Patent Application: US-68260996-A

Abstract:
a method of processing magnetic field measurements to obtain magnetic dipole density image data is provided . the process involves making measurements of total magnetic field at a plurality of locations over an area of interest , dividing the area of interest into a plurality of narrow parallel strips , performing a strip integral and forming and solving a system of linear equations to obtain a generalized projection . the process is then repeated to produce a plurality of generalized projections from the original measured data by performing strip integrals for a plurality of different orientations of the strips . the process is then completed by performing an inverse radon transformation on the generalized projections to obtain the magnetic dipole moment density image data .

Description:
as described above , the task at hand is to reconstruct a magnetic dipole density image of the cross section of s &# 39 ; from which the boundary of s can be determined . the following describes a method for accomplishing this task using techniques related to computerised tomography . computerised tomography ( ct ) and tomographic image reconstruction techniques provide a well - defined automatic method for three - dimensional modelling using anomaly data . such techniques have been extensively studied and developed for applications in medicine and in some forms of non - destructive testing . they have also been used for seismic data processing in the petroleum industry , and for studying hidden or buried objects . briefly , computerised tomography relates to the reconstruction of an unknown function from line integrals . in particular , it relies upon the property that an unknown function ƒ in a region s &# 39 ; can be uniquely reconstructed from line integrals of ƒ taken along all directions through s &# 39 ;. generally speaking , modern ct methods are applicable to imaging situations where line integrals or strip integrals of a parameter , such as x - ray attenuation in medical imaging or wave slowness in seismic tomography , are available as collected data . to define the operation of computerised tomography more precisely , consider a plane in two dimensions in which any point can be represented in terms of its cartesian coordinates x and y , or its polar coordinates τ and φ . for an arbitrary straight line in the plane , let φ denote the angle between the x - axis and the direction normal to the line , and let τ denote the signed perpendicular distance from the origin to the line . the line can then be described by the equation let f ( τ , θ ) be the radon transform 11 !, line integral , or projection of ƒ ( x , y ). then ƒ in polar coordinates can be reconstructed uniquely from f ( τ , θ ) by the inverse transformation ## equ13 ## where f 1 ( τ , θ ) is the partial derivative of f ( τ , θ ) with respect to τ . in practice , f ( τ , θ ) is only known at discrete points along lines , i . e ., for τ = τ k = kη for nonnegative integer values of k , and θ = θ j = jπ / n ; j = 0 , 1 , 2 . . . , n - 1 . here , each value of θ defines one of n views taken around a semicircle surrounding the object , and η is the separation between two adjacent parallel lines in each view . hence , in practical implementations , the integrals in equation ( 12 ) are replaced by summations to obtain an approximate reconstruction formula for ƒ . sub . ξ , depending on ξ , given by ## equ14 ## here , δθ is the angle between two adjacent views , ξ is a filter function introduced by regularisation related to approximation theory and l is the number of line integrals within a view . it can be shown 13 ! that l and n are related optimally by the expression l = n / π . image reconstructing techniques based on equation ( 13 ) are known to as back projection methods . now apply the principles of computerised tomography to the anomaly interpretation task at hand . fig1 a illustrates a schematic cross - section of object 10 beneath earth &# 39 ; s surface 12 . object 10 is contained within area of interest s &# 39 ;. as shown in fig1 a , an x - axis is defined to be oriented parallel to the earth &# 39 ; s surface , and a z - axis is defined to be oriented vertically . object 10 has a magnetic dipole moment density m whose direction is defined by its direction cosines p = cosα and q = cosβ where α is m &# 39 ; s angle from the x - axis and β is m &# 39 ; s angle from the z - axis . the earth &# 39 ; s magnetic field is measured along a line parallel to the x - axis at elevation z o above the earth &# 39 ; s surface giving an anomaly as illustrated by curve 14 in fig1 b . to apply the principles of computerised tomography , instead of dividing the area of interest s &# 39 ; directly into a number of pixels as is conventionally done , divide it into a number of narrow strips as shown in fig2 . for a given angle θ with respect to the x - axis , draw parallel lines 16 with the same interval η between two adjacent lines , drawn parallel lines being normal to all lines making an angle θ with the x - axis such that they can all be described by equation ( 11 ). let the area between two adjacent lines define a