2012 2nd International Conference on Industrial Technology and Management (ICITM 2012)
IPCSIT vol. 49 (2012) © (2012) IACSIT Press, Singapore
DOI: 10.7763/IPCSIT.2012.V49.40

College of Mechatronics Engineering and Automation, National University of Defense Technology,
Changsha, China
Based on integral iteration idea and iterative formula in wave number domain, an improved iteration method is proposed for downward continuation of potential fields. We present a method for estimating iterative number, according to the expression of absolute value of difference of two successive iterations. The improved method is tested on both model and real observation data. The results show that new method is faster than iteration method. The cost is almost equal to that of iteration method within one iterative process. New method also gives good results for downward continuation.
potential field, downward continuation, space domain, wave number domain, iteration method, iterative number
Geomagnetic aided navigation is a brand new navigation technology without accumulation error, which could correct the position errors of inertial navigation system. The integrated geomagnetic and inertial navigation system will possess the advantages of high precision and long duration, all of which makes it a very hot issue at home and abroad in navigation field [1-7]. Spatial geomagnetic database is a precondition when implementing geomagnetic navigation. Potential field continuation is an effective method for constructing geomagnetic database. Xu[8] proposed an integral iteration method for downward continuation, which can stably continue by 20 sampling intervals compared with the traditional fast Fourier transform method [9]. Zhang[10] proved the convergence of this method mathematically. Yu[11] also analyzed the influence of noise. According to the integral iteration idea and iterative formula in wave number domain [12], we propose an improved method. Model and real observation data tests both show that new method gives a good result.
It is known that in spatial domain potential field data and has relationship as follows:
( , , )

 z
2

   
 [( x
Where z axes positive directly downward, z

0
,
 
)
2
T

(
 
, 0)
( y
 
)
2  z
2 3 2
] d
  (1) denotes potential field data on observed plane at altitude z=0, denotes potential field data on target plane.

Corresponding author. Tel.: + 13667352074; fax: +073184576305-8212
E-mail address : markbluesky@163.com
218

Apparently, Equation (1) is two dimensional convolution between
T respect to x and y
. and h
 x 2  y 2  h
with
For convenience, redefine the variables by new symbols
( , )

( , , 0) (2)
( , )

( , , ) (3) and let

1
 z
2

( x
2  y
2  z
(4)
Applying new symbol, equation (1) can be rewritten as

   

Simply written as:
 
, y
     
(5)
( , )

( , )

( , ) (6) where “*” denotes convolution operation.
Applying Fourier transform to ( , ) , ( , )
 and ( , )
   

   

respectively, we can get

2

(

) dxdy

2

(

) dxdy
(7)
(8) and
   


2

(

) dxdy
 e 2
 u
2  v z (9)
In wave number domain, (6) can be expressed as
( , )

( , ) ( , ) (10)
It is clear that formula (10) is the expression of upward continuation of potential field data in wave number domain.
Aiming at solving ill-posed problems in downward continuation of potential field, Liu[13] proposed an iteration method for downward continuation in wave number domain. Here we give the basic process of this method as follows:
Without any prior information, the first step of the estimation is
( , ) may be a good choice as the prior estimated value of ( , ) . So
F u v

G u v G u v

H u v G u v (11)
According to the new estimated value , the second iterative process can obtain
ˆ
2
( , )
 ˆ
1
  ˆ
1
(12)
Formula (12) can be rewritten as:
ˆ ( , )
2

G u v

 
H u v
 
H u v 2

(13)
The same as above process, the n-th iterative process can get:
ˆ n
( , )
 ˆ n

1 u v

G u v

( , )
ˆ n

1
( , )] (14)
(14) can be rewritten as:
ˆ ( , ) n

G u v

 
H u v
 
H u v n

(15)
If the following condition is satisfied max
ˆ n
( , )
 ˆ n

1
( , )
 
1
(16) where

1 is a small positive number, we can obtain the result as
( , )

F

1
[ F u v where
F

1 denotes inverse Fourier transform.
(17)
219

According to the properties of geometric progression, (15) can be expressed as:
F u v

( , )
 
H u v n

1
 
( , )]

 
H u v n

1
(18)
Apparently, 0

( , ) 1 when z

0 , so we can define a new operator as follows,
R n

 
H u v n

1
(19)
Then the improved method for the downward continuation of potential field can be written as:
( , )

F

1
[ n
( , )] (20)
From formula (19), it is clear that the new operator of the improved method gradually amplifies the useful signal with the increasing of iterative number n. At the same time, noise is accumulated during iterative process. The precision of continuation will be degraded by it. Therefore, too large iterative number is inadvisable.
Here we give a new iterative termination rule as follows
F
ˆ n

