Pure Appl. Geophys. 173 (2016), 1265–1278 Ó 2015 Springer Basel DOI 10.1007/s00024-015-1168-9
Pure and Applied Geophysics
A least-squares minimization approach for model parameters estimate by using a new magnetic anomaly formula E. R. ABO-EZZ1,2 and K. S. ESSA1 Abstract—A new linear least-squares approach is proposed to interpret magnetic anomalies of the buried structures by using a new magnetic anomaly formula. This approach depends on solving different sets of algebraic linear equations in order to invert the depth (z), amplitude coefficient (K), and magnetization angle (h) of buried structures using magnetic data. The utility and validity of the new proposed approach has been demonstrated through various reliable synthetic data sets with and without noise. In addition, the method has been applied to field data sets from USA and India. The best-fitted anomaly has been delineated by estimating the rootmean squared (rms). Judging satisfaction of this approach is done by comparing the obtained results with other available geological or geophysical information. Key words: New magnetic anomaly formula, a linear leastsquares, model parameters, rms.
1. Introduction The main objective of magnetic surveying is identification of regions of the Earth’s crust that have anomalous magnetizations (DOBRIN and SAVIT 1988). In applied geophysics, the anomalous magnetizations might be associated with local mineralization that is potentially of commercial interest or they could be due to subsurface structures that have a bearing on the location of oil deposits (SCHUMACHER and ABRAMS 1996). The estimation of model parameters of a buried structure is an essential target in interpretation of magnetic data. Thus, many published methods have presented for interpreting magnetic data, for example,
1 Geophysics Department, Faculty of Science, Cairo University, P.O. 12613, Giza, Egypt. E-mail:
[email protected];
[email protected] 2 Prince Sattam bin Abdulaziz University, Al-Kharj, Kingdom of Saudi Arabia.
Hilbert transforms (MOHAN et al. 1982), Euler deconvolution (THOMPSON 1982), monograms (PRAKASA RAO et al. 1986), Walsh transform technique (SHAW and AGARWAL 1990), 3D inversion (LI and OLDENBERG 1996), NFG method (BEREZKIN et al. 1994), SPI (THURSTON and SMITH 1997), neural networks (HAJIAN et al. 2012), continuous curves (ABDELRAHMAN and ESSA 2005), LWN method (SALEM et al. 2005), wavelets (COOPER 2006), nonlinear leastsquares minimization (ABDELRAHMAN et al. 2007), DEXP method (FEDI 2007), tilt depth (SALEM et al. 2007), multi-scale methods (CELLA et al. 2009), analytical signal method (MA and DU 2012), parametric curves (ABDELRAHMAN et al. 2012). However, the drawback with these techniques is that some of them generate large number of invalid solutions due to sensitivity to noise, does not distinguish between adjacent causative bodies, window sizes are highly subjective, used few data points on profile. Also, some of them require knowledge about the structural index parameter of the source and model parameter dependent. In the present paper, we have developed an efficient approach in order to estimate the magnetic parameters as best as possible, e.g., depth (z), an effective angle of magnetization (h), and amplitude coefficient (K) of simple shaped bodies such as vertical cylinder, horizontal cylinder and sphere. Firstly, various forward models are linearized by a transformation of the model parameters. Secondly, the linear least-squares inversion of an overdetermined system is used in order to obtain the least-squares estimates of the transformed parameters. Finally, the model parameters of the buried structures are subsequently estimated from magnetic data measured along a profile by an inverse transformation of the solution
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parameters. The efficiency and stability of this approach have been tested through synthetic examples with and without random errors and have been demonstrated by three field examples from USA and India. In all cases, the estimated model parameters are consistent with the actual ones.
2. The method 2.1. The forward modeling The magnetic anomaly (T) due to most simple geologic models like a sphere, a horizontal cylinder and a thin sheet at any point on the free surface along the principle profile in a Cartesian coordinate system is given by ABDELRAHMAN and ESSA (2015) as: Tðxi ; zÞ ¼ K
Az2 þ Bxi þ Cx2i ; ðx2i þ z2 Þq
ð1Þ
i ¼ 0; 1; 2; 3; . . .; N;
Pure Appl. Geophys.
effective magnetization angle, FHD and SHD denote the first and the second horizontal derivatives of the magnetic anomaly, respectively, N is the number of data points, and q is the shape factor. For example, the shape factor is as follows: for sphere it is 2.5, for horizontal cylinder 2, and for thin sheet 1.0. Examples of K and h for the case of the sphere are the magnetic moment and effective angle of magnetization in the plane of the vertical profile coinciding with the xdirection, respectively (RAO et al. 1973; RAO and SUBRAHMANYAM 1988). For the case of thin sheet and horizontal cylinder, we use theoretical expressions of K and h (GAY 1965, 1966) which are given in Table 1. In this table, c is magnetic susceptibility contrast; I0 is the true inclination of the geomagnetic field; T0i and I0i are the effective intensity and effective inclination of the geomagnetic field in the vertical plain normal to the strike of the body; t and d are the thickness and the dip of the thin sheet; S is the cross-sectional area of the horizontal cylinder; and a is the azimuth of the body measured in the clockwise direction from
where
8 8 > 3 sin2 h 1 3z sin h > > > > > > > > > > 2 sin h 3z cos h > > > > > > < 3z sin h < cos h ; ; B ¼ A¼ > > cos h 2z sin h > > > > > > > > > > > > > > : > : cos h sin h z 8 2 3 cos h 1 > > for a sphere ðtotal field Þ > > > > sin h > for a sphere ðvertical fieldÞ > > > > < 2 cos h for a sphere ðhorizontal fieldÞ C¼ ; for a horizontal cylinder, FHD of thin sheet, > cos h > > > > > > and SHD of geological contact ðall fieldsÞ > > > > : for a thin sheet ðthin dikeÞ and FHD of geological contact ðall fieldsÞ: 0
In Eq. (1), z is the depth from surface to the center of the body, xi is the horizontal distance from the observation to the source, K is the amplitude coefficient (effective magnetization intensity), h is an
magnetic north. However, the examples of K and h in the case of the spheres are the magnetic moment and effective angle of magnetization in the plain of the principle profile coinciding with the x-direction.
