpolation, and the 1985 Yellow Sea height can be obtained through subtracting the height anomaly by the GPS ground height of the point. The method provided by the invention has the advantages that the continuity and smoothness problems of a current geoid grid interpolation are thoroughly solved, and the interpolation result is sole and is very high in computation accuracy. Therefore, the method can be widely used for modern height measurement.

Description

Geoid's model nine grid five Spline Interpolation Method
Technical field
The present invention is a kind of method calculated for geoid surface or quasigeoid grid model spline interpolation, belongs to " physical geodesy " technical field in " Geodesy and Survey Engineering " subject.
Figure viewed from behind technology
The normal level that the GPS geodetic height that certain any height anomaly is numerically equal to this point deducts this point is high.In Surveying Engineering application, the universal GPS of the making geodetic height of GPS is converted into GPS survey becomes a kind of urgent need of production, and height anomaly is the parameters necessary of this conversion process.The method combined by gravity and geometry, first refines according to tight theory and survey the geoid's model in district, then the height anomaly calculating unknown point is current prefered method.The present invention mainly solves the problem calculating unknown point height anomaly on the basis of geoid surface or quasigeoid grid model.Up to now, this problem has proposed three technical schemes, but these three technical schemes all exist deficiency in various degree.As everyone knows, geoid surface is a continuous print, smooth, complicated curved surface.First technical scheme carries out unknown point height anomaly interpolation with inverse distance-weighting in single geoid surface graticule mesh, and this method exists serious problem, and namely interpolation curved surface gross distortion can occur, and curved surface disconnects, discontinuous, rough; Second technical scheme is a kind of easy interpolation method, during its interpolation calculation remains and carries out in single graticule mesh, be not but adopt inverse distance weighted interpolation, but adopt linear interpolation, this method also has problems, namely interpolation result is not smooth curved surface, but pinch plane, there is obvious cut-off rule in mesh edge, is similar to folding line, and cusp is formed at node place, cause curved surface rough; 3rd technical scheme is put forward by applicant, the interpolation calculation of the method is carried out in four graticule mesh, adopt Biquadratic Polynomial matching, the height anomaly of unknown point is obtained through continuous cubic interpolation, but in use, we find that Biquadratic Polynomial matching exists " imperial lattice phenomenon ", and namely curve exists unnecessary swing, and interpolation result may not uniquely, cause reducing Security and feasibility in use.Substantially be all adopt these three kinds of technical schemes at present both at home and abroad.
Patent of the present invention is the 4th technical scheme that we propose, its interpolation is carried out in nine graticule mesh, interpolation curve is five smoothing splint curves, its advantage is the continuity and the slickness problem that solve geoid surface graticule mesh interpolation up hill and dale, interpolation result is unique, and computational accuracy is very high.
Summary of the invention
A kind of geoid's model nine grid five Spline Interpolation Method, is characterized in that, comprise the following steps:
L) regularization of geoid surface grid model form:
(1.1) regularization of indication of the present invention, comprise grid model from east to west, by north orientation south equidistant arrangement, graticule mesh is neat continuously;
(1.2) lower left corner of graticule mesh, the terrestrial coordinate in the upper right corner is indicated;
(1.3) spacing of graticule mesh east-west direction and the spacing of North and South direction is indicated;
2) by interpolation point terrestrial coordinate, the graticule mesh participating in calculating is determined:
(2.1) according to the coordinate of interpolation point, determine which graticule mesh interpolation point drops into, and writes down its numbering i;
(2.2) eight graticule mesh adjacent with graticule mesh i are determined;
(2.3) write down this step 2) in (2.1), (2.2) determined nine graticule mesh node numbering;
3) coordinate, the height anomaly of 16 numbering nodes of nine graticule mesh is determined; Wherein, the numbering of 16 nodes of nine graticule mesh from top to bottom, from left to right number consecutively be 1,2,3 ..., 16;
4) by its height anomaly of interpolation point Geodetic Coordinate Calculation, concrete steps are as follows:
(4.1) cross the coordinate of interpolation point, parallel parallel draws parallel lines;
(4.2) four nodes a, b, c, d that parallel lines are crossing with graticule mesh are determined;
(4.3) by the grid line at a, b, c, d tetra-node places, spline curve fitting is formed;
(4.3.1) by equidistant 4 structure minimum curvature battens, parameters, comprises the number of times of splines number of times, batten correction; Read in the coordinate of known node, and be " batten number of times adds 1 " individual known point at end points place to extension number, the coordinate of extension point is determined according to being calculated by two or a consecutive point coordinate;
(4.3.2) set 4 nodes of wish formation curve as r i(x i, y i), i=1 ~ 4, then the δ spline function curve connecting k batten k-1 the polishing of each node is: wherein r ifor the height anomaly of node; R kh () is the height anomaly of some h matching polishing, put h and 4 node on the same line, and 1≤h≤4;
(4.3.3) Ω kx () is δ Gene radiotherapy: Ω k ( x ) = Σ j = 0 k + 1 ( - 1 ) j k + 1 j ( x + k + 1 2 - j ) + k 1 k ! , Character expression wherein calculates by following: n m = n ! ( n - m ) ! m ! , Z + k = Z k , Z > 0 0 , Z ≤ 0
(4.3.4) height anomaly of polishing match point is calculated, and matching is obtained comparing at the calculation level height anomaly of site position and the original height anomaly of node, if less than original node abnormal elevation value, then its difference is added on original height anomaly; Otherwise, difference is reduced from the original height anomaly of node, namely the original height anomaly of node is revised;
