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

zk

yj

xi

R

dvr

,

1MF̂CΔFF̂ΔT m

( IGRF)

1. Understanding magnetic total field

2) General form of total field anomaly due to causative body (R)

Where Cm 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.

M

F̂

(E0)

Measured total fielddata (amplitude)

2 magnetic total field measurement device— (horizontal) triangle frame on airplane

M1 M2

M3

15.73 m

11.8

8 m

x (E)

y (N)

)M()M()M(

0

88.11

0

0

0

7.865

0

0

7.865

]MMM[

321

321

0

Sensor location:

(E1a)

Case 1: triangle frame with reference to the geographic north

Case 2: triangle frame wrt the survey line direction

)M()M()M(

0

88.11

0

0

0

7.865

0

0

7.865

100

0)cos()sin(

0)sin()cos(

]MMM[

321

321

2. Triangle measurement device (continued)

M1

M2M3

15.73 m

11.88 m

Sensor location:

x (E)

y (N)

Survey l

ine directi

on

φ

(E1b)

3. Four steps of rotation from the NED frame to the triangle frame

(1/4) rotation around E-Axis with Pitch = 10o

)cos()sin(0

)sin()cos(0

001

R E

pp

pp (E2a)

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

)cos(0)sin(

010

)sin(0)cos(

R N

rr

rr(E2b)

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

100

0)cos()sin(

0)sin()cos(

R E yy

yy(E2c)

Combination of the above three rotations about axis in sequence of D-N-E

DNEEND RRRR

100

0)cos()sin(

0)sin()cos(

)cos(0)sin(

010

)sin(0)cos(

)cos()sin(0

)sin()cos(0

001

yy

yy

rr

rr

pp

pp

Orthogonal rotation from NED (cyan) to pink frame by REND

(E3)

Three sensor locations before and after rotation—for the model forward calculation

)m()m()m(

0

88.11

0

0

0

7.865

0

0

7.865

R mmm

321

END321

Case 1: NED frame – flight heading with reference to the geographic north

Case 2: Survey line frame – flight heading wrt the survey line direction

)m()m()m(

0

88.11

0

0

0

7.865

0

0

7.865

100

0)cos()sin(

0)sin()cos(

R mmm

321

DNE321

Where φ is azimuth angle of the survey line;

in RDNE angle YAW is amended by subtracting φ

Their three ground positions defined by

321 mmm},,{

obsobsobs zyx

(E4a)

(E4b)

3. Four steps of rotation (continued)(4/4) rotation from airplane (NED’) frame to triangle (T) frame

0)sin()cos(

0)sin()cos(

001

R

31

32

12

T

v

v

v

M1 M2

M3

15.73 m

11.8

8 m

X (E)

Y (N)

0

V12 (TE’)

V32V31

Directional gradients:

V12 = (M2 - M1) / d12

V32 = (M2 - M3) / d32

V31 = (M1 - M3) / d31

Where θ = atan(2 d30 / d12)

θ

Magnetometer triangle device based on the rotated airplane

Gradients defined in triangle frame i.e. airplane board plane

0

D

N

E

T

T

T

0)sin()cos(

0)sin()cos(

001

V

V

V

31

32

12

(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

vrfh

rfvhrfrfh

)()()(

lim)(

0

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)

cos

cos

cos

v

z

rf

y

rf

x

rfrf

)()()()(

(E6b)

(E6c)x

y

zv

α

β

4. Directional derivatives: general formBased on E6, three wanted directional derivatives can be created

3

3

3

3

2

2

2

2

1

1

1

1

cos

cos

cos)()()()(

cos

cos

cos)()()()(

cos

cos

cos)()()()(

z

rf

y

rf

x

rfrf

z

rf

y

rf

x

rfrf

z

rf

y

rf

x

rfrf

)()()(

cos

cos

cos

cos

cos

cos

cos

cos

cos)()()(

321

3

3

3

2

2

2

1

1

1

321

zyx fffrfrfrf

(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

;

0

)sin(

)cos(

;

0

)sin(

)cos(

;

0

0

1

313322121

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.

)()()(

0

)sin(

)cos(

0

)sin(

)cos(

0

0

1)()()(

313212

313212

zyx fffrfrfrf

(E8b)

313212

(E8a)

0

)sin(

)cos(

0

)sin(

)cos(

0

0

1

100

0)cos()sin(

0)sin()cos()()()(

313212

zyx fffrfrfrf

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 frameDirection of the derivatives in (E8b) can be rotated from the levelled into the (rotated)

flying airplane frame based on rotation form defined in (E3)

)()()(

0

)sin(

)cos(

0

)sin(

)cos(

0

0

1

R)()()(

313212

END313212

zyx fff

rfrfrf

(E9a)

313212 313212

z

rfy

rfx

rf

rf

rf

rf

)(

)(

)(

R

0

0

0

)sin(

)sin(

0

)cos(

)cos(

1

)(

)(

)(

)(

)(

)(

TEND

31

32

12

31

32

12

(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.

