# De-Noising of Magnetotelluric Signals by Discrete Wavelet Transform and SVD Decomposition

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Multiscale Dispersion Entropy

- (1)
- Time series u
_{j}(j = 1, 2, …, N) are taken via a coarse graining preprocessing for non-overlapping segments x_{b}and are obtained as follows:$${x}_{b}=\frac{1}{\tau}{\displaystyle \sum _{j=\left(b-1\right)*\tau +1}^{b*\tau}{u}_{j},\left(1\le b\le L\right)}$$ - (2)
- x
_{b}is mapped to c classes from 1 to c. First, the normal cumulative distribution function (NCDF) is employed to map the segment to y_{b}from 0 to 1:$${y}_{b}=\frac{1}{\sigma \sqrt{2\pi}}{\displaystyle \underset{-\infty}{\overset{{x}_{b}}{\int}}{e}^{\frac{-{\left(t-rms\right)}^{2}}{2{\sigma}^{2}}}}dt$$_{b}^{c}= round(c• y_{b}+ 0.5) is used for y_{b}to z_{b}^{c}from 1 to c. Then, the embedding dimension m and time delay d are introduced to reconstruct z_{i}^{m,c}as follows:$${z}_{i}{}^{m,c}=\left\{{z}_{i}{}^{c},{z}_{i+d}{}^{c}\cdots {z}_{i+\left(m-1\right)d}{}^{c}\right\}$$_{i}^{m,c}is equal to c^{m}, since the signal has m members and each member can be an integer from 1 to c.Finally, for each c^{m}potential dispersion pattern, the relative frequency is defined as follows:$$p\left({\pi}_{{v}_{0}\cdots v}{}_{{}_{m-1}}\right)=\frac{Number\left\{i|i\le L-\left(m-1\right)d,\begin{array}{cccc}{z}_{i}{}^{m,c}& has& type& {\pi}_{{v}_{0}\cdots v}{}_{{}_{m-1}}\end{array}\right\}}{L-\left(m-1\right)d}$$_{i}^{m,c}divided by the total number of embedding signals with embedding dimension m. ${v}_{0}={z}_{i}^{c}$, ${v}_{1}={z}_{i+d}^{c}$…, ${v}_{m-1}={z}_{i+\left(m-1\right)d}^{c}$. - (3)
- The MDE is obtained as follows:$$MD{E}_{rms}=E\left(x,\tau ,m,c,d\right)=\left[{e}_{1},{e}_{2}\cdots ,{e}_{\tau}\right]$$

#### 2.2. Phase Space Reconstruction

_{1}, u

_{2}… u

_{N}}, where N is the length of the signal, is reconstructed to different phase space vectors x

_{i}= {u

_{i}, u

_{i}

_{+}

_{τ}… u

_{i}

_{+ (m−1)}

_{τ}} (i = 1, 2, …, m) through the method of delay, where τ denotes time delay and m is the embedding dimension. Thus, the two parameters are necessary for the reconstruction