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Table 1 Characteristic amplitude coefficient K and an effective angle of magnetization h for vertical, horizontal, and total magnetic field anomalies due to thin sheets and horizontal cylinders (after Gay 1963, 1965) Field
Thin sheets
Vertical Horizontal Total
Horizontal cylinders
Amplitude coefficient (K)
Magnetization parameter (h)
Amplitude coefficient (K)
Magnetization parameter (h)
2 c t T0i 2 c t T0i sin a 2 c t T0i sin Io = sin I0i
I0i d I0i d 90 2I0i d 90
2 c T0i S 2 c T0i S = sin a 2 c t T0i S =ðsin Io =I0i Þ
I0i 90 I0i 180 2I0i 180
In all cases, the values of K and h can be used for detailed interpretation as shown in Table 1.
p23 ; p12 ¼ p2 p4 ; p13 ¼ p3 p4 and p14 ¼ p24 in Eq. (5c), we get the following equations: Ti x2i þ Ti p1 p2 p3 xi p4 x2i ¼ 0 ;
2.2. The inverse problem solution Multiplying the both sides of Eq. (1) by the q mathematical term x2i þ z2 the following equation is obtained: q Ti x2i þ z2 K A z2 K B xi KC x2i ¼ 0 ; ð2Þ i ¼ 1; 2; 3; . . .; N : Equation (2) is a nonlinear function of the parameters z, K, A, B, and C. This nonlinearity can be overcome by introducing new set of p variables as follows: p1 ¼ z2 ; p2 ¼ KAz2 ; p3 ¼ KB ; and p4 ¼ KC : ð3Þ Using Eq. (3) and substituting in Eq. (2), we get the following linear form: q Ti x2i þ p1 p2 p3 xi p4 x2i ¼ 0 : ð4Þ By supposing q = 1 (thin sheet model), q = 2 (H. cylinder model) and q = 2.5 (sphere model) in Eq. (2), we get the following linear equations: Ti x2i þ Ti p1 p2 p3 xi p4 x2i ¼ 0 ;
ð5aÞ
Ti x4i þ 2Ti x2i p1 þ Ti p21 p2 p3 xi p4 x2i ¼ 0 ; ð5bÞ 2 8 2 6 2 2 4 3 2 2 4 Ti2 x10 i þ 5Ti xi p1 þ 10Ti xi p1 þ 10Ti xi p1 þ 5Ti xi p1 2 5 þ Ti p 1 ¼ p22 þ 2p2 p3 þ p23 x2i þ 2p2 p4 x2i þ 2p3 p4 x3i þ p24 x4i :
ð5cÞ p31 ;
Putting p5 ¼ p21 in Eqs. (5b) and (5c) and p6 ¼ p7 ¼ p41 ; p8 ¼ p51 ; p9 ¼ p22 ; p10 ¼ p2 p3 ; p11 ¼
ð6aÞ
Ti x4i þ 2Ti x2i p1 þ Ti p5 p2 p3 xi p4 x2i ¼ 0 ; ð6bÞ 2 8 2 6 2 4 Ti2 x10 i þ 5Ti xi p1 þ 10Ti xi p5 þ 10Ti xi p6
þ 5Ti2 x2i p7 þ Ti2 p8 p9 2p10 xi p11 x2i ð6cÞ 2p12 x2i 2p13 x3i p14 x4i ¼ 0 : The global optimal solutions of the linear system of Eqs. (6a), (6b), and (6c) are found by minimizing the following mathematical objective functions onto the real space R. The mathematical form can be written as follows: p ¼ arg min uðpÞ ¼
N X
½uðpÞ2 ¼ min:
ð7Þ
i¼1
To solve this mathematical form use a minimization approach, i.e.: P o Ni¼1 ½uðpÞ2 ¼ 0; i ¼ 1; 2; 3; . . . ð8Þ opi This system of simultaneous linear equations could be written in matrix form as: G:m ¼ d ;
ð9Þ
where G is a square matrix, m is a vector matrix including the unknowns, and d is a vector matrix. The linear system of algebraic Eqs. (9) can be easily solved either by one of the direct method (Gauss) or by one of the iterative methods (Jacobi and Gauss–Seidel) (COLLINS 2003; GOLUB and MEURANT 1983). In each case, the magnetic parameters
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800
20 16
Magnetic anomaly (nT)
700
Magnetic anomaly (nT)
Noisy anomaly Predicated anomaly
Model parameters q = 1.0 z = 10 m θ = -30° K = 800 nT*m
600
500
400
300
Noisy anomaly Predicated anomaly
Model parameters q = 2.0 z = 12 m θ = 45° K = 2000 nT*m2
12 8 4 0 -4
200
-8 100 -20
-15
-10
-5
0
5
10
15
20
-20
x (m)
-10
-5
0
5
10
15
20
x (m)
Figure 1 A noisy and predicted magnetic anomaly profiles over a magnetized thin sheet model (q = 1) with model parameters are: z = 10 m, h = -30o, K = 800 nT m, and profile length = 40 m
Figure 2 A noisy and predicted magnetic anomaly profiles over a magnetized horizontal cylinder model (q = 2) with model parameters are: z = 12 m, h = 45o, K = 2000 nT m2, and profile length = 40 m
related to the causative body are computed as shown in Appendix.
