CN106908057A - A kind of earth-magnetism navigation reference map construction method - Google Patents

Abstract

A kind of earth-magnetism navigation reference map construction method, it is characterised in that：By setting up multifractal model, the relation with yardstick is estimated in fitting when most preferably approaching, so that it is determined that estimating for each regional area, solves the correction coefficient of point to be inserted, and build reference map using multi-fractal Kriging regression method.Description of the effectively enhancing to GEOMAGNETIC FIELD local singularity of the invention, effectively improves the pattern accuracy of conventional method.

Description

A kind of earth-magnetism navigation reference map construction method

Technical field

The present invention relates to a kind of earth-magnetism navigation technology, more particularly to a kind of earth-magnetism navigation reference map construction method.

Background technology

Earth's magnetic field is the intrinsic resource of the earth, for Aeronautics and Astronautics, navigation provide natural coordinate system.Geomagnetic chart and ground Magnetic model is the Main Means for describing magnetic field of the earth, is also the technical tool for carrying out earth-magnetism navigation and its Military Application, and its is accurate Degree is the key factor for determining earth-magnetism navigation feasibility.The structure of present invention research high accuracy earth magnetism reference map is high-precision to realizing It is significant for degree earth-magnetism navigation positioning.

The content of the invention

It is an object of the invention to provide a kind of earth-magnetism navigation reference map construction method, accurately magnetic navigation is realized Important prerequisite.

The object of the present invention is achieved like this：

A kind of earth-magnetism navigation reference map construction method, it is characterised in that：By setting up multifractal model, fitting is most preferably forced The relation with yardstick is estimated when near, so that it is determined that estimating for each regional area, solves the correction coefficient of point to be inserted, and is utilized Multi-fractal Kriging regression method builds reference map.

Described a kind of earth-magnetism navigation reference map construction method, it is characterized in that described sets up multifractal model method For：Research object is divided into N number of different regional area, is gone to cover this N number of partial zones with the regular grid that size is δ × δ Domain, the distribution probability of regional area is p in i-th grid_i(δ), and meetWith different scaling exponent a_iTable Show, mathematical linguistics is described as：

Q powers are multiplied by formula (1) both sides simultaneously and are taken and is obtained：

When yardstick level off to zero when

Described a kind of earth-magnetism navigation reference map construction method, it is characterized in that described fitting is estimated and chi when most preferably approaching The relational approach of degree is：

Estimate and meet following exponential relationship with yardstick

m∝δ^D, D=1,2,3... (4)

In formula, m is estimating under yardstick δ,

Actual conditions also need to consider the influence of factors, yardstick and corresponding estimate that can not to comply fully with formula (4) this Preferable state, being fitted relational expression when most preferably approaching is

M=b δ^a (5)

In formula, a, b are constant.Because fit procedure Central Plains power exponent D has been unsatisfactory for determining for strict correlation fractal dimension Justice, therefore replace D, a to represent concavo-convex characteristic of the interpolation point in its small neighbourhood with a.

Described a kind of earth-magnetism navigation reference map construction method, it is characterized in that described each regional area of determination is estimated Method is：For the curved surface in three dimensions, from the physical significance of double integral, under the conditions of per unit aerial density, its survey Degree can be understood as curved surface and reference axis into figure volume.Then

M (s, δ)=f (s, δ) S_δ (6)

In formula：F (s, δ) is height, is being denoted herein as geomagnetic field intensity value.Understood centered on s points by formula (6), be with δ The square field inner measure of yardstick is f (s, δ) δ², while understanding that it is b δ to estimate according to fractal theory formula (5)^a, simultaneous Can obtain

f(s,δ)·δ²=b (s) δ^a(s) (7)

Described a kind of earth-magnetism navigation reference map construction method, it is characterized in that the correction coefficient side of described solution point to be inserted Method is：Two-dimensional space is asked when estimating, if the discrete point in a certain region is uniformly distributed according to unique step, can be with interpolation point Centered on, the odd-multiple of minimum range sets up some square cartridges for the length of side between consecutive number strong point, then the face of interpolation point is close Degree is equivalent to the geomagnetic field intensity average value of all discrete points in box, then be multiplied by box area with surface density according to formula (6) can Seek measure value m.Yardstick δ and box number N are set in box dimension, that is, be also given value, then be able to can be asked according to formula (5) Singular coefficient a and correction coefficient N^2-a。

