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Semi-vectorization: an efficient technique for synthesis and analysis of gravity gradiometry data

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Abstract

The harmonic synthesis and analysis of the elements of gravitational tensor can be done in few minutes if a suitable programming algorithm is used. Vectorization is an efficient technique for such processes, but the size of matrices will increase when the resolution of synthesis or analysis is high; say higher than 0.5° × 0.5°. Here, we present a technique to manage the computer memory and computational time by excluding one computational loop from the matrix products and we call this method semi-vectorization. Based on this technique, we synthesize the gravitational tensor using the EGM96 geopotential model and after that we analyze the tensor for recovering the geopotential coefficients. MATLAB codes are provided which are able to analyze 224 millions gradiometric data, corresponding to a global grid of 2.5′ × 2.5′ on a sphere in 1,093 s by a personal computer with 2 Gb RAM.

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Acknowledgment

The first author is thankful to the Swedish National Space Board (SNSB) for financial support of the project 63:07:1.

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Authors and Affiliations

  1. Division of Geodesy, Royal Institute of Technology, Stockholm, Sweden

    Mehdi Eshagh & Makan Abdollahzadeh

Authors
  1. Mehdi Eshagh
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  2. Makan Abdollahzadeh
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Corresponding author

Correspondence to Mehdi Eshagh.

Additional information

Communicated by: H. A. Babaie

Appendices

Appendix A

The constant coefficients of Eqs. 2b–2f (Eshagh 2009c):

