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REGIONAL GRAVITATIONAL FIELD MODELLING BY THE SPECTRAL COMBINATION �OF SATELLITE HIGHER-ORDER RADIAL DERIVATIVES OF THE GRAVITATIONAL POTENTIAL AND A GLOBAL GEOPOTENTIAL MODEL

EGU 2023, Vienna, Austria, 23-28 April 2023

Martin Pitoňák

Michal Šprlák

Pavel Novák

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Regional gravitational field modelling by the spectral combination of satellite higher-order radial derivatives …

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Outline:

1

Motivation

2

Theory

3

Numerical experiment

4

Results

5

Conclusion

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Motivation

 

mean orbital sphere (x)

mean Earth sphere (y)

Earth surface (y)

 

 

 

 

 

 

 

 

 

 

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Motivation

Solution of the spherical vertical boundary value problem (BVP) in the spectral domain for the unknown gravity disturbances is defined as (Grafarend 2001):

Solution of the vertical-vertical spherical gradiometry BVP for the same quantity is (Martinec 2003):

Solution of the vertical-vertical-vertical gravitational curvature BVP is (Šprlák and Novák 2016):

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Theory: Method of spectral weighting (an example for VVV)

This method is based on the minimisation of the error between the estimated value and the true value:

where a general gravity disturbance estimator in the spectral form from the third-order radial derivative is given by the following series:

A symbol an is the spectral weight and other symbols used in the previous equation are:

We can rewrite the error between the general gravity disturbance estimator and its theoretical value to the following form:

The global root mean square error is:

where

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Regional gravitational field modelling by the spectral combination of satellite higher-order radial derivatives …

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Theory: Method of spectral weighting (an example for VVV)

This method is based on the minimisation of the error between the estimated value and the true value:

where a general gravity disturbance estimator in the spectral form from the third-order radial derivative is given by the following series:

A symbol an is the spectral weight and other symbols used in the previous equation are:

We can rewrite the error between the general gravity disturbance estimator and its theoretical value to the following form:

The global root mean square error is:

where

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Regional gravitational field modelling by the spectral combination of satellite higher-order radial derivatives …

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Theory: Method of spectral weighting (an example for VVV)

This method is based on the minimisation of the error between the estimated value and the true value:

where a general gravity disturbance estimator in the spectral form from the third-order radial derivative is given by the following series:

A symbol an is the spectral weight and other symbols used in the previous equation are:

We can rewrite the error between the general gravity disturbance estimator and its theoretical value to the following form:

The global root mean square error is:

where

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Pitoňák et al.

Regional gravitational field modelling by the spectral combination of satellite higher-order radial derivatives …

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2

Theory: Method of spectral weighting (an example for VVV)

This method is based on the minimisation of the error between the estimated value and the true value:

where a general gravity disturbance estimator in the spectral form from the third-order radial derivative is given by the following series:

A symbol an is the spectral weight and other symbols used in the previous equation are:

We can rewrite the error between the general gravity disturbance estimator and its theoretical value to the following form:

The global root mean square error is:

where

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Pitoňák et al.

Regional gravitational field modelling by the spectral combination of satellite higher-order radial derivatives …

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2

Theory: Method of spectral weighting (an example for VVV)

This method is based on the minimisation of the error between the estimated value and the true value:

where a general gravity disturbance estimator in the spectral form from the third-order radial derivative is given by the following series:

A symbol an is the spectral weight and other symbols used in the previous equation are:

We can rewrite the error between the general gravity disturbance estimator and its theoretical value to the following form:

The global root mean square error is:

where

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Pitoňák et al.

Regional gravitational field modelling by the spectral combination of satellite higher-order radial derivatives …

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2

Theory: Method of spectral weighting (an example for VVV)

This method is based on the minimisation of the error between the estimated value and the true value:

where a general gravity disturbance estimator in the spectral form from the third-order radial derivative is given by the following series:

A symbol an is the spectral weight and other symbols used in the previous equation are:

We can rewrite the error between the general gravity disturbance estimator and its theoretical value to the following form:

The global root mean square error is:

where

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Pitoňák et al.

Regional gravitational field modelling by the spectral combination of satellite higher-order radial derivatives …

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2

Theory: Method of spectral weighting (an example for VVV)

This method is based on the minimisation of the error between the estimated value and the true value:

where a general gravity disturbance estimator in the spectral form from the third-order radial derivative is given by the following series:

A symbol an is the spectral weight and other symbols used in the previous equation are:

We can rewrite the error between the general gravity disturbance estimator and its theoretical value to the following form:

The global root mean square error is:

where

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2

We now take the derivative of GRMSE with respect to the spectral weight:

Then we equate the result to zero and after simplification, we get:

Theory: Method of spectral weighting (an example for VVV)

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We now take the derivative of GRMSE with respect to the spectral weight:

Then we equate the result to zero and after simplification, we get:

Theory: Method of spectral weighting (an example for VVV)

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We now take the derivative of GRMSE with respect to the spectral weight:

Then we equate the result to zero and after simplification, we get:

Theory: Method of spectral weighting (an example for VVV)

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Theory: Method of spectral weighting

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Theory: Method of spectral weighting – local integration

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Theory: Method of spectral weighting – local integration

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Theory: Method of spectral weighting – local integration

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Numerical experiment:

  • Input model: GO_CONS_GCF_2_TIM_R6e d/o 250,

​

- Input data: Tz, Tzz, Tzzz,

​

- regular equiangular grid located at the satellite level r = 6633850 m and limited by � with the regular step of 0.2°,

​

  • Outputs: δg located at the Brillouin’s sphere R = 6383850 m over Himalayas,

​

  • Integration radii: ψo = 3°, 5°, 10°, 15°, 20°, 25°.