1
( , )
 ˆ
( , ) max
 
2
(21) where

2
is a small positive number, 0.01 ~ 0.05 can be satisfied.
From formula (15), the absolute value of difference of two successive iterations can be expressed as
ˆ n

1
( , )
 ˆ n
( , )

( , ) [1

( , )] n

1
(22)
According to formula (18), (22) and (21),we can get
( , )[1

( , )] n

1 max
 
H u v n

1
 
2
(23)
It can be easily proved that
( , )[1

( , )] n

1
 
H u v n

1
is monotonically decreasing with iterative number n, if ( , )

1 . So, formula (23) can be equally expressed as arg max J

( , )
( , )[1

( , )]
 
( , )]
( , )

1 ， ( , )

N
 
2
(24)
By further simplification, we can obtain arg max J

( , )
. .
( , )
 log

2
 log

H u v
 
( , )

 
2
( , )

1
，
( , )

N

From formula (25), we can easily estimate the iterative number n.
In summary, using formula (9), (19), (20) and (25), we can get the procedures of the improved method as follows:
(1) Perform Fourier transform to observed potential field data ( , ) .
(2) Calculate the upward continuation operator ( , ) by formula (9)
(3) The iterative number n is estimated by formula (25)
(4) Then the new downward continuation operator R n is calculated by formula (19).
(5) Reverse Fourier transform to yield the object field in space by formula (20).
In this part, we use spherical gravity anomaly model to test the presented method. The theory formula to compute gravity data is as follows [13]
220



G
4
 r 3
3
  h
 z
[( o x
 x ) 2

( o y
 y ) 2 ( h z ) ]
3
2
(26) where ( , y
, ) denotes the position of spherical center. r is the spherical radius. We choose the gravitation constant G

6.67
,residual density 0.2 g/cm
3
,the z axes positive directly downward.
Suppose there are five spherical gravitational bodies underground,and the position of five spherical centers
(m): (0, 0,5000) , (6000, 6000, 6000) , (0, 0, 6000) , to Cartesian coordinate. Spherical radiuses are locate
( 5000, 7000,5000) , (10000, 10000, 4000) r
1
    
2 3 4 5
0.5 km in respectively according
. The observation area spread both from -25000 m to 25000 m in x direction and y directions. Thus the total number of observed data is direction respectively. Sample interval is 50 m in both
1001 1001 .
The theoretical gravity data on the plane of z=0 m and z=-500 m height are given, as shown in Fig.1 and
Fig.2a, respectively. Choose the small positive number obtain the iterative number n

23

2

0.05
. Using the method in this paper, we can
and gravity data of downward continued from -500m height to 0m without smoothing operation, as shown in Fig. 2b. The further comparison between the central line data of Fig. 2a and
Fig. 2b is shown in Fig.3. It is clear that result from the presented method is satisfactory. The improved method possesses the advantage of high precision in downward continuation of potential field. We also carry out a contrast experiment on the same model data with iteration method. The results show that new method is faster than iteration method, and almost achieves the same good results for downward continuation.
25000
20000
15000
10000
5000
0
-5000
-10000
-15000
-20000
-25000
-25000 -15000 -5000 5000 15000 25000 x/m
Fig. 1: Contour of theoretical gravity data (in mGal) at z=-500m
(a)
25000
20000
15000
10000
5000
0
-5000
-10000
-15000
-20000
-25000
-25000 -15000 -5000 x/m
5000 15000 25000
(b)
25000
20000
15000
10000
5000
0
-5000
-10000
-15000
-20000
-25000
-25000 -15000 -5000 x/m
5000 15000 25000
Fig. 2: Contour of gravity data (in mGal) at z=0m: (a)Contour of theoretical gravity data at z=0m ； (b)Contour of gravity data at z=0m from downward continuing gravity data at z=-500m ；
221

30
20
10
0
50
40
theoretical data
downward continued data
-10
-25,000 -20,000 -15,000 -10,000 -5,000 0 x/m
5,000 10,000 15,000 20,000 25,000
Fig. 3: Comparison between the central line data of Fig. 2a and 2b
Next, we use real magnetic anomaly data to test the presented method. Figure 4a shows the magnetic anomaly data on observed plane at altitude z=0. Sample interval is 200 m in both x and y directions. First, using the traditional FFT method for upward continuation, we can get the magnetic anomaly data at altitude z=-2000m, as shown in Fig. 4b. Choose the small positive number