$$ a_{nm}^1 = \frac{1}{4}\sqrt {{n + \left| m \right|}} \sqrt {{n + \left| m \right| - 1}} \sqrt {{n - \left| m \right| + 1}} \sqrt {{n - \left| m \right| + 2}} \sqrt {{\frac{{2 - {\delta_{\left| m \right|,0}}}}{{2 - {\delta_{\left| m \right| - 2,0}}}}}} $$
(A.1)
$$ a_{nm}^2 = - \frac{1}{4}\left[ {\left( {n + \left| m \right|} \right)\left( {n - \left| m \right| + 1} \right) + \left( {n - \left| m \right|} \right)\left( {n + \left| m \right| + 1} \right)} \right] - \left( {n + 1} \right) $$
(A.2)
$$ a_{nm}^3 = \frac{1}{4}\sqrt {{n + \left| m \right| + 2}} \sqrt {{n + \left| m \right| + 1}} \sqrt {{n - \left| m \right|}} \sqrt {{n - \left| m \right| - 1}} \sqrt {{\frac{{2 - {\delta_{\left| m \right|,0}}}}{{2 - {\delta_{\left| m \right| + 2,0}}}}}} $$
(A.3)
$$ a_{nm}^4 = a_{nm}^2 + \left( {n + 1} \right)\left( {n + 2} \right) $$
(A.4)
$$ b_{nm}^1 = \frac{1}{4}\sqrt {{\frac{{2 - {\delta_{\left| m \right|,0}}}}{{2 - {\delta_{\left| m \right| - 2,0}}}}}} \sqrt {{n + \left| m \right|}} \sqrt {{n - \left| m \right| + 1}} \sqrt {{n - \left| m \right| + 2}} \sqrt {{n + \left| m \right| - 1}} $$
(A.5)
$$ b_{nm}^2 = \frac{1}{4}\left[ {\left( {n + \left| m \right|} \right)\left( {n + \left| m \right| - 1} \right) + \frac{{\left| m \right| - 1}}{{\left| m \right| + 1}}\left( {n - \left| m \right|} \right)\left( {n - \left| m \right| - 1} \right) + \frac{{2\left( {n + \left| m \right| + 1} \right)\left( {n + \left| m \right| + 2} \right)}}{{\left| m \right| + 1}}} \right] - \left( {n + 1} \right) $$
(A.6)
$$ b_{nm}^3 = \frac{1}{4}\sqrt {{\frac{{2 - {\delta_{\left| m \right|,0}}}}{{2 - {\delta_{\left| m \right| + 2,0}}}}}} \sqrt {{n - \left| m \right|}} \sqrt {{n - \left| m \right| - 1}} \sqrt {{n + \left| m \right| + 2}} \sqrt {{n + \left| m \right| + 1}} $$
(A.7)
$$ b_{nm}^4 = b_{nm}^2 + \left( {n + 1} \right)\left( {n + 2} \right) $$
(A.8)
$$ c_{nm}^1 = \frac{m}{{4\left| m \right|}}\sqrt {{\frac{{\left( {2 - {\delta_{\left| m \right|,0}}} \right)\left( {2n + 1} \right)}}{{\left( {2 - {\delta_{\left| m \right| - 2,0}}} \right)\left( {2n - 1} \right)}}}} \sqrt {{n + \left| m \right|}} \sqrt {{n + \left| m \right| - 1}} \sqrt {{n + \left| m \right| - 2}} \sqrt {{n - \left| m \right| + 1}} $$
(A.9)
$$ c_{nm}^2 = \frac{m}{2}\sqrt {{n + \left| m \right|}} \sqrt {{n - \left| m \right|}} \sqrt {{\frac{{2n + 1}}{{2n - 1}}}} $$
(A.10)
$$ c_{nm}^3 = \frac{m}{{4\left| m \right|}}\sqrt {{\frac{{\left( {2 - {\delta_{\left| m \right|,0}}} \right)\left( {2n + 1} \right)}}{{\left( {2 - {\delta_{\left| m \right| + 2,0}}} \right)\left( {2n - 1} \right)}}}} \sqrt {{n - \left| m \right|}} \sqrt {{n - \left| m \right| - 1}} \sqrt {{n - \left| m \right| - 2}} \sqrt {{n + \left| m \right| + 1}} $$
(A.11)
$$ d_{nm}^1 = \frac{m}{{4\left| m \right|}}\sqrt {{\frac{{\left( {2 - {\delta_{\left| m \right|,0}}} \right)\left( {2n + 1} \right)}}{{\left( {2 - {\delta_{\left| m \right| - 2,0}}} \right)\left( {2n - 3} \right)}}}} \sqrt {{n + \left| m \right|}} \sqrt {{n - \left| m \right| + 1}} \sqrt {{n - \left| m \right| + 2}} \sqrt {{n - \left| m \right| + 3}} $$
(A.12)
$$ d_{nm}^2 = \frac{m}{{4\left| m \right|}}\left[ {\left( {n + \left| m \right|} \right)\left( {n - \left| m \right| + 1} \right) - \left( {n - \left| m \right|} \right)\left( {n - \left| m \right| + 1} \right)} \right]\sqrt {{\frac{{\left( {2n + 1} \right)\left( {n + \left| m \right| + 1} \right)}}{{\left( {2n + 3} \right)\left( {n - \left| m \right| + 1} \right)}}}} $$
(A.13)
$$ d_{nm}^3 = - \frac{m}{{4\left| m \right|}}\sqrt {{\frac{{\left( {2 - {\delta_{\left| m \right|,0}}} \right)\left( {2n + 1} \right)}}{{\left( {2 - {\delta_{\left| m \right| + 2,0}}} \right)\left( {2n + 3} \right)}}}} \sqrt {{n - \left| m \right|}} \sqrt {{n + \left| m \right| + 1}} \sqrt {{n + \left| m \right| + 2}} \sqrt {{n + \left| m \right| + 3}} $$
(A.14)
$$ e_{nm}^1 = \frac{{n + 2}}{2}\sqrt {{n + \left| m \right|}} \sqrt {{n - \left| m \right| + 1}} \sqrt {{\frac{{2 - {\delta_{\left| m \right|,0}}}}{{2 - {\delta_{\left| m \right| - 1,0}}}}}} $$
(A.15)
$$ e_{nm}^2 = - \frac{{n + 2}}{2}\sqrt {{n - \left| m \right|}} \sqrt {{n + \left| m \right| + 1}} \sqrt {{\frac{{2 - {\delta_{\left| m \right|,0}}}}{{2 - {\delta_{\left| m \right| + 1,0}}}}}} $$
(A.16)
$$ f_{nm}^1 = \frac{{m\left( {n + 2} \right)}}{{2\left| m \right|}}\sqrt {{n + \left| m \right|}} \sqrt {{n + \left| m \right| - 1}} \sqrt {{\frac{{\left( {2 - {\delta_{\left| m \right|,0}}} \right)\left( {2n + 1} \right)}}{{\left( {2 - {\delta_{\left| m \right| - 1,0}}} \right)\left( {2n - 1} \right)}}}} $$
(A.17)
$$ f_{nm}^2 = \frac{{m\left( {n + 2} \right)}}{{2\left| m \right|}}\sqrt {{n - \left| m \right|}} \sqrt {{n - \left| m \right| - 1}} \sqrt {{\frac{{\left( {2 - {\delta_{\left| m \right|,0}}} \right)\left( {2n + 1} \right)}}{{\left( {2 - {\delta_{\left| m \right| + 1,0}}} \right)\left( {2n - 1} \right)}}}} $$
(A.18)
$$ g_{nm}^1 = \frac{{m\left( {n + 2} \right)}}{{2\left| m \right|}}\sqrt {{n - \left| m \right| + 1}} \sqrt {{n - \left| m \right| + 2}} \sqrt {{\frac{{\left( {2 - {\delta_{\left| m \right|,0}}} \right)\left( {2n + 1} \right)}}{{\left( {2 - {\delta_{\left| m \right| - 1,0}}} \right)\left( {2n + 3} \right)}}}} $$
(A.19)
$$ g_{nm}^2 = \frac{{m\left( {n + 2} \right)}}{{2\left| m \right|}}\sqrt {{n + \left| m \right| + 1}} \sqrt {{n + \left| m \right| + 2}} \sqrt {{\frac{{\left( {2 - {\delta_{\left| m \right|,0}}} \right)\left( {2n + 1} \right)}}{{\left( {2 - {\delta_{\left| m \right| + 1,0}}} \right)\left( {2n + 3} \right)}}}} . $$
(A.20)

