1) Magnetic total field (T) obtained from airborne survey (see R.J.Blakely, 1995) (ΔT) Total field anomaly (IGRF removal), which satisfy potential theory, only when Theory ( IGRF) 1. Understanding magnetic total field 2) General form of total field anomaly due to causative body ( R ) Where C m is mag constant; is a unit vector in direction of the regional field; is vector of magnetisation; r is distance between observation and integral element. (E0) Measured total field data (amplitude)

2 magnetic total field measurement device — (horizontal) triangle frame on airplane M1M2 M m m x (E) y (N) 0 Sensor location: (E1a) Case 1: triangle frame with reference to the geographic north

Case 2: triangle frame wrt the survey line direction 2. Triangle measurement device (continued) M1M1 M2 M m m Sensor location: x (E) y (N) Survey line direction φ (E1b)

3. Four steps of rotation from the NED frame to the triangle frame (1/4) rotation around E-Axis with Pitch = 10 o (E2a)

3. Four steps of rotation (continued): (2/4) rotation around N-Axis with Roll = 10 o (E2b)

3. Four steps of rotation (continued): (3/4) rotation around D-Axis with Yaw = 10 o (E2c)

Combination of the above three rotations about axis in sequence of D-N-E Orthogonal rotation from NED (cyan) to pink frame by R END (E3)

Three sensor locations before and after rotation —for the model forward calculation Case 1: NED frame – flight heading with reference to the geographic north Case 2: Survey line frame – flight heading wrt the survey line direction Where φ is azimuth angle of the survey line; in R DNE angle YAW is amended by subtracting φ Their three ground positions defined by (E4a) (E4b)

3. Four steps of rotation (continued) (4/4) rotation from airplane (NED ’ ) frame to triangle (T) frame M1M2 M m m X (E) Y (N) 0 V 12 (T E ’ ) V 32 V 31 Directional gradients: V 12 = (M 2 - M 1 ) / d 12 V 32 = (M 2 - M 3 ) / d 32 V 31 = (M 1 - M 3 ) / d 31 Where θ = atan(2 d 30 / d 12 ) θ Magnetometer triangle device based on the rotated airplane Gradients defined in triangle frame i.e. airplane board plane 0 (TN’)(TN’) θ (E5a) or (E5b)

4. Method regarding relationship of directional derivatives (gradients) in XYZ (or NED) frame and the rotated airplane triangle frame Part 4 is to generate rotation of a point location in two different co-ordinate systems and rotation of directional derivatives represented in different co-ordinate systems Theory

4. Directional derivatives: definition Directional derivatives in 3D Cartesian coordinate system Where vector r = [ x y z ] representing an observation station and v is a unit vector of directional cosines with three angles, α, β and γ, between the unit vector and x, y and z axis respectively, (E6a) (E6b) (E6c) x y z α β

4. Directional derivatives: general form Based on E6, three wanted directional derivatives can be created (E7b) (E7a) (E7a) can be written in matrix form, Note that (E7b) can be applied in both orthogonal and none-orthogonal rotation cases.

4. Directional derivatives: levelled triangle frame With (E7b) and the triangle directions (V12, V32m V31) defined in (E5), let Again, three column vectors,, represents the triangle sensors gradient directions, M1-to-M2, M3-to-M2 and M3-to-M1, respectively. (E8b) (E8a) Case 1: NED frame – flight heading with reference to the geographic north Case 2: Survey line frame – flight heading wrt the survey line direction Where φ is azimuth angle of the survey line (E8c)

4. Directional derivatives: rotated triangle frame Direction of the derivatives in (E8b) can be rotated from the levelled into the (rotated) flying airplane frame based on rotation form defined in (E3) (E9a) (E9a) can be written in the transposed form, (E9b) (E9b) is a final result representing relationship of directional derivatives between XYZ (or NED) frame and the rotated airplane triangle frame, in which the left hand elements are measured data and the right hand derivatives are to be estimated. (R T )

Summary 1: Rotations of gradients from NED frame to the airplane triangle frame Orthogonal rotation from NED (cyan) to pink frame by R END Rot = R T R T END = None-orthogonal rotation from the pink to the triangle frame by R T (E10) R END is also refers to directional cosine matrix converted with roll, pitch and yaw. The result has been verified completely same as that from Matlab “aerospace toolbox”.