strip . then η is the width of the strips , and is chosen ( to ensure that a desired spatial resolution is obtained in the reconstructed image 10 !) such that η ≦ min ( δx , δz ), where δxδz is the size of a pixel to be produced in the reconstructed . let s &# 39 ; k denote the area of the kth strip , and let δ k denote the intersection of the source body s with s &# 39 ; k ( i . e . δ k = s &# 39 ; k ∩ s ). then ## equ15 ## where t ik represents the contribution of the source body from the kth strip to the measured field intensity at the ith measuring point , and n is the total number of measuring points . it follows that ## equ16 ## where n is the total number of the strips . ( for the sake of simplicity we set the number of strips equal to the number of measuring points ). let a ik denote the strip integral ## equ17 ## where w &# 39 ; is a constant and w &# 39 ;≈ m since a ik and t ik are numbers , it will always be true that where y ik is an unknown real number and a k = a kk . substituting equation ( 17 ) into equation ( 15 ) results in ## equ18 ## this can be written in the matrix form where ## equ19 ## and , for consistency , the field intensity b a ( i ) is denoted by b i . we now have a system of linear equations ( 19 ) in which the unknown is the matrix y . unlike equation ( 10 ), this system theoretically has no approximation error ( ignoring round off errors ) because it is not a discrete form of a fredholm integral of the first kind . if the ith row of y is computed from equation ( 19 ) then , according to ( 18 ), the contribution of the source body to the ith measuring point from the kth strip is simply t ik = y ik a k . it is then possible to determine the contribution of the source body from all of these strips . it is evident that ## equ20 ## that is , that y ik a k is the strip integral of the image function ƒ ( x , z ), and we refer to it as the generalised radon transform . since all of these strips are divided in parallel in a direction specified by the angle θ , the set { y ik a k | k = 1 , 2 , . . . , n } can be considered to be a projection of an image function ƒ ( x , z ) according to the definition used in computerised tomography as described above . for each value of θ , there is a set of strips and a corresponding projection of the image function as defined in equation ( 21 ). construct and solve a system of linear equations for each value of θ given by θ j = jπ / n , j = 0 , 1 , . . . , n - 1 to produce n projections of the image function ƒ ( x , z ) along n directions . with these n projections the image ƒ ( x , z ) is readily reconstructed through the back projection method of equation ( 13 ). once the image ƒ ( x , z ) is reconstructed , it is possible to find the distribution function w ( x , z ), since the data kernel k ( x i , z o , x , z ) is known . recall from its definition in equation ( 4 ) that the data kernel is considered to be a function of the argument ( x &# 39 ;, z 0 , x , z ), while p , q , p and q are all considered as constants . for a given source body , p and q are , indeed , constants . however , p and q , the directional cosines of the unit vector t , are variable ; when t is parallel to the total magnetic field intensity , the measured magnetic field data is the magnitude of the total magnetic field intensity , otherwise the data is the component of the total field in the direction of t . since p and q satisfy the data kernel k can be considered to be a function of the extended argument ( x &# 39 ;, z 0 , x , z , p ). that is ## equ21 ## for a given direction t j , ( i . e ., given p j ) define vector a j as where the elements a k j are computed using ## equ22 ## for a given p j , denote the measured anomaly field data ( a component of the total field in the direction of t j ) by the vector b j which is defined as defining the ( n × n ) matrices a and b by juxtaposing vectors as follows equation ( 30 ) has a unique solution if a is a non - singular or , equivalently , if the vector set { a j | j = 1 , 2 , . . . , n } is a linearly independent set . from the definition of a j , it is clear that the linearity of this vector set is determined by the linearity of the data kernel k ( x k , z o , x , z , p j ) with respect to the variable p j . however , k ( x k , z o , x , z , p j ) is a nonlinear function of p because , from ( 23 ), q =√ 1 - p 2 . therefore , a 1 , a 2 , . . . , a n are linearly independent vectors , a is non - singular , and ( 30 ) has a unique solution given by it was indicated before that only one row of the solution matrix is required . if one transposes equation ( 30 ) and multiplies the transposed matrix equation by the vector e i where e i is an n - dimensional vector with a one in its ith position and zeros elsewhere , then one has where y i is the ith row of the unknown matrix y and b i is the ith row of the matrix b . if we rewrite equation ( 30 ) explicitly as ## equ23 ## then we can see that the elements of b i , the ith row of the matrix b , are the components in the direction p j of the total field intensity measured at the same point . that is the problem that arises here is how to obtain the vector sets { a j | j = 1 , 2 , . . . , n } and { b j | j = 1 , 2 , . . . , n } in practice . the magnetic field intensity measured by most modern three - component magnetometers is that produced by vector summation of the earth &# 39 ; s main field and the anomaly field introduced by the magnetic body . this vector summation is shown schematically in fig3 where h represents the earth &# 39 ; s main field , c represents the anomaly field , t represents the total field measured by the magnetometer , and γ is the angle between h and t . the earth &# 39 ; s main field at a surveying point can reasonably be considered to be a known constant . thus , the anomaly field c can be obtained simply from where b i is the magnitude of c . we have where δα is the difference between angles p j and p j + 1 and ( j - 1 ) δα is the biasing angle of the p j from p 1 . it follows that therefore , b i can be obtained from the magnitude of the total field intensity and direction cosines cos ( j - 1 ) δα ! at the measuring point ( x i , z o ). by substituting { p j | j = 1 , 2 , . . . , n } into equation ( 24 ), the vector set { a j | j = 1 , 2 . . . , n } defined in equation ( 25 ) is readily obtained by equation ( 26 ). the problem has now been reduced to that of solving an ordinary system of linear equations ( 34 ). however , the vector set { a j | j = 1 , 2 . . . , n } may be far from orthogonal , even though it is linearly independent . therefore , a may be ill - conditioned or rank deficient . thus , standard methods of linear algebra for solving ( 34 ), such as lu , cholesky , or qr factorization , may not be reliable if one uses them in a straightforward manner to compute the solution to the problem . the most common methods used to solve systems of linear equations involving ill - conditioned matrices are regularization methods . the simplest of these is the so - called damped least squares method in the marquardt - levenberg &# 39 ; s formulation where λ is a damping factor or regularization parameter , and i is the identity matrix . an important issue in the use of this method is the choice of an optimal λ , and the generalised cross validation ( gcv ) method is commonly used for this purpose 12 !. quite often , the gcv method fails to find an optimal λ , and there now exists a more powerful l - curve method for choosing an optimal λ . the following describes a numerical example that illustrates the use of the method for interpreting magnetic anomaly profiles . the object considered here is a square object s within area of interest s &# 39 ; as shown in fig4 . the labels 1 , 2 , . . . , 28 , . . . , 66 surrounding area of interest s &# 39 ; refer to strip numbers within s &# 39 ; . the dimensions of the pixels were set at δxδz = 0 . 5m × 0 . 5m . the total number , n , of pixels within area of interest s &# 39 ; is 61 × 21 = 1281 . this requires s &# 39 ; to be divided into 66 strips for a given angle θ and the total number of views is set to be 33 . for the chosen object , with strips in a direction θ ≈ 54 . 5 °, only strips 28 to 40 intersect with the object , as shown in fig4 . measurements were simulated at an elevation z o = 0 . 5m above the surface of the earth using equation ( 3 ). this simulated anomaly field data is shown as line 16 in fig5 where the simulated magnetic field measurements are plotted against the strip number . the subscript i in equation ( 39 ) was chosen to be i = 32 . random errors δ i were added to this perfect data to give b i 1 = b 32 + δ 32 . a corresponding projection of the object - related image function as defined in ( 21 ) is shown as line 18 in fig6 with a typical λ chosen using the gcv method . in fig6 the projection is a plot of strip integrals of the image function against the strip number , and it can be seen that the strip integrals are significantly different from zero at least for strips 30 to 37 . fig7 shows the reconstructed image function ƒ ( x , z ) defined in equation ( 21 ). this is accomplished by using a back projection algorithm with projections obtained by solving 33 systems of linear equations . although in the conventional application of ct methods , the image of an object is obtained from projections directly measured at different angles around . the object , the present invention illustrates how the principles of ct can be used to obtain the image of an object using only the potential field measured on a plane located some distance away from the object . some advantages of the method presented here are that : ( 1 ) it can find a physically acceptable solution to a fredholm integral equation of the first kind without any a priori knowledge about the solution , ( 2 ) its computational cost is low , and ( 3 ) it can be extended easily to three - dimensional problems . thus , there has been described a method for interpreting magnetic anomaly profiles . those skilled in the art will appreciate that various modifications can be made without departing from the scope of the invention as , broadly described , and the embodiment described is presented for purposes of illustration and not limitation . 1 ! w . m . telford , l . p . geldart & amp ; 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