Described a kind of earth-magnetism navigation reference map construction method, it is characterized in that described utilization multi-fractal Kriging regression Method builds benchmark drawing method：

Z_δ=N^2-a·Z_Nδ (8)

Z in formula_Nδ、Z_δThe average of geomagnetic field intensity respectively in N δ, δ neighborhoods, N is the ratio of out to out and smallest dimension Value.Kriging regression is carried out to this region, that is, estimates that yardstick is the average value of geomagnetic field intensity in the square scope of N δ.It is many The transformational relation between the estimation region of fractal theory and the region of search of Kriging regression method is again：

(N·δ)²=π R² (9)

In formula, R is the radius in the round region of Kriging regression method search.Searching for Kriging regression can be obtained by formula (9) Rope radius.It is identical to ensure the data point number included in estimation region because data point is distributed by uniform grid, take The search radius of Kriging regression are

I.e. yardstick for the square area of N δ be Kriging regression search circle domain in the largest square to be included.Gram In golden interpolation formula be：

Formula (11) is substituted into formula (8) to obtain

Interpolation problem is tieed up for n

Brief description of the drawings

Fig. 1 is that earth-magnetism navigation reference map builds flow chart；

Fig. 2 regions earth magnetism reference map simulation result.

The essence of technical scheme is：Multi-fractal Theory and Kriging technique are combined, multi-fractal is proposed Kriging regression method, the method corrected using successive interpolation carries out the structure of earth magnetism reference map.Test result indicate that, the method energy Enough description of the effectively enhancing to GEOMAGNETIC FIELD local singularity, effectively improve the pattern accuracy of conventional method.

The advantage of the technical scheme is：

(1) the LPF of traditional interpolation method compensate for；

(2) more protuberances can be shown or the singularity characteristics such as sink, and with good interpolation, composition essence Degree is high.

Specific embodiment

Step 1, set up multifractal model：

Research object is divided into N number of different regional area, is gone to cover this N number of office with the regular grid that size is δ × δ Portion region, the distribution probability of regional area is p in i-th grid_i(δ), and meetFor different partial zones Domain, the value of distribution probability is also differed, and can use different scaling exponent a_iRepresent, mathematical linguistics is described as：

A is defined as in multi-fractal Theory the singular coefficient of regional area, because the number of N is a lot, therefore can use not Same a constitutes spectral sequence f (a), i.e. multifractal spectra.F (a)~a curves are a models for describing multi-fractal.

Q powers are multiplied by formula (1) both sides simultaneously and are taken and is obtained：

When yardstick level off to zero when

The relation with yardstick is estimated in step 2, fitting when most preferably approaching：

In two-dimensional space, will be in s points, the quality definition in δ fields is estimating under δ yardsticks, with time, spatial distribution Function f can be understood as line density.Its continuous expression formula is

Its discrete expression is

Function f can be understood as the surface density in the region in three dimensions, will be centered on s points, and δ is the square of the length of side Quality definition in neighborhood is estimating under δ yardsticks, and its continuous expression formula is

M (s, δ)=∮ f (s, δ) dS_δ (6)

S in formula_δThe area of δ neighborhoods is represented, its discrete expression is

It can be seen from definition according to fractal dimension, estimate and meet following exponential relationship with yardstick

m∝δ^D (8)

Actual conditions also need consider factors influence, yardstick and it is corresponding estimate can not comply fully with formula (4-12) this Preferable state is planted, relational expression when fitting is most preferably approached is

M=b δ^a (9)

In formula：A, b are constant.Because fit procedure Central Plains power exponent D has been unsatisfactory for determining for strict correlation fractal dimension Justice, therefore replace D with a.A represents concavo-convex characteristic of the interpolation point in its small neighbourhood, is also called singular coefficient, if formula (9) exists First derivative is zero or is not changed on a certain location point, then function is at this point unusual.The exponential relationship of formula (9) Straight line can be converted into log-log coordinate system, but may not be a strict straight line due to factors such as errors, needed It is fitted with least square method, a is the slope of straight line after fitting.For two-dimensional space, work as a<When 2, interpolation curved surface is convex； As a=2, interpolation curved surface is at this point unusual；Work as a>When 2, interpolation curved surface is recessed.