2

0.01
. Using the method in this paper, we can obtain the iterative number n

123 and magnetic anomaly data downward continued from -2000m height to 0m without smoothing operation, as shown in Fig. 4c. The further comparison between the central line data of Fig. 4a and Fig. 4c is shown in Fig. 5. It is clear that the presented method gives a good result.
The new method possesses high precision and stability in downward continuation of potential field.
Moreover, it achieves a high computation speed. It takes about 1.2 s to continue 1024 1024 potential values on a Pentium(R) 2.41GHz PC. The cost is almost equal to that of iteration method within one iterative process.
90000
80000
70000
60000
50000
40000
30000
20000
10000
(a)
90000
80000
70000
60000
50000
40000
30000
20000
10000
(b)
10000 20000 30000 40000 50000 60000 70000 80000 90000
10000 20000 30000 40000 50000 60000 70000 80000 90000
90000
80000
70000
60000
50000
40000
30000
20000
10000
(c)
10000 20000 30000 40000 50000 60000 70000 80000 90000 x/m
Fig.4: Contour of the tested real magnetic anomaly data (in nT): (a)Contour of the real magnetic anomaly data in the observation plane
；
(b)Contour of 2km upward continuation for the magnetic anomaly data showed in Fig. 4a
；
(c)
Contour of 2km downward continuation for the magnetic anomaly data showed in Fig. 4b
222

theoretical data
downward continued data
300
200
100
0
-100
-200
-300
0 10,000 20,000 30,000 40,000 50,000 x/m
60,000 70,000 80,000 90,000 100,000
Fig. 5: Comparison between the central line data of Fig. 4a and 4c
In this paper, we present an improved method for downward continuation of potential field based on integral iteration idea and iterative formula in wave number domain, and we give a method for iterative number estimation. Model and real observation data tests show that new method is faster than iteration method, and almost achieves the same good results for downward continuation. In the future, more attention should be paid to the research of method of more effective iterative number estimation, since iterative number plays a very important role in the downward continuation like regularization parameter.
[1] Behzad Kamgar-Parsi and Behrooz Kamgar-Parsi, "Vehicle localization on gravity maps," Proc. SPIE Conf. on
Unmanned Vehicle Technology, vol. 3693, p. 182~191, 1999.
[2] Garner C. B, "Gravitational field maps and navigational errors," IEEE JOURNAL of OCEANIC ENGINEERING, vol. 27, p. 726~737, 2002.
[3] Xu D X, "Using gravity anomaly matching techniques to implement submarine navigation," Chinese J. Geophys.
(in Chinese), vol. 48 (4), p. 812~816, 2005.
[4] Peng F Q, "Geomagnetic model and geomagnetic navigation," Hydrographic surveying and charting, vol. 26, p.
73~75, 2006.
[5] Xu S Z, Zhang X D, and Zhang C D, "Several problems of underwater magnetic location," Journal of Dalian
Naval Academy, vol. 30, p. 4~6, 2007.
[6] Chen L W, Zhang H, Zheng Z Q, and Wu M P, "Technique of geomagnetic field continuation in underwater geomagnetic aided navigation," Journal of Chinese Inertial Technology, vol. 15, p. 693~697, 2007.
[7] Hao Y L, Zhao Y F, and Hu J F, "Preliminary analysis on the application of geomagnetic field matching in underwater vehicle navigation," Process in Geophysics, vol. 23, p. 594~598, 2008.
[8] Xu S Z, "The integral-iteration method for continuation of potential fields," Chinese J. Geophys. (in Chinese), vol.
49, p. 1176~1182, 2006.
[9] Xu S Z, "A comparison of effects between the iteration method and FFT for downward continuation of potential fields," Chinese J. Geophys. (in Chinese), vol. 50 p. 285~289, 2007.
[10] Zhang H, Chen L W, and Ren Z X, et al, "Analysis on convergence of iteration method for potential fields downward continuation and research on robust downward continuation method," Chinese J . Geophys. (in
Chinese), vol. 52, p. 1107~1113, 2009.
[11] Yu B, Zhai G J, and Liu Y C, et al, "Analysis of noise effect on the calculation error of downward continuation with iteration method," Chinese J . Geophys. (in Chinese), vol. 52, p. 2182~2188, 2009.
[12] Liu D J, Hong T Q, and Jia Z H, et al, "Wave number domain iteration method for downward continuation of potential fields and its convergence," Chinese J . Geophys. (in Chinese), vol. 52, p. 1599~1605, 2009.
[13] Chen S C and Xiao P F, "Wavenumber domain generalized inverse algorithm for potential field downward continuation," Chinese J . Geophys. (in Chinese), vol. 50, p. 1816~1822, 2007.
223