CHINESE JOURNAL OF GEOPHYSICS  2015, Vol. 58 Issue (2): 207-212   PDF    
Citation
BIAN Gang, XIA Wei, JIN Shao-Hua, WU Di, XIAO Fu-Min, SUN Xin-Xuan, WANG Mei-Na. 2015. DATUM REDUCTION FOR CORRECTION OF MULTI-STATION GEOMAGNETIC DIURNAL VARIATIONS USING THE LEAST SQUARES FITTING METHOD. CHINESE JOURNAL OF GEOPHYSICS, 58(2): 207-212.   
DATUM REDUCTION FOR CORRECTION OF MULTI-STATION GEOMAGNETIC DIURNAL VARIATIONS USING THE LEAST SQUARES FITTING METHOD
BIAN Gang1, 2, 3, XIA Wei1, 2, 3, JIN Shao-Hua1, 2, 3, WU Di1, 2, XIAO Fu-Min1, 3, SUN Xin-Xuan1, 3, WANG Mei-Na1, 3    
1. Department of Hydrography and Cartography, Dalian Naval Academy, Dalian 116018, China;
2. Institute of Navigation Engineering, Naval University of Engineering, Wuhan 430033, China;
3. Key Laboratory of Hydrography and Cartography of PLA, Dalian 116018, China
Abstract: The geomagnetic diurnal variation(GDV) is the main influence in marine magnetic survey,especially in offshore and large areas.The multi-station method is the main technique to deal with the GDV correction.However,to diminish the difference between the magnetic field data and multiple stations data,the correction datum of the secondary stations must be reduced to the main station.In marine sounding the least squares fitting(LSF) method has been applied in the interpolation of the height of the tide.Considering the similarity of the influence mechanism of the tidal variation and GDV,the LSF method has been introduced in the datum reduction.The datum of the secondary station can be corrected,which is decided by the expansion and contraction of the amplitude,the translation of the time(phase) of the diurnal variation of the main station,and the residual error of the diurnal variation correction of the main and secondary station is minimum with the LSF method.The validity of the method has been testified by the observed synchronous data of the multiple stations.The results show that the LSF method can be applied when the characteristics of the diurnal geomagnetic variations of the stations are similar,which requires a short synchronous observation period.Otherwise the synchronous correction(SC) method should be qualified.It is concluded that the LSF method can be effectively applied to the datum reduction with multi-station GDV correction,therefore the difference of the magnetic field which is caused by the datum can be diminished and the correction precision can be improved in magnetic survey of offshore and large areas.Moreover the reduction method must be adjusted to the need of the practice.In addition,the LSF method can be applied to the determination of the time difference and the correction value interpolation.
Key words: Marine magnetic survey    Multi-station diurnal variation correction    Datum reduction    Least squares fitting method(LSF)    Synchronous correction method(SC)    
1 INTRODUCTION

The influence of the GDV can be 10~40 nT in quiet days and 100~1000 nT in disturbed days in marine magnetic survey(Xu, 2003; Guan, 2005). In a given region, the GDV field is synchronous, which means the variation is correlative in space(Yuan et al., 2003; Gao et al., 2009). The synchronous region is 400~500 km in l and areas and 100~200 km in marine areas. Especially in sea areas, affected by the electromagnetic induction of the current, the conductivity structure of the seabed and coastal areas and the shape of the shoreline, the GDV in sea areas is more complex and the synchronous range can be less than 100~200 km(Qi, 1975). Meanwhile influenced by the coastal effect, the amplitude and the phase of the GDV can be different to a large extent when the base station is constructed in coastal areas(Auld, 1979; Whellams, 1996; Riddihough, 2002; Wang et al., 2011). When the survey area is in a large area or in offshore, restricted by the effective functional distance of the base station, the whole survey area can not be completely controlled by single base stations. Therefore the multi-station technique should be applied to deal with the GDV correction of large and offshore areas(Guo, 1999; Xu, 2007; Bian G, 2009). However the datum of the each base station must be reduced to the main station in multi-station GDV correction(DZ/T 0142-94, 1995; GB/T 13909-92, 1993; GJB. 7537-2012, 2012). The synchronous correction(SC)method is the common tool, in which the difference between the mean value of the GDV and the datum at the main station and the sub-station are assumed to be the same during the synchronous observation period. So the datum of the sub station must be reduced to the main station. However the influenced factors of the GDV are the same to each station. To obtain the stable datum, the synchronous observation period must be longer(Han, 1994; Bian, 2003; Bian, 2009).