​

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Results: gravity disturbances (in mGal)

​

global

ψo = 3°

ψo = 3° + FZ

​

Tz

Tzz

Tzzz

Tz

Tzz

Tzzz

Tz

Tzz

Tzzz

STD

1,288

1,113

1,115

25767,292

2003,757

225,566

3522,616

190,390

11,985

MIN

-4,260

-4,478

-4,489

-16688,263

-5716,626

-935,892

-13156,723

-484,692

-52,919

MAX

5,155

4,233

4,213

95870,708

5401,053

704,540

2776,983

601,592

53,706

MEAN

0,007

-0,007

-0,007

43692,647

732,121

24,287

-5776,466

-72,364

-8,148

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Results: gravity disturbances (in mGal)

​

global

ψo = 5°

ψo = 5° + FZ

​

Tz

Tzz

Tzzz

Tz

Tzz

Tzzz

Tz

Tzz

Tzzz

STD

1,288

1,113

1,115

12614,389

895,985

101,516

4341,293

275,535

30,824

MIN

-4,260

-4,478

-4,489

-56201,634

-2400,797

-333,355

2511,405

-702,816

-153,872

MAX

5,155

4,233

4,213

-7922,940

1873,944

445,595

19437,510

738,899

94,734

MEAN

0,007

-0,007

-0,007

-31408,013

-447,105

4,531

10345,955

123,944

-6,001

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Results: gravity disturbances (in mGal)

​

global

ψo = 10°

ψo = 10° + FZ

​

Tz

Tzz

Tzzz

Tz

Tzz

Tzzz

Tz

Tzz

Tzzz

STD

1,288

1,113

1,115

6837,723

256,531

38,430

6290,127

119,964

16,988

MIN

-4,260

-4,478

-4,489

-80,469

-686,815

-172,825

-30382,329

-442,654

-62,961

MAX

5,155

4,233

4,213

23521,102

889,381

134,802

-5379,978

300,352

64,270

MEAN

0,007

-0,007

-0,007

14143,701

258,600

10,609

-11513,860

-129,721

-7,978

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Results: gravity disturbances (in mGal)

​

global

ψo = 15°

ψo = 15° + FZ

​

Tz

Tzz

Tzzz

Tz

Tzz

Tzzz

Tz

Tzz

Tzzz

STD

1,288

1,113

1,115

3078,984

140,658

16,972

1767,498

72,348

9,788

MIN

-4,260

-4,478

-4,489

-17327,018

-635,114

-83,758

2314,516

-142,119

-51,516

MAX

5,155

4,233

4,213

-4602,447

253,200

96,216

10159,511

314,158

27,620

MEAN

0,007

-0,007

-0,007

-11292,327

-218,182

0,386

6566,150

110,460

-1,599

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Results: gravity disturbances (in mGal)

​

global

ψo = 20°

ψo = 20° + FZ

​

Tz

Tzz

Tzzz

Tz

Tzz

Tzzz

Tz

Tzz

Tzzz

STD

1,288

1,113

1,115

2759,373

97,185

10,675

1512,032

48,978

7,598

MIN

-4,260

-4,478

-4,489

1590,783

-126,513

-28,839

-7017,367

-200,605

-26,638

MAX

5,155

4,233

4,213

12719,359

389,529

59,954

-878,765

61,585

19,389

MEAN

0,007

-0,007

-0,007

7525,224

154,026

8,582

-4111,809

-76,951

-0,583

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Results: gravity disturbances (in mGal)

​

global

ψo = 25°

ψo = 25° + FZ

​

Tz

Tzz

Tzzz

Tz

Tzz

Tzzz

Tz

Tzz

Tzzz

STD

1,288

1,113

1,115

1931,206

60,688

5,106

972,392

26,959

4,551

MIN

-4,260

-4,478

-4,489

-8136,754

-229,260

-16,753

-197,162

-43,800

-7,484

MAX

5,155

4,233

4,213

0,269

71,232

19,520

4201,874

110,059

18,174

MEAN

0,007

-0,007

-0,007

-4414,228

-86,729

1,915

2215,656

40,990

7,609

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Conclusion

  • The spectral weighting method with the limited integration radius was tested,

​

  • The worst fit was obtained from the first-order radial derivatives of the disturbing gravitational potential,

​

  • The superior results were achieved from the third-order radial derivatives of the disturbing gravitational potential.

​

​

​

Thank you for your attention

pitonakm@ntis.zcu.cz

​