Aeromagnetic Compensation Algorithm Based on Levenberg-Marquard Neural Network

doi:10.11947/j.JGGS.2021.0406

Abstract

Abstract:

The magnetic compensation of aeromagnetic survey is an important calibration work, which has a great impact on the accuracy of measurement. In an aeromagnetic survey flight, measurement data consists of diurnal variation, aircraft maneuver interference field, and geomagnetic field. In this paper, appropriate physical features and the modular feedforward neural network (MFNN) with Levenberg-Marquard (LM) back propagation algorithm are adopted to supervised learn fluctuation of measuring signals and separate the interference magnetic field from the measurement data. LM algorithm is a kind of least square estimation algorithm of nonlinear parameters. It iteratively calculates the jacobian matrix of error performance and the adjustment value of gradient with the regularization method. LM algorithm’s computing efficiency is high and fitting error is very low. The fitting performance and the compensation accuracy of LM-MFNN algorithm are proved to be much better than those of TOLLES-LAWSON (T-L) model with the linear least square (LS) solution by fitting experiments with five different aeromagnetic surveys’ data.

Key words: modular feedforward neural network; aeromagnetic compensation; LM back propagation algorithm; regularization

Li LIU,Qingfeng XU,Hui GU,Lei ZHOU,Zhenfu LIU,Lili CAO. Aeromagnetic Compensation Algorithm Based on Levenberg-Marquard Neural Network[J]. Journal of Geodesy and Geoinformation Science, 2021, 4(4): 74-83.

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LM algorithm flow"

	Input training set, and set MSE goal, minimum gradient, regularization coefficient μ, etc.
	Initialize network connection weights and thresholds.
Iterate	① Compute the output error E_r of all inputs with the current weights and biases.
	② Compute the sum of the square differences S_se corresponding to all inputs.
	③ Compute the jacobian matrix:H=J(se)^T×J(se).^*
	④ Compute the gradient:g=J(se)^T×E_r.
	⑤ Compute the adjustment value of the gradient:Δg=(H+Iμ)^-1g. ^**.
	⑥ Recompute S_se with g+Δg:if S_se goes down then decreases μ and goes to step ①, and if S_se goes up then increases μ and goes to step ⑤.
↑No	Whether the stop condition MSE goal, minimum gradient, or maximum epochs is reached.
Yes→	Stop training, and output the weights, biases and training parameters.

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References 24

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Subnetwork	Function	Features	Input layer dimension	Hidden layer dimension
Z₁	H_sq	t	3	1
Z₂	H_loc	φ×θ ^*	6	3
Z₃	H_perm	cosα, cosβ, cosγ	9	4
Z₄	H_ind	cosα², cosαcosβ, cosαcosγ, cosβ²,cosβcosγ	15	7
Z₅	H_eddy	cosαcosα', cosαcosβ', cosαcosγ', cosβcosα', cosβcosβ', cosβcosγ', cosγcosα', cosγcosβ'	24	12
Z₆	H_g	dφ,dθ	6	3

Data set	Best MSE	Regression rate
Training set	2.9846e-4	0.99946
Test set	2.9426e-4	0.99946
Validation set	2.8862e-4	0.99947

Method	Standard Deviation/nT	IR
Raw data	11.7035	—
Data compensated by LS	0.4399	26.606
Data compensated by MFNN	0.3836	30.5121

Experiment code	Date of measurement	Platform
UAV-2	Sep. 20, 2019	Unmanned helicopter
UAV-3	Nov. 14, 2019	Unmanned helicopter
South China Sea Survey(SCSS)-1	Apr. 19, 2019	Y-8
SCSS-2	May 9, 2019	Y-8

Experiment code	Standard Deviation of Raw data	Standard Deviation of LS compensation	Standard Deviation of networks Z₃—Z₅ compensation	Standard Deviation of all networks compensation
UAV-2	17.8126	1.1256	0.9324	0.5004
UAV-3	41.1538	0.7028	0.4856	0.4855
SCSS-1	191.8949	23.0391	0.7030	0.6018
SCSS-2	118.0897	45.7260	5.6801	1.0443