According to the time and space distribution of the GDV, the influenc of the GDV on marine magnetic survey is similar to that of tide on marine sounding(Liang, 1996; Bian, 2008). At present, the tide correction techniques have been effectively applied in marine sounding, including the correction method of tidal zoning, time difference method and the least squares fitting(LSF)method(Liu, 2006). Especially in the LSF method, the tidal data have been adequately applied to calculate the ratio of tidal range, the time difference and the datum deviation between two tidal stations. Furthermore the water level correction precision has been evidently improved(Liu, 2003). Based on the time and space characteristics of the GDV, this work introduced the LSF method into the base value reduction of the multi-station GDV correction, analyzed the feasibility of this method and testified its validity by the observed synchronous data of multiple stations.

2 THE LEAST SQUARES FITTING METHOD

In 1992, Liu firstly introduced the LSF method into the tide interpolation in marine sounding, with which the water level correction precision has been evidently improved. Because of the similarity of the geomagnetic diurnal variation to the marine magnetic survey and the tidal variation to the marine sounding, this method can be applied to the base value reduction in multi-station diurnal variation correction. The basic principle is stated below.

The method is built on that the GDV in the survey area varies with the planar position linearly. So the amplitude of the GDV at the main station is exp and ed and contracted, and the time(phase)is translated. Furthermore to the GDV correction values, the sum of the square of the residual errors for both the main and sub station is minimum with the least square constraints conditions. Firstly, both the main and sub station, the GDV data are fitted by the least square method to determine the transfer parameters γAB, δAB, and εAB:

where γAB is the ratio of the amplitude, δAB is the time difference and εAB is the base value deviation between two stations A and B.

As shown in Eq.(1), the main problem of the reduction of the base value in multi-station GDV is to obtain the ratio of the amplitude γAB(or γAB), the time difference δAB(or δBA) and the base value deviation εAB(or εBA). The determination of these parameters are shown as below.

In Fig. 1, the discrete sampling series of the GDV curve of main station A and sub station B can be expressed as

where t0 is the starting time, △t0 is the sampling interval, △t0=1, 5, 10, 20, 30, 60 s; and N is the total sampling number.

Fig.1 Sketch of the least squares fitting method

The error equation of the main station and sub station can be expressed as

Given the starting values γ0, δ0 and ε0, the expression(2)can be linearized and the matrix form is
where V is the closure error vector, A is the designed matrix, the row element is [TA(t0 +n△t0 +δ0), γ0T' A(t0 + n△t0+δ0), 1], n = 0, 1, · · ·, N. T' A(t0+n△t0+δ0)is the derivative of T to δ, X = [△γ, △δ, △ε]T is the unknown parameter vector, L is the constant vector, and the row element is [γ0TA(t0+n△t0+δ0)+ε0-TB(t0+n△t0)], n = 0, 1, · · ·, N.

According to the least squares rule [V TV ] = min,

It further yields
Therefore to any sub station i, applied the GDV value both main station A and sub station i, the correlative parameters between two stations can be obtained by the least squares method. At last the base value of the sub station i can be obtained as
where T'i is the starting reduction value of sub station i, which can be selected arbitrarily and the result is not affected by the value. In this paper, the GDV value is used as the starting value, that means T'i = 0, εAi is the base value deviation which is obtained by the GDV value for both main station A and sub station i with the LSF method.

We can note that this method is based on the linear change of the GDV between the stations with the planar positions. The relationship of the GDV between the stations can be really reflected through the expansion and contraction of the amplitude, and the translation of the time(phase)of the GDV of the main station. And also the synchronous observation period can be less than that the SC method. When the difference of the diurnal variation between the stations is big, the method is inadvisable.

3 ANALYSIS AND COMPARISON WITH EXAMPLES

To prove the validity of the LSF method, the synchronous observed data of the GDV has been applied to analyze and compare. Furthermore the maximum comparative error(MCE)has been selected as the evaluating index(Bian, 2009), and the results have been compared with the synchronous correction method(Bian, 2008).

The synchronous observed GDV data has been acquired in a marine magnetic survey in 2006. The sketch map of the base station is shown in Fig. 2. The distances between the stations are respectively DAB = 97.69 km, DAC = 925.20 km, DBC = 828.72 km. The synchronous observational time period is Aug. 25-Aug. 28 2006. The high precision of magnetometer HC-90D is adopted, the sensibility is 0.001 nT. The sample interval is 1 minute, and average of the GDV is selected as the base value of each station. The GDV curve is shown in Fig. 3.