Appendix B

The constant coefficients of Eqs. 4a–4d:

$$ h_{nm}^1 = \frac{1}{2}\sqrt {{\frac{{2 - {\delta_{\left| m \right|,0}}}}{{2 - {\delta_{\left| m \right| - 2,0}}}}}} \sqrt {{n + \left| m \right|}} \sqrt {{n + \left| m \right| - 1}} \sqrt {{n - \left| m \right| + 1}} \sqrt {{n - \left| m \right| + 2}} $$
(B.1)
$$ h_{nm}^2 = {m^2} $$
(B.2)
$$ h_{nm}^3 = \frac{1}{2}\sqrt {{\frac{{2 - {\delta_{\left| m \right|,0}}}}{{2 - {\delta_{\left| m \right| + 2,0}}}}}} \sqrt {{n - \left| m \right|}} \sqrt {{n - \left| m \right| - 1}} \sqrt {{n + \left| m \right| + 1}} \sqrt {{n + \left| m \right| + 2}} $$
(B.3)
$$ k_{nm}^1 = \frac{m}{{2\left| m \right|}}\sqrt {{\frac{{\left( {2 - {\delta_{\left| m \right|,0}}} \right)\left( {2n + 1} \right)}}{{\left( {2 - {\delta_{\left| m \right| - 2,0}}} \right)\left( {2n + 3} \right)}}}} \sqrt {{n + \left| m \right|}} \sqrt {{n - \left| m \right| + 3}} \sqrt {{n - \left| m \right| + 2}} \sqrt {{n - \left| m \right| + 1}} $$
(B.4)
$$ k_{nm}^2 = \frac{m}{2}\sqrt {{\frac{{\left( {2n + 1} \right)\left( {n + m + 1} \right)}}{{\left( {2n + 3} \right)\left( {n - m + 1} \right)}}}} \left[ {\left( {n + m + 2} \right)\left( {n - m + 1} \right) - \left( {n - m + 2} \right)\left( {n - m + 1} \right)} \right] $$
(B.5)
$$ k_{nm}^3 = - \frac{m}{{2\left| m \right|}}\sqrt {{\frac{{\left( {2 - {\delta_{\left| m \right|,0}}} \right)\left( {2n + 1} \right)}}{{\left( {2 - {\delta_{\left| m \right| + 2,0}}} \right)\left( {2n + 3} \right)}}}} \sqrt {{n - \left| m \right|}} \sqrt {{n + \left| m \right| + 3}} \sqrt {{n + \left| m \right| + 2}} \sqrt {{n + \left| m \right| + 1}} $$
(B.6)
$$ l_{nm}^1 = \frac{m}{{2\left| m \right|}}\sqrt {{\frac{{\left( {2 - {\delta_{\left| m \right|,0}}} \right)\left( {2n + 1} \right)}}{{\left( {2 - {\delta_{\left| m \right| - 2,0}}} \right)\left( {2n - 1} \right)}}}} \sqrt {{n + \left| m \right|}} \sqrt {{n + \left| m \right| - 3}} \sqrt {{n + \left| m \right| - 2}} \sqrt {{n - \left| m \right| + 1}} $$
(B.7)
$$ l_{nm}^2 = - m\sqrt {{\frac{{2n + 1}}{{2n - 1}}}} \sqrt {{n - \left| m \right|}} \sqrt {{n + \left| m \right|}} $$
(B.8)
$$ l_{nm}^3 = - \frac{m}{{2\left| m \right|}}\sqrt {{\frac{{\left( {2 - {\delta_{\left| m \right|,0}}} \right)\left( {2n + 1} \right)}}{{\left( {2 - {\delta_{\left| m \right| + 2,0}}} \right)\left( {2n - 1} \right)}}}} \sqrt {{n - \left| m \right|}} \sqrt {{n - \left| m \right| - 1}} \sqrt {{n - \left| m \right| - 2}} \sqrt {{n + \left| m \right| + 1}} $$
(B.9)

and \( e{\prime}_{nm}^1 = e_{nm}^1/\left( {n + 2} \right) \), \( e{\prime}_{nm}^2 = e_{nm}^2/\left( {n + 2} \right) \), \( g{\prime}_{nm}^1 = g_{nm}^1/\left( {n + 2} \right) \), \( g{\prime}_{nm}^2 = g_{nm}^2/\left( {n + 2} \right) \), \( f{\prime}_{nm}^1 = f_{nm}^1/\left( {n + 2} \right) \), \( f{\prime}_{nm}^2 = f_{nm}^2/\left( {n + 2} \right) \).

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Eshagh, M., Abdollahzadeh, M. Semi-vectorization: an efficient technique for synthesis and analysis of gravity gradiometry data. Earth Sci Inform 3, 149–158 (2010). https://doi.org/10.1007/s12145-010-0062-3

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