25
3. Theoretical examples In this study, the advantages of the proposed technique were examined through three synthetic magnetic anomalies, thin sheet, a horizontal cylinder and a sphere model, which are generated using Eq. (1). The thin sheet model (q = 1) parameters is: z = 10 m, h = -30°, K = 800 nT m, and profile length = 40 m. The horizontal cylinder model (q = 2) parameters is: z = 12 m, h = 45o, K = 2000 nT m2, and profile length = 40 m. The sphere model (q = 2.5) parameters is: z = 8 m, h = 60o, K = 6500 nT m3, and profile length = 40 m. Random Gaussian noise is added to the theoretical magnetic anomalies with a maximum error of 10 nT (Figs. 1, 2, 3), computed by:
Magnetic anomaly (nT)
20
DTrand ðxi Þ ¼ DTðxi Þ þ 10ðRANðiÞ 0:5Þ:
-15
Noisy anomaly Predicated anomaly
Model parameters q = 2.5 z=8m θ = 60° K = 6500 nT*m3
15
10
5
0
-5
-10 -20
-15
-10
-5
0
5
10
15
20
x (m) Figure 3 A noisy and predicted magnetic profile over a spherical magnetized model (q = 2.5) with model parameters are: z = 8 m, h = 60o, K = 6500 nT m3, and profile length = 40 m
ð10Þ
The random Gaussian distributions are intentionally used in order to regenerate magnetic anomalies as close as possible to the real observed field measurements. The regenerated magnetic anomalies are
interpreted by using the procedures of the proposed method, whereas the evaluated magnetic parameters are presented in Table 2. Table 2 clearly shows that the error (e) in the obtained parameters in case of noise-free data is equal
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Table 2 Numerical results of the present method applied to synthetic examples: the thin sheet model (z = 10 m, h = -30°, K = 800 nT m), the horizontal cylinder model (z = 12 m, h = 45o, K = 2000 nT m2) and the sphere model (z = 8 m, h = 60o, K = 6500 nT m3) with profile length = 40 m, and corrupted data with ± 10 % of random noise
Thin sheet Horizontal cylinder Sphere
Using noise-free data
Using noisy data
z (m)
e (%)
h (°)
e (%)
K
e (%)
z (m)
e (%)
h (°)
e (%)
K
e (%)
10 12 8
0 0 0
-30 45 60
0 0 0
800 2000 6500
0 0 0
10.12 12.47 8.35
1.2 3.9 4.4
-28.42 43.93 64.4
5.3 2.4 4.0
789 2051 6513
1.4 2.6 0.2
to zero in all examined cases. However, using contaminated noisy data will lead to different errors in each case. In case of thin sheet model, the errors are 1.2, 5.3, and 1.4 % for z, h, and K, respectively. In case of horizontal cylinder model, the errors are 3.9, 2.4, and 2.6 % for z, h, and K, respectively. In addition, in case of sphere model, the errors are 4.4, 4.0, and 0.2 % for z, h, and K, respectively. Modeling results presented in Table 2 also show a good agreement between assumed and evaluated magnetic parameters. Finally, Table 2 shows that the method can produce results within a reasonable tolerance given noisy data.
160 Observed anomaly Thin sheet model fit Horizontal cylinder model fit Sphere model fit
140
Magnetic anomaly (nT)
Type of model
120 100 80 60 40 20 0
4. Field examples
-20 -200
-150
-100
-50
0
50
100
150
200
x (m)
We discuss three published field examples, from mineral exploration in southern Illinois, Arizona from USA and in Manjampali from India with various depths of burial and geologic complexity are analyzed in order to examine the applicability of the developed inversion technique. These particular data sets were selected in this research paper for two main reasons. First, the field data sets produced by a body that can be represented by a simple source. Second, these data sets were measured from sites with available drilling information; hence we can compare our numerical results yielded from the inversion to those confirmed from drilling. 4.1. A peridotitic dike field example, southern Illinois, USA A prominently observed magnetic anomaly has been measured over an altered peridotitic dike in
Figure 4 A residual magnetic anomaly profile over a buried causative dike, southern Illinois, USA (KIRKHAM 2001)
southern Illinois, USA (KIRKHAM 2001). Figure 4 shows the pertinent residual magnetic anomaly profile (east–west direction), and the strike of the buried causative dike is northwards (KIRKHAM 2001). From closely-spaced drilling done by the coal mining company, it is known that the depth to the top of the dike (from the surface) is about 10 m and its width is also about the same. This is resulted from the analysis of the high-resolution aeromagnetic survey carried out by the US Geological Survey (HILDENBRAND and RAVAT 1997). In addition, the coal company was interested in precisely locating these dikes without expensive drilling because the dikes obstruct their mining operations. This anomaly profile has a length of 400 m, and was digitized at intervals of 5.43 m.