Step 3, determine estimating for each regional area：

Formula (7) can calculate some region of fractal measure with (9) simultaneous.For the curved surface in three dimensions, by double The physical significance of integration understands, under the conditions of per unit aerial density, its estimate can be understood as curved surface and reference axis into figure Volume.Then formula (7) is write as

M (s, δ)=f (s, δ) S_δ (10)

In formula：F (s, δ) is height, is being denoted herein as geomagnetic field intensity value.Understood centered on s points by formula (4-14), δ For the square field inner measure of yardstick is f (s, δ) δ², while understanding that it is b to estimate according to fractal theory formula (4-13) δ^a, simultaneous can obtain

f(s,δ)·δ²=b (s) δ^a(s) (11)

Step 4, the correction coefficient for solving point to be inserted：

Two-dimensional space is asked when estimating, if the discrete point in a certain region is uniformly distributed according to unique step, can be with to be inserted Centered on value point, the odd-multiple of minimum range sets up some square cartridges for the length of side between consecutive number strong point, then interpolation point Surface density is equivalent to the geomagnetic field intensity average value of all discrete points in box, then is multiplied by box face with surface density according to formula (10) Product can seek measure value m.Yardstick δ and box number N be in box dimension set, that is, be also given value, then can according to formula (9), Singular coefficient a and correction coefficient N can be sought^2-a。

Step 5, using multi-fractal Kriging regression method build reference map：

From formula (11), centered on the square with the length of side as δ, yardstick in the square of N δ estimate for

Z_Nδ(N·δ)²=b (N δ)^a (12)

Yardstick for δ it is foursquare estimate for

Z_l·δ²=b δ^a (13)

Simultaneous formula (12) and formula (13) can be obtained

Z_δ=N^2-a·Z_Nδ (14)

Z in formula_Nδ、Z_δThe average of geomagnetic field intensity respectively in N δ, δ neighborhoods, N is the ratio of out to out and smallest dimension Value.Kriging regression is carried out to this region, that is, estimates that yardstick is the average value of geomagnetic field intensity in the square scope of N δ.By It is square in the estimation region of multi-fractal Theory, and the region of search of Kriging regression method is the circle with R as radius, the two Between transformational relation be：

(N·l)²=π R² (15)

The search radius of Kriging regression can be obtained by formula (15).Because data point is distributed by uniform grid, it is Ensure that the data point number included in estimation region is identical, the search radius for taking Kriging regression are

I.e. yardstick for the square area of N δ be Kriging regression search circle domain in the largest square to be included.Gram In golden interpolation formula be：

Formula (17) is substituted into formula (14) to obtain

Interpolation problem is tieed up for n

Claims (6)

1. a kind of earth-magnetism navigation reference map construction method, it is characterised in that：By setting up multifractal model, fitting is most preferably approached When estimate relation with yardstick, so that it is determined that estimating for each regional area, solves the correction coefficient of point to be inserted, and using many Shape Kriging regression method is divided to build reference map again.

2. a kind of earth-magnetism navigation reference map construction method according to claim 1, it is characterized in that described sets up multiple point Shape model method is：Research object is divided into N number of different regional area, is gone to cover this with the regular grid that size is δ × δ N number of regional area, the distribution probability of regional area is p in i-th grid_i(δ), and meetWith different marks Degree index a_iRepresent, mathematical linguistics is described as：

$p_i\left(\mathrm{\&delta;}\right)={\mathrm{\&delta;}}^{a_i},i=1,2,...,N---\left(1\right)$

Q powers are multiplied by formula (1) both sides simultaneously and are taken and is obtained：

$\underset{i=1}{\overset{N}{\&Sigma;}}{p}_i^q=\underset{i=1}{\overset{N}{\&Sigma;}}{\mathrm{\&delta;}}^{a_{i}q}---\left(2\right)$

When yardstick level off to zero when

$\underset{\mathrm{\&delta;}\&RightArrow;0}{\lim }\frac{1}{q-1}\&CenterDot;\frac{ln\underset{i=1}{\overset{N}{\&Sigma;}}{p}_i^q}{ln\mathrm{\&delta;}}=D_q---\left(3\right)$