(4.3.5) again calculate final polishing match point height anomaly with revised original node height anomaly, obtain than the front polishing matched curve once more approaching original node;
(4.4) in nine graticule mesh, by 1,2,3,4 four node structure Article 1 SPL, the height anomaly of interpolation a point; By 5,6,7,8 four node structure Article 2 SPL, the height anomaly of interpolation b point; By 9,10,11,12 4 node structure Article 3 SPL, the height anomaly of interpolation c point; By 13,14,15,16 4 node structure Article 4 SPL, the height anomaly of interpolation d point;
(4.5) according to the height anomaly of a, b, c, d tetra-nodes, according to (4.3.1) in step (4.3) to the step in (4.3.5), structure Article 5 SPL, matching a, b, c, d tetra-nodes, the height anomaly of interpolation P point.
2., because geoid surface grid model and quasigeoid grid model have identical form, its difference is only node height anomaly slightly difference.Therefore, geoid's model nine grid five Spline Interpolation Method according to claim 1, also can be applicable in quasigeoid grid model interpolation calculation.
Accompanying drawing explanation
Fig. 1 geoid's model nine grid five Spline Interpolation Method schematic diagram
Embodiment
Geoid's model nine grid five Spline Interpolation Method, its elementary object is based on grid model, is gone out the abnormal elevation value of position, arbitrfary point by the abnormal elevation value interpolation calculation of node.Two innovative points of the present invention are: one is that the world proposes employing nine graticule mesh first to carry out interpolation calculation at home, completely avoid the unique and rough discontinuous problem of curved surface of interpolation result; Two is in the calculating of a unknown point height anomaly, and introduced by five smoothing splint curves in the interpolation calculation of geoid surface grid model, interpolation result is reliable, and interpolation precision is very high.Principle constructs smooth SPL according to 4 known points equidistantly distributed, and obtained the height anomaly of impact point position by five interpolation.Comprise the following steps in concrete enforcement:
One, the regularization of geoid surface grid model form;
The grid model regularization of indication of the present invention, refers to that grid model forms the file layout of specifying: the lower left corner of graticule mesh, the terrestrial coordinate in the upper right corner and graticule mesh east-west direction spacing and North and South direction spacing are indicated in file first trip; Second row starts the abnormal elevation value for node, and often row from east to west, often arranges by the north orientation equidistant arrangement in south, and graticule mesh is neat continuously.
Two, by interpolation point terrestrial coordinate, the height anomaly of interpolation point is calculated;
1) if interpolation point terrestrial coordinate overlaps with certain node, then its height anomaly is exactly the height anomaly of node.
2) if interpolation point falls on the line of certain two node, then the SPL that its height anomaly is formed by four nodes is determined, these four nodes are located along the same line, and nearest with interpolation point.
3) if interpolation point falls into graticule mesh blank space, then determine the graticule mesh of participating in calculating, according to the coordinate of interpolation point, determine which graticule mesh interpolation point drops into, and writes down its numbering i; Determine eight graticule mesh adjacent with graticule mesh i; Write down the numbering of nine mesh node; Determine coordinate and the height anomaly of 16 numbering nodes of nine graticule mesh.
4) interpolation calculation height anomaly according to the following steps, implementation step is as follows:
(4.1) cross the coordinate of interpolation point, parallel parallel draws parallel lines, and these parallel lines stride across three nearest graticule mesh of interpolation point, and intersect at four intersection points a, b, c, d with graticule mesh;
(4.2) four collinear and nearest with putting a nodes form spline fit curve, thus determine the height anomaly of an a; In like manner obtain the height anomaly of other three intersection points such as some b, c, d etc.
5) spline curve construction step is as follows:
(5.1) by equidistant 4 structure minimum curvature battens, parameters, comprises the number of times of splines number of times, batten correction; Read in the coordinate of known node, and be " batten number of times adds 1 " individual known point at end points place to extension number, the coordinate of extension point calculates according to two consecutive point coordinates to be determined;
(5.2) set the node of wish formation curve as r i(x i, y i), i=1 ~ 4, then the δ spline function curve connecting k batten k-1 the polishing of each node is: wherein r ifor the height anomaly of node; R kh () is the height anomaly of some h matching polishing, put h and 4 node on the same line, and 1≤h≤4;
(5.3) Ω kx () is δ Gene radiotherapy: Ω k ( x ) = Σ j = 0 k + 1 ( - 1 ) j k + 1 j ( x + k + 1 2 - j ) + k 1 k ! , Character expression wherein calculates by following: n m = n ! ( n - m ) ! m ! , Z + k = Z k , Z > 0 0 , Z ≤ 0
(5.4) height anomaly of polishing match point is calculated, and matching is obtained comparing at the calculation level height anomaly of site position and the original height anomaly of node, if less than original node abnormal elevation value, then its difference is added on original height anomaly; Otherwise, difference is reduced from the original height anomaly of node, namely the original height anomaly of node is revised;
(5.5) again calculate final polishing match point height anomaly with revised original node height anomaly, obtain than the front polishing matched curve once more approaching original node.
6) as shown in Figure of description, in nine graticule mesh, by 1,2,3,4 four node structure Article 1 SPL, the height anomaly of interpolation a point; By 5,6,7,8 four node structure Article 2 SPL, the height anomaly of interpolation b point; By 9,10,11,12 4 node structure Article 3 SPL, the height anomaly of interpolation c point; By 13,14,15,16 4 node structure Article 4 SPL, the height anomaly of interpolation d point.Interpolation point P and a, b, c, d tetra-intersection points are on the same line, according to the height anomaly of a, b, c, d tetra-nodes, according to step 5) in the method that describes to (5.5) of sub-step (5.1), structure Article 5 SPL, the height anomaly of interpolation P point.