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Table 3 Modeling results of the buried causative dike, southern Illinois, USA Type of model
Computed parameters
Thin sheet Horizontal cylinder Sphere
rms (nT)
z (m)
h (°)
K
10.42 88.5 45.88
50.5 40.26 52.3
192.02 nT m 2313.17 nT m2 2433.2 nT m3
8.73 31.6 28.59
The bold values indicate the accurate model parameters estimation Table 4 Comparative results of the southern Illinois field example, USA, and the % of error from each method Parameters Drilling information Using KIRKHAM (HILDENBRAND method (2001) and RAVAT 1997) e (%)
Using SALEM and RAVAT Using SALEM method (2003) method (2005)
Using ORUC method (2012)
Using the present method
e (%)
e (%)
e (%)
z (m) h (°) K (nT m)
9.17 – –
10 – –
0.0 – –
For each shape factor, the proposed inverse technique has then been applied to the observed data to determine the model parameters z, h, and K (Table 3). Using the estimated model parameters for each shape factor, we computed the root-mean square (rms) of the differences between the observed and the predicted magnetic anomalies (Fig. 4). This demonstrates that the shape of the buried structure resembles a thin sheet-like structure model buried at a depth of 10.42 m. The shape and the depth of the ore body obtained through the present method is consistent very well with those obtained by HILDENBRAND and RAVAT (1997), KIRKHAM (2001), SALEM and RAVAT (2003), SALEM (2005), and ORUC (2012) (Table 4).
-8.3 – –
9.3 – –
7.0 – –
7.25 – –
-27.5 – –
10.42 50.5 192
4.2 – –
600 Observed anomaly Thin sheet model fit Horizontal cylinder model fit Sphere model fit
500
Magnetic anomaly (nT)
10 – –
e (%)
400
300
200
100
0
-100 -400
-300
-200
-100
0
100
200
300
400
x (m)
4.2. The Pima copper mine field example, Arizona, USA The Pima mining district is an old and one of the major sources of the copper in the United States. It is located south-southwest of Tucson, along the eastern pediment of the Sierrita Mountains in Pima County, Arizona, USA. A vertical component of magnetic
Figure 5 A vertical magnetic anomaly for the Pima cooper mine, Pima County, Arizona, USA (Gay 1963; his Figure 10)
anomaly was shown in Fig. 5 for the Pima cooper mine (GAY 1963; his Figure 10, I0i ¼ 62 ). This vertical magnetic anomaly profile of long 750 m was
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Table 5 Modeling results of Pima copper mine anomaly, Arizona, USA Type of model
Computed parameters
Thin sheet Horizontal cylinder sphere
rms (nT)
z (m)
h (°)
K
61.5 410.3 308.75
266.4 76.43 35.87
1219.2 nT m 24564.7 nT m2 163095.2 nT m3
40.5 188.9 165.79
The bold values indicate the accurate model parameters estimation
Table 6 Comparative results of the Pima copper mine anomaly, Arizona, USA, and the % of error from each method Drilling information (GAY 1963)
Using GAY method (1963)
Using ABDELRAHMAN et al. (2003)
e (%) z (m) h (°) K (nT m) d (°)
64 – – 45 S
69.8 – – 55 S
9.1 – – –
Using ABDELRAHMAN and ESSA method (2015)
e (%) 68 -52 1611 –
digitized at an interval of 25 m. The digitized values were subjected to the proposed inverse technique to determine the model parameters z, h, and K for each shape factor (Table 5). The model parameters obtained are: z = 61.5 m, h = -66.4°, K = 1219 nT m and amount of dip (d) = 52° S which have the minimum rms. This suggests that the shape of the buried structure resembles a thin sheet (dike)-like body (Fig. 5). The shape and the depth of the ore body obtained by the present method are compatible with those obtained from drilling information and those obtained by other published results as in GAY (1963), ABDELRAHMAN et al. (2003), and ABDELRAHMAN and ESSA (2015) (Table 6).