(RT)

Summary 1: Rotations of gradients from NED frame to the airplane triangle frame

Orthogonal rotation from NED (cyan) to pink frame by REND

)sin()cos()sin()sin()cos()sin()cos()sin()sin()sin()cos()cos(

)sin()cos()sin()sin()sin()cos()cos()sin()cos()sin()sin()cos(

)sin()sin()cos()cos()cos(

rpyrppyypryp

pryrpypyprpy

ryryr

0)sin()cos(

0)sin()cos(

001

Rot = RT • RTEND =

None-orthogonal rotation from thepink to the triangle frame by RT

(E10)REND 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

D

N

E

T

T

TTENDT

31

32

12

RR

V

V

V

D

T

S

T

T

TTSTDT

31

32

12

RRR

V

V

V

Where RSTD 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

100

0)cos()sin(

0)sin()cos(

R

φ is azimuth angle of the survey line

5. Solution to Equation 11 — Understanding features of E11a

• REND 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,

REND = REND’ and REND = REND-1

• RT 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 RT, inverse of (RT· RSTD ) is normally close to singular and hence results in badly solution to (Te,Tn,Td). It also suggests that solution to TE and TN can only be contained excepted TD.

Rewriting (E11a),

D

N

E

T

T

TTENDT

31

32

12

RR

V

V

V

5. Solution to E11a via estimation approach

Method I: by Singular Value Decomposition (SVD) method, e.g.

Let A = (RT· RTDST ) A = U·S·V’

A-1 = V· S-1·U’ (reducing trace)

31

32

121

V

V

V

A

D

N

E

T

T

T

No solution to Td due to

reducing matrix trace

5. Solution to E11a via estimation approachMethod II: Based on (E11a) of orthogonal rotation between

survey frame and airplane frame

D

N

E

D

N

E

T

T

T

T

T

TTENDR

airplane

frame

NED

frame

D

N

E

D

N

E

T

T

T

T

T

T

ENDR

airplane

frame

NED

frame

31

32

12

'31

'32

'12

ENDR

V

V

V

V

V

V

airplane

frame

NED

frame

(E5b)

D

N

E

T

T

T

0)sin()cos(

0)sin()cos(

001

V

V

V

31

32

12

survey frame

)sin()cos(VV

)sin()cos(VV

V

1231

1232

12

N

N

E

T

T

Tsolving

equation

approximate

orthogonal

vectors to

triangle’s

6. Estimation to TD

Method 1: Grid transformationTD may be estimated in different ways and with the TE and TN 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, TE and TN , into TD ,

iv

WvuT

iu

WvuTvuvuT N

NE

ED ),(),(),( 22

Where u and v are FFT domain wavenumbers; WE and WN are weights generated due to TE and TN 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,

0),(),(),(

0),(),(),(

2222

2222

KvuWvuTiv

WvuT

iu

WvuTvu

Kvuiv

WvuT

iu

WvuTvu

vuT

TN

NE

E

NN

EE

D

Where K0 is wavenumber criteria; weights generated with satisfying, WE + WN + WT = 1,

and normally setting WT >> { WE , WN } when (u ,v) → 0.

(E12a)

(E12b)

6. Estimation to TD (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).

n

22

),,(),,(

nzyxTn

zyxT

D

N

E

DNE

n

n

n

TTTnzyxTn

zyxT][),,(

),,(

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

2222DDNNEEDNE nTnTnTTTT let

)sin();cos()cos();sin()cos( InDInDIn DNE

After solving the quadratic equation for TD, we have

2

222222

1

)1(2)1()(

D

NNDNENEEEDNNEEDD n

TnnTTnnTnnTnTnnT

(E13a)

(E13b)

6. Estimation to TD (continued)Method 3: Solving Laplacian equation

)(

)sin(

)cos()cos(

)sin()cos(

b

Iy

TUzy

y

Tz

DIy

TUyy

y

Ty

DIy

TUxy

y

Tx

)sin(

)cos()cos(

)sin()cos(

ITTz

DITTy

DITTx

Method 3: With the above definition, total filed can be divided into three components,

(E14b)

TzT

yx

z

I

DTyTx

Applying partial derivatives to (E14a) wrt x, y and z gives three groups of results,

(E14a)

)(

)sin(

)cos()cos(

)sin()cos(

a

Ix

TUzx

x

Tz

DIx

TUyx

x

Ty

DIx

TUxx

x

Tx

)(

)sin(

)cos()cos(

)sin()cos(

c

Iz

TUzz

z

Tz

DIz

TUyz

z

Ty

DIz

TUxz

z

Tx

Wanted TD

)sin(

)cos()cos()sin(

;0)sin()cos()cos()sin()cos(;0

I

ID

y

TD

x

T

z

TT

Iz

TDI

y

TDI

x

TUzzUyyUxx

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.