Claims (2)

1. geoid's model nine grid five Spline Interpolation Method, is characterized in that, comprise the following steps:
L) regularization of geoid surface grid model form:
(A) regularization of indication of the present invention, comprise grid model from east to west, by north orientation south equidistant arrangement;
(B) lower left corner of graticule mesh, upper right corner terrestrial coordinate is indicated;
(C) each spacing of graticule mesh east-west direction and the spacing of North and South direction is indicated;
2) by interpolation point terrestrial coordinate, the graticule mesh of participating in calculating is determined:
I (), according to the coordinate of interpolation point, determines which graticule mesh interpolation point drops into, and writes down its numbering i;
(ii) eight graticule mesh adjacent with graticule mesh i are determined;
(iii) write down this step 2) in (i), (ii) nine graticule mesh the numbering of node;
3) coordinate, the height anomaly of 16 nodes of nine graticule mesh is determined; Wherein, the numbering of 16 nodes of nine graticule mesh from top to bottom, from left to right number consecutively be 1,2,3 ..., 16;
4) by its height anomaly of interpolation point Geodetic Coordinate Calculation, concrete steps are as follows:
(A) cross the coordinate of interpolation point, parallel parallel draws parallel lines;
(B) four nodes a, b, c, d that parallel lines are crossing with graticule mesh are determined;
(C) by the grid line at a, b, c, d tetra-node places, by following (C-1) to (C-5) five steps, spline curve fitting is formed;
(C-1) by equidistant 4 structure minimum curvature battens, parameters, comprises the number of times of splines number of times, batten correction; Read in the coordinate of known node, and be " batten number of times adds 1 " individual known point at end points place to extension number, the coordinate of extension point calculates according to the coordinate of two consecutive point to be determined;
(C-2) set 4 nodes of wish formation curve as r i(x i, y i), i=1 ~ 4, then the δ spline function curve connecting k batten k-1 the polishing of each node is: wherein r ifor the height anomaly of node; R kh () is the height anomaly of some h matching polishing, put h and 4 node on the same line, and 1≤h≤4;
(C-3) Ω kx () is k δ Gene radiotherapy: Ω k ( x ) = Σ j = 0 k + 1 ( - 1 ) j k + 1 j ( x + k + 1 2 - j ) + k 1 k ! , Character expression wherein calculates by following: n m = n ! ( n - m ) ! m ! , Z + k = Z k , Z > 0 0 , Z ≤ 0 ;
(C-4) height anomaly of polishing match point is calculated, and matching is obtained comparing at the calculation level height anomaly of site position and the original height anomaly of node, if less than original node abnormal elevation value, then its difference is added on original height anomaly; Otherwise, difference is reduced from the original height anomaly of node, namely the original height anomaly of node is revised;
(C-5) with the polishing match point height anomaly that revised original node height anomaly iterative computation is final, obtain than the front polishing matched curve once more approaching original node;
(D) in nine graticule mesh, by 1,2,3,4 four node structure Article 1 SPL, the height anomaly of interpolation a point; By 5,6,7,8 four node structure Article 2 SPL, the height anomaly of interpolation b point; By 9,10,11,12 4 node structure Article 3 SPL, the height anomaly of interpolation c point; By 13,14,15,16 4 node structure Article 4 SPL, the height anomaly of interpolation d point;
(E) according to the height anomaly of a, b, c, d tetra-nodes, according to (C-1) in step (C) to the step in (C-5), structure Article 5 SPL, the height anomaly of interpolation P point.
2. geoid's model nine grid five Spline Interpolation Method according to claim 1, it is characterized in that, the method can be applicable in quasigeoid grid model interpolation calculation.
CN201210478131.1A 2012-11-22 2012-11-22 Nine-grid five-spline interpolation method for geoid model Expired - Fee Related CN102930176B (en)

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