60 – – –
Figure 6 shows the vertical magnetic anomaly over a narrow band of quartz–magnetite in Manjampalli near Karimnagar town, India (MURTHY et al. 1980). Twenty-five equi-spaced observations are extended along this profile at an interval of 5.86 m. All anomaly points are used to calculate the model
e (%)
6.25 – – –
61.5 -66.4 1219 52 S
3.9 – – 15
10000 Observed anomaly Thin sheet model fit Horizontal cylinder model fit Sphere model fit
8000
6000
4000
2000
0
-2000
-4000 -75
4.3. A narrow band of quartz–magnetite field example, Manjampalli, India
Using the present method
e (%)
6.25 – – –
Magnetic anomaly (nT)
Parameters
-50
-25
0
25
50
75
x (m) Figure 6 A vertical magnetic anomaly component over a narrow band of quartz–magnetite in Manjampalli near Karimnagar town, India (MURTHY et al. 1980)
parameters for each shape factor (q = 1, q = 2, and q = 2.5) (Table 7). The calculated parameters are used to determine the best fitting geometrical body
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Table 7 Modeling results of the narrow band of quartz–magnetite in Manjampalli near Karimnagar town, India Type of model
Computed parameters
Thin sheet Horizontal cylinder Sphere
rms (nT)
z (m)
h (°)
K
12.22 25.67 100.81
85.34 52.66 65.42
7890 nT m 155,514 nT m2 1,058,923 nT m3
1980 997 2937
The bold values indicate the accurate model parameters estimation
Table 8 Comparative results of a narrow band of quartz–magnetite anomaly, Manjampalli, India Parameters
Using RADHAKRISHNA MURTHY et al. Method (1980)
Using SUDHAKAR et al. method (2004)
Using AL-GARNI method (2011)
Using the present method
z (m) h (°) K (nT m2)
23.71 49 –
26.52 37 –
25.26 – –
25.67 52.66 15,514
(shape factor) by using the rms criterion. The model with the minimum rms was for a horizontal cylinder. Its parameters are: z = 25.67 m, h = 52.66°, and K = 15514 nT m2 (Fig. 6). The shape and the model parameters of the ore body obtained by the present method calibrated with the results published in other scientific literatures (MURTHY et al. (1980), SUDHAKAR et al. (2004), and AL-GARNI (2011)) (Table 8).
5. Conclusions We propose a linear least-squares approach to interpret magnetic data caused by simple geometrically structures like a thin sheet (thin dike), a horizontal cylinder and a sphere using a new magnetic anomaly formula. In this approach, the forward equation is analyzed in order to remove successively nonlinearities of the model parameters. After simple manipulations, the nonlinear parameters (depth z, magnetization angle h, and amplitude coefficient K) become linear because the estimation of z, h, and K is formulated as linear functions. The present method has the advantages of being simple, fast, and independent of an initial guess. In addition, the major advantage of using this approach is that not
only can we determine the model parameters (z, h, and K), but we can also estimate the source of the caused magnetic anomaly (shape factor q). Also, our inversion technique has been developed under the assumptions that the surface magnetic anomaly is produced by the induced magnetization only and that there are no remanent magnetization. The synthetic examples show that the feasibility of the proposed method is accurate and stable. Furthermore, applying the proposed technique to different real cases from USA and India has confirmed that the obtained results are consistent with the information from drilling and results previously published in other journals.
Acknowledgments We would like to thank the Colin Farquharson, Editor, and three capable reviewers for their excellent suggestions and thorough review that enhanced our original manuscript. Also, we would like to thank Dr. Sayed Ismail, Ph.D. in linguistics and translation for proofreading and Dr. Salah Mehanee, Geophysics Department, Faculty of Science, Cairo University, for his constant help and encouragement.
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A least-squares minimization approach for model parameters estimate
Appendix Problem formulation due to thin sheet model According to Eq. (1), the general expression of an arbitrary magnetized thin sheet-like structure (q = 1) in a Cartesian coordinate system is given by: Tðxi ; zÞ ¼ K
Az2 þ Bxi þ Cx2i ; ðx2i þ z2 Þ