Fig.2 Sketch map of basic stations

Fig.3 Synchronous curves of GDV at three base stations

As shown in Fig. 3, the amplitude of the GDV during the synchronous period at each station is less than 100 nT. The type of the GDV is of the quiet day according to the specifications GJB.7537-2012. The difference between the stations will be augmented with the increase of distance. The synchronous curves of stations A and station B are consistent. The big differences are present between the synchronous curves of station A or station B and station C, which are caused by the attitude difference between the stations.

The above synchronous observed data have been applied to validate the feasibility of the LSF method. And the MCE of the base value has been selected as the evaluating index.

The station A has been selected as the main station, with base value 46994.27 nT. The station B and station C are sub stations, and the base value can be obtained by different methods as shown in Tables 1-4.

Table 1 Results of the SC method for base station B(unit: nT)

Table 2 Results of the LSF method for base station B(unit: nT)

Table 3 Results of the SC method for base station C(unit: nT)

Table 4 Results of the LSF method for base station C(unit: nT)

For sub base station B, although the daily means of the GDV are different on varied days, with the maximum MCE 16.38 nT, the reduction values from different methods are almost equal(the MCE of SC method is 0.82 nT, the MCE of LSF is 0.33 nT). Therefore, after reduction of the base value, the magnetic field difference caused by the base value can be greatly decreased. Furthermore the magnetic field level in the survey area can be unified. However, the maximum of MCE between the base values on the same day is 0.53 nT(Aug. 27), while the maximum of MCE between the SC method is 0.82 nT and the LSF method is 0.33 nT. For sub base station B, the distance is less(DAB = 97.69 km), the types of GDV of two stations are similar, the reduction parameters in different time are stable, the base values are also the same. In this case the LSF method will be better than the SC method.

For sub base station C, the mean values of the GDV are different on different days, and the maximum MCE is 18.21 nT. But the MCE of different methods are greatly decreased(the MCE of the SC method is 4.51 nT, the MCE of the LSF method is 9.10 nT). Therefore, after reduction of the base value, the magnetic field difference caused by the base value can be greatly decreased. Furthermore the magnetic field level in the survey area can be unified. Moreover the maximum of MCE between the base values on the same date is 3.41 nT(Aug. 28). And the maximum of MCE between the SC method is 4.51 nT and the LSF method on the same day is 9.10 nT. For sub base station C, the distance is bigger(DAC = 925.20 km), the types of GDV of two stations are different, the reduction parameters change with time, and the base values are different. In this case the SC method is superior to the LSF method.

Comparing the results of sub station B to sub station C, the shorter of the distance between two base stations, the less will be the MCE of the reduction value. Therefore the consistency exists in different reduction methods with shorter distance. Then the LSF method will be better than the SC method(0.33 nT<0.82 nT). Otherwise, the SC method will be superior to the LSF method(4.51 nT<9.10 nT)over large distances.

We can summarize that the magnetic field difference caused by the base value can be greatly decreased through the reduction of the base value in multi-station GDV correction. Furthermore the GDV precision can be guaranteed. For different types of GDV, the reduction method can be different and must be selected suitably.

4 CONCLUSIONS

(1)The influence of GDV on marine magnetic survey is similar to that of the tide on marine bathymetric measurement. The LSF method is built on that the GDV in the survey area varies linearly with the planar position. So the amplitude of the diurnal variation at the main station can be exp and ed and contracted, and the time(phase)can be translated. After such processing to the GDV correction values of both main station and sub-station, the sum of the squares of their residual errors is minimum under the least square constraint. The magnetic field difference caused by the base value can be greatly decreased through the reduction of the base value in multi-station GDV correction. Furthermore the magnetic field level in the survey area can be unified.

(2)Analysis of synchronous observational data from multiple stations shows that the LSF method can yield best results when the main station and sub-station have similar GDV characteristics, and the requirement of the time period can be lower. Therefore the SC method is superior to the LSF when GDV characteristics are much different. In multi-station GDV correction, the method for datum reduction should be selected appropriately according to real needs.

(3)The LSF method can also applied to the determination of time differences and interpolation of the GDV correction values, which can greatly improved the GDV correction precision for marine magnetic survey in marine areas.

ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China(41374108, 41476087).

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