3. a kind of earth-magnetism navigation reference map construction method according to claim 1, it is characterized in that described fitting is most preferably forced Estimate when near and be with the relational approach of yardstick：

Estimate and meet following exponential relationship with yardstick

m∝δ^D, D=1,2,3... (4)

In formula, m is estimating under yardstick δ,

Actual conditions also need consider factors influence, yardstick and it is corresponding estimate can not comply fully with formula (4) this ideal State, the relational expression that fitting is optimal when approaching is

M=b δ^a (5)

In formula, a, b are constant.The definition of strict correlation fractal dimension has been unsatisfactory for due to fit procedure Central Plains power exponent D, because This replaces D, a to represent concavo-convex characteristic of the interpolation point in its small neighbourhood with a.

4. a kind of earth-magnetism navigation reference map construction method according to claim 1, it is characterized in that each office of described determination The Measurement Method in portion region is：

For the curved surface in three dimensions, from the physical significance of double integral, under the conditions of per unit aerial density, it is estimated can Be interpreted as curved surface and reference axis into figure volume.Then

M (s, δ)=f (s, δ) S_δ (6)

In formula：F (s, δ) is height, is being denoted herein as geomagnetic field intensity value.Understood centered on s points by formula (6), with δ as yardstick Square field inner measure be f (s, δ) δ², while understanding that it is b δ to estimate according to fractal theory formula (5)^a, simultaneous can obtain

f(s,δ)·δ²=b (s) δ^a(s) (7)

5. a kind of earth-magnetism navigation reference map construction method according to claim 1, it is characterized in that described solution point to be inserted Correction factor method be：

Two-dimensional space is asked when estimating, if the discrete point in a certain region is uniformly distributed according to unique step, can be with interpolation point Centered on, the odd-multiple of minimum range sets up some square cartridges for the length of side between consecutive number strong point, then the face of interpolation point is close Degree is equivalent to the geomagnetic field intensity average value of all discrete points in box, then be multiplied by box area with surface density according to formula (6) can Seek measure value m.Yardstick δ and box number N are set in box dimension, that is, be also given value, then be able to can be asked according to formula (5) Singular coefficient a and correction coefficient N^2-a。

6. a kind of earth-magnetism navigation reference map construction method according to claim 1, it is characterized in that multiple point of described utilization Shape Kriging regression method builds benchmark drawing method：

Z_δ=N^2-a·Z_Nδ (8)

Z in formula_Nδ、Z_δThe average of geomagnetic field intensity respectively in N δ, δ neighborhoods, N is the ratio of out to out and smallest dimension.It is right This region carries out Kriging regression, that is, estimate that yardstick is the average value of geomagnetic field intensity in the square scope of N δ.Multiple point Transformational relation between the theoretical estimation region of shape and the region of search of Kriging regression method is：

(N·δ)²=π R² (9)

In formula, R is the radius in the round region of Kriging regression method search.The search half of Kriging regression can be obtained by formula (9) Footpath.Because data point is distributed by uniform grid, to ensure that the data point number that is included is identical in estimation region, in taking gram The search radius of golden interpolation are

$R=\frac{\sqrt{2}}{2}(N\&CenterDot;\mathrm{\&delta;})---\left(10\right)$

I.e. yardstick for the square area of N δ be Kriging regression search circle domain in the largest square to be included.Ke Lijin Interpolation formula is：

$Z_{Nl}(x_0,y_0)=\underset{i=1}{\overset{N}{\&Sigma;}}({\mathrm{\&lambda;}}_i\&CenterDot;Z(x_i,y_i\left)\right)---\left(11\right)$

Formula (11) is substituted into formula (8) to obtain

$Z_l(x_0,y_0)=N^{2-a}\&CenterDot;\underset{i=1}{\overset{N}{\&Sigma;}}({\mathrm{\&lambda;}}_i\&CenterDot;Z(x_i,y_i\left)\right)---\left(12\right)$

Interpolation problem is tieed up for n

$Z_l(x_0,y_0)=N^{n-a}\&CenterDot;\underset{i=1}{\overset{N}{\&Sigma;}}({\mathrm{\&lambda;}}_i\&CenterDot;Z(x_i,y_i\left)\right)---\left(13\right)$