For simplification, Ti is used instead of Tðxi ; zÞ in the rest of this paper. Multiplying the both sides of Eq. (11) by ðx2i þ z2 Þ we obtain the following: Ti x2i þ Ti z2 K A z2 K B xi KC x2i ¼ 0; i ¼ 1; 2; 3; . . .N: ð12Þ Equation (12) is not linear in the function of parameters z, k A, B and C. This nonlinearity is avoided by introducing new variables p1, p2, p3 and p4, defined as follows: p1 ¼ z2 ;
ð13Þ
p2 ¼ KAz2 ;
ð14Þ
p3 ¼ KB;
ð15Þ
p4 ¼ KC:
ð16Þ
and
Introducing these new variables into Eq. (12), it can be concluded that Ti x2i þ Ti p1 p2 p3 xi p4 x2i ¼ 0 :
ð17Þ
Equation (17) is now linear in function of variables p1, p2, p3 and p4. The global optimal solution of the linear system of this equation is found by minimizing the following mathematical objective function onto the real space R: In mathematical form, it can be written as: p ¼ arg min uðpÞ N X ¼ ðTi x2i þ Ti p1 p2 p3 xi p4 x2i Þ2 : Subject to p 2 R
To solve this mathematical nonlinear program, it is sufficient to find the unique solution of the following system of linear equations: ouðpÞ ¼ 0 ð i ¼ 1; 2; 3; and 4 Þ: oðpi Þ This system of linear equations could be written in matrix form as: G m ¼ d;
i ¼ 1; 2; 3. . .N: ð11Þ
i¼1
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ð18Þ
where G is an invertible squared matrix of 4 9 4 dimensions given as follows: 3 2 N N N N P 2 P P P 2 T T T x T x i i i i i 7 i 6 i¼1 i¼1 i¼1 7 6 i¼1 6 N N N P P 2 7 6 PT N xi xi 7 i 7 6 7 6 i¼1 i¼1 i¼1 G ¼ 6 N 7 N N N P P P 7 6 P 2 3 6 Ti xi xi xi xi 7 7 6 i¼1 i¼1 i¼1 i¼1 7 6 N N N N 5 4 P P P P 2 2 3 4 Ti xi xi xi xi i¼1
i¼1
i¼1
i¼1
m and d are vectors of four dimensions given as: 2 3 p1 6 p2 7 7 m ¼ 6 4 p3 5; p4 and 2
N P
Ti2 x2i
3
7 6 7 6 1¼1 7 6 P N 6 27 Ti xi 7 6 7 6 d ¼ 6 1¼1 7 N 7 6 P 6 Ti x3i 7 7 6 1¼1 7 6 N 5 4 P Ti x4i 1¼1
The depth to the center of the body (z) is found by using Eq. (13), the effective magnetization angle (h) is obtained by Eqs. (14), (15) and (16) as: p2 B h ¼ Arc tan ; ð19Þ p3 A z 2 by knowing the value of h, the values of A and B can be calculated as follows: cos ðhÞ A¼ ; 2 ð20Þ and B ¼ sin ðhÞ:
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Knowing the values of z, A and B, the amplitude coefficient (K) can be obtained as follows: 1 p3 p4 þ k¼ : ð21Þ 2 A z2 B
Problem formulation due to a horizontal cylinder model
Tðxi ; zÞ ¼ K
ðx2i þ z2 Þ2
;
Arranging Eq. (23) we obtain: Ti x4i þ 2Ti x2i z2 þ Ti z4 K A z2 K B xi KC x2i ¼ 0 ; i ¼ 1; 2; 3; . . .N :
i¼1
Ti2
i¼1 N P i¼1 N P i¼1
Ti Ti xi
Ti x2i
ð29Þ
ð31Þ
where G is a square matrix of 5 9 5 dimensions given as follows:
p5 ¼ KC:
G m ¼ d;
Ti2 x2i
i¼1 N P
ð28Þ
This system of linear equations could be written in matrix form as:
Equation (24) is not linear in the function of parameters z, K A, B and C. To avoid this nonlinearity we introduce a new variables p1, p2, p3, p4 and p5, defined as follows:
i¼1 N P
p4 ¼ KB;
ouðPÞ ¼ 0 ði ¼ 1; 2; 3; 4 and 5Þ: oðpi Þ
ð24Þ
6 2 6 i¼1 6 N P 6 6 2 Ti x2i 6 i¼1 6 N 6 P 2 G ¼ 6 6 2 Ti xi i¼1 6 6 N 6 2 P Ti x3i 6 6 i¼1 6 N P 4 2 Ti x4i
ð27Þ
Equation (30) is now linear in function of variables p1, p2, p3, p4 and p5. The global optimal solution of the linear system of this equation is found by minimizing the following mathematical objective function onto the real space R:p ¼ arg min uðPÞ ¼ PN 4 2 2 2 i¼1 ðTi xi þ 2Ti xi p1 þ p2 Ti p3 p4 xi p5 xi Þ . Subject to P 2 R To solve this nonlinear program, it is sufficient to find the unique solution of the following system of linear equations:
ð23Þ
N P
p3 ¼ KAz2 ;
ð30Þ
i ¼ 1; 2; 3. . .N:
Ti2 x4i
ð26Þ
Ti x4i þ 2Ti x2i p1 þ p2 Ti p3 p4 xi p5 x2i ¼ 0 :
Multiplying both sides of Eq. (22) by ðx2i þ z2 Þ2 , we obtain the following: T i x4i þ 2z2 x2i þ z4 ¼ K A z2 þ K B xi þ KC x2i :
N P
p2 ¼ z4 ;
Introducing these new variables into Eq. (24), it can be concluded that
ð22Þ
2
ð25Þ
and
The mathematical expression for the magnetic anomaly (vertical, horizontal, and total) observed at a point F (x) along the x-axis due to an infinitely extended horizontal cylinder (q = 2) as presented in Eq. (1) is given as: Az2 þ Bxi þ Cx2i
p1 ¼ z2 ;
N P i¼1 N P
Ti x2i
Ti
i¼1
i¼1 N P i¼1
i¼1 N P
Ti x3i Ti xi
i¼1 N P
N N P
N P
xi x2i
xi
i¼1 N P i¼1 N P i¼1
x2i
x3i
N P
3 Ti x4i
7 7 7 7 Ti x2i 7 7 i¼1 7 N 7 P x2i 7 7 i¼1 7 N P 3 7 xi 7 7 7 i¼1 7 N P 5 4 xi i¼1 N P
i¼1
Vol. 173, (2016)