Summary 2: Relationship between measuring gradients and wanted gradients Where R STD is for rotation from orthogonal survey frame to variation airplane frame in which angle YAW is adjusted by subtracting an angle of the survey line direction (E11b) (E11a) Case 1: NED frame – flight heading with reference to the geographic north Case 2: Survey line frame – flight heading wrt the survey line direction φ is azimuth angle of the survey line

5. Solution to Equation 11 — Understanding features of E11a R END is a directional cosine matrix of rotating NED to airplane frame which is defined by flight attitude (roll, pitch, yaw) at each observation point. It is orthogonal matrix and thus satisfying, R END = R END ’ and R END = R END -1 R T is an fixed element matrix subject to the triangle device with trace of 2 (rather than 3). Due to the none orthogonal and trace number of R T, inverse of (R T · R STD ) is normally close to singular and hence results in badly solution to (Te,Tn,Td). It also suggests that solution to T E and T N can only be contained excepted T D. Rewriting (E11a),

5. Solution to E11a via estimation approach Method I: by Singular Value Decomposition (SVD) method, e.g. Let A = (R T · R T DST ) A = U·S·V ’ A -1 = V· S -1 ·U ’ (reducing trace) No solution to Td due to reducing matrix trace

5. Solution to E11a via estimation approach Method II: Based on (E11a) of orthogonal rotation between survey frame and airplane frame airplane frame NED frame airplane frame NED frame airplane frame NED frame (E5b) survey frame solving equation approximate orthogonal vectors to triangle’s

6. Estimation to T D Method 1: Grid transformation T D may be estimated in different ways and with the T E and T N obtained from the above steps, which can be categorized into three methods. Method 1: By means of (grid data) FFT transformation of the two derivative components, T E and T N, into T D, Where u and v are FFT domain wavenumbers; W E and W N are weights generated due to T E and T N noise levels, defaults are 0.5 representing equally weighting. To reduce transforming artifacts from the wavenumber ( u, v) simultaneously closing to zero, the above form may be modified by adding total field (T) term, Where K0 is wavenumber criteria; weights generated with satisfying, W E + W N + W T = 1, and normally setting W T >> { W E, W N } when ( u, v) → 0. (E12a) (E12b)

6. Estimation to T D (continued) Method 2: Solving the quadratic equation Where is an unit vector of directional cosine of Earth magnetic field, defined by local Earth mag (RGRF) declination ( D ) and inclination ( I ). Method 2: Based on the triangle device (i.e. based-point) and E6a, total filed gradient in Earth magnetic filed direction (total gradient) can be written as let After solving the quadratic equation for T D, we have (E13a) (E13b)

6. Estimation to T D (continued) Method 3: Solving Laplacian equation Method 3: With the above definition, total filed can be divided into three components, (E14b) Tz T y x z I D Ty Tx Applying partial derivatives to (E14a) wrt x, y and z gives three groups of results, (E14a) Wanted T D

Conclusion 1)Starting from initial definition of mag total field intensity (TFI, in E0), forward calculation due to the sensors in airplane triangle locations is created with employing airplane attitude information (E4a-b). It constructs a valid comparison between true and estimated gradients created in three frames, NED, levelled triangle and airplane triangle. 2)Formula have been established based on the triangle device and the attitude information, leading to a final form in (E9b) that represents the forward rotation of XYZ gradients from NED frame to the airplane triangle frame. It is a foundation of the study. 3)Due to less information obtained about TFI vertical variation, which results in inversion of the rotation matrix being singular, the inversed rotation, from the triangle gradients to the NED gradients, has to be sorted off by estimation, shown in part 5 of slides.