A least-squares minimization approach for model parameters estimate
m and d are vectors of five dimensions given as: 2 3 p1 6 7 6 p2 7 6 7 7 m ¼6 6 p3 7 ; 6 7 4 p4 5 p5
1275
Problem formulation due to a sphere model The mathematical expression for the magnetic anomaly (vertical, horizontal, and total) observed at a point F (x) along the x-axis due to a sphere model as presented in Eq. (1) is given as: Tðxi ; zÞ ¼ K
Az2 þ Bxi þ Cx2i ; ðx2 þ z2 Þ5=2
i ¼ 1; 2; 3. . .N:
i
and 3
ð36Þ
T 2 x6 7 6 6 1¼1 i i 7 7 6 P 6 N 2 47 6 Ti xi 7 7 6 1¼1 7 6 N 7 6 P 2 7 d¼6 T i xi 7 6 7 6 1¼1 7 6 P 6 N T x5 7 7 6 6 1¼1 i i 7 7 6 N 5 4 P Ti x6i
Multiplying both sides of Eq. (36) by ðx2i þ z2 Þ2 we obtain the following: 5 Ti x2i þ z2 2 ¼ KA z2 þ KB xi þ KC x2i : ð37Þ
2
N P
5
1¼1
The solved linear system of algebraic Eqs. (31) has a unique solution p, which allows us easily to obtain the parameters related to the causative body as follows: The depth to the center of the body (z) is found by using Eqs. (25) and (26) and given as: pffiffi pffiffi z þ 4 z z¼ : ð32Þ 2 The magnetization angle (h) is obtained by Eqs. (28), (29) and (32) as: 2 z p5 h ¼ Arc tan : ð33Þ p4 By knowing the values of z and h, the values of A, B and C can be calculated as follows: A ¼ cosðhÞ; B ¼ 2z sinðhÞ;
ð34Þ
and C ¼ cosðhÞ: By using the values of z, A, B, and C, the amplitude coefficient (K) can be computed by the following relation: 1 p3 p4 p5 K¼ þ þ : ð35Þ 3 A z2 B C
By taking the squared and factorizing of both sides we obtain the following: 8 2 6 4 4 6 2 8 10 Ti2 x10 i þ 5xi z þ 10xi z þ 10xi z þ 5xi z þ z ¼ K 2 A2 z4 þ 2A B xi z2 þ B2 x2i þ 2AC x2i z2 þ 2BCx3i þ C2 x4i Þ : ð38Þ Arranging Eq. (38) we obtain: 2 8 2 2 6 4 2 4 6 Ti2 x10 i þ 5Ti xi z þ 10Ti xi z þ 10Ti xi z
þ 5Ti2 x2i z8 þ Ti2 z10 K 2 A2 z4 2A B K 2 xi z2 x2i B2 K 2 þ 2ACK 2 z2 2BCK 2 x3i C 2 K 2 x4i ¼ 0:
ð39Þ Equation (39) is not linear in the function of parameters z, K A, B and C. To avoid this nonlinearity, we introduce a new variables p1, p2, p3, p4, p5, p6, p7, p8, p9, and p10 defined as follows: p1 ¼ z2 ;
ð40Þ
p2 ¼ z4 ;
ð41Þ
p3 ¼ z 6 ;
ð42Þ
p4 ¼ z 8 ;
ð43Þ
p5 ¼ z10 ;
ð44Þ
p6 ¼ K 2 A2 z 4 ;
ð45Þ
p7 ¼ AB K 2 z2 ; p8 ¼ K 2 B2 þ 2ACz2 ;
ð46Þ ð47Þ
1276
E. R. Abo-Ezz, K. S. Essa
p9 ¼ K 2 B C;
ð48Þ
p10 ¼ K 2 C 2 :
ð49Þ
To solve this nonlinear equation, it is sufficient to find the unique solution of the following system of linear equations: ouðPÞ oðpi Þ ¼ 0 ði ¼ 1; 2; 3. . . and 10Þ: This system of linear equations could be written in matrix form as:
and
Introducing these new variables into Eq. (39), it can be concluded that
G m ¼ d,
þ
N P 25 Ti4 x16 i i¼1 N P 50 T 4 x14 i i i¼1 N P 50 Ti4 x12 i i¼1 N 25 P T 4 x10 i i i¼1 N 5P Ti4 x8i i¼1 G ¼ N P 5 Ti2 x8i i¼1 N 10 P T 2 x9 i i i¼1 N P 2 10 5 Ti xi i¼1 N P 10 Ti2 x11 i i¼1 N P 2 12 5 T x i i i¼1
þ
Ti2 p5
x4i p10
¼ 0:
N P
50
Ti4 x12 i
100
Ti4 x10 i
100
100
i¼1 N P
50
N P
10
i¼1 N P
i¼1
10
25
Ti4 x10 i
50
Ti4 x8i
50
i¼1 N P
Ti4 x6i
25
i¼1 N P
Ti4 x4i
5
i¼1 N P
i¼1 N P
i¼1
Ti4 x8i
50
Ti4 x6i
10
i¼1 N P
10
Ti2 x7i
20
Ti2 x8i
10
Ti2 x9i
20
Ti2 x10 i
10
i¼1 N P i¼1 N P i¼1 N P i¼1 N P
i¼1
i¼1 N P
Ti2 x6i
10
N P
Ti4 x12 i
N P
i¼1 N P
20
N P
50
i¼1 N P
10
ð50Þ
Ti4 x14 i
100
20
x2i p8
; p6 2xi p7
i¼1 N P i¼1 N P i¼1 N P i¼1 N P i¼1
i¼1 N P
5
Ti4 x8i
10
Ti4 x6i
10
i¼1 N P
5
Ti2 x5i
10
5
i¼1 N P
Ti2 x6i
5
Ti2 x7i
10
Ti2 x8i
5
Ti2 x3i
2
Ti2 x4i
Ti2 x5i
2
Ti2 x6i
i¼1 N P
i¼1 N P i¼1
10
Ti4 x4i
10
Ti4 x2i Ti4
i¼1 N P
Ti2 x2i
i¼1 N P
Ti4 x6i
i¼1
Ti4 x4i
i¼1 N P
5
i¼1 N P
N P
i¼1 N P
i¼1 N P
p ¼ arg min uðpÞ 2 2 10 Ti xi þ 5Ti2 x8i p1 þ 10Ti2 x6i p2 N X 6 ¼ 4 þ 10Ti2 x4i p3 þ 5Ti2 x2i p4 þ Ti2 p5 p6 2xi p7 x2i p8 2x3i p9 x4i p10
Subject to P 2 R:
Ti2 xi
i¼1 N P
Ti2 x3i
Ti2 x4i
32 7 5 :
Ti2 x8i
10
Ti2 x6i
20
Ti2 x4i
20
i¼1 N P
i¼1 N P i¼1
N P
5
i¼1 N P
Ti2 x2i
10
Ti2
2
i¼1
Ti2
Ti2 x2i
i¼1 N P
i¼1
N P
Ti4 x8i
i¼1 N P
Ti4 x2i
i¼1 N P
Ti2 x4i
N P
Ti4 x10 i
Equation (50) is now linear in function of variables p1, p2, p3, p4, p5, p6, p7, p8, p9, and p10. The global optimal solution of the linear system of this equation is found by minimizing the following mathematical objective function onto the real space R. In mathematical form, it can be written:
i¼1
ð51Þ
where G is a square matrix of 10 9 10 dimensions given as follows:
2 8 2 6 2 4 Ti2 x10 i þ 5Ti xi p1 þ 10Ti xi p2 þ 10Ti xi p3
5Ti2 x2i p4 2x3i p9
Pure Appl. Geophys.
N 2
N P
xi
i¼1 N P
x2i
i¼1 N P
2
i¼1 N P i¼1
x3i
x4i
N P i¼1 N P i¼1 N P i¼1 N P
2 4
i¼1 N P
2
i¼1 N P
4
i¼1 N P
2
i¼1 N P i¼1
5
Ti2 x7i
10
Ti2 x5i
10
Ti2 xi
xi x2i x3i x4i x5i
10
Ti2 x8i
20
Ti2 x6i
20
Ti2 x4i
10
Ti2 x2i
2
i¼1 N P i¼1
5
N P
i¼1 N P i¼1 N P
x2i
i¼1 N P
2
i¼1 N P
x3i
x4i
i¼1 N P
2
i¼1 N P i¼1
N P
Ti2 x10 i
i¼1 N P
Ti2 x3i
i¼1 N P i¼1 N P
N P
Ti2 x9i
x5i
x6i
i¼1 N P
Ti2 x11 i
i¼1 N P i¼1 N P
i¼1 N P
2
i¼1 N P
4
i¼1 N P
2
i¼1 N P
4
i¼1 N P
2
i¼1 N P i¼1
Ti2 x9i Ti2 x7i Ti2 x5i
Ti2 x3i x3i x4i x5i x6i x7i
Ti2 x12 i i¼1 N P 10 Ti2 x10 i i¼1 N P 10 Ti2 x8i i¼1 N P 2 6 5 Ti xi i¼1 N P Ti2 x4i i¼1 N P x4i i¼1 N P 2 x5i i¼1 N P 6 xi i¼1 N P 7 2 xi i¼1 N P 8 xi 5
N P
i¼1
m and d are vectors of four dimensions given as: 3 2 p1 6 p2 7 7 6 6 p3 7 7 6 6 p4 7 7 6 6 p5 7 7; 6 m ¼ 6 7 6 p6 7 6 p7 7 7 6 6 p8 7 7 6 4 p9 5 p10
Vol. 173, (2016)
A least-squares minimization approach for model parameters estimate
and N P 4 18 5 T x i i 1¼1 N 10 P T 4 x14 1¼1 i i N P 10 Ti4 x14 i 1¼1 N P 4 12 5 Ti xi 1¼1 P N T 4 x10 i i d ¼ 1¼1N P 2 10 T x 1¼1 i i N 2 P T 2 x11 i i 1¼1 P N Ti2 x12 i 1¼1 N P 2 13 2 T x i i 1¼1 N P 2 14 T x i i
1277
8 8 2 > > < 3z sin h < 3 sin h 1 ; B ¼ 3z cos h ; A ¼ 2 sin h > > : : 3z sin h cos h 8 2 for total field > < 3 cos h 1 ; for vertical field C ¼ sin h > : for horizontal field 2 cos h ð54Þ Finally, knowing the values of z, A, B, and C the amplitude coefficient (K) can be determined as follows: 2 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi p7 p10 5 1 4 p6 p8 p9 ; þ K¼ þ pffiffiffiffiffiffi þ þ 5 A z2 B2 þ 2ACz2 BC C z AB ðall fieldsÞ:
ð55Þ
1¼1
The solved linear system of algebraic Eq. (51) has a unique solution p, which allows us easily to obtain the parameters related to the causative body as follows: The depth to the center of the body (z) is found by using Eqs. (40–47) and given as: pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi p1 þ 4 p2 þ 6 p3 þ 8 p4 þ 10 p5 z¼ : 5 ð52Þ Once z is known, the magnetization angle (h) can be determined as follows: 3 p6 h ¼ Arc tan ; ðVertical fieldÞ 2zp7 ð53Þ z p7 ; ðHorizontal fieldÞ: h ¼ Arc tan 3p6 Knowing h, the values of A, B, and C can be determined as follows:
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(Received January 22, 2015, accepted August 17, 2015, Published online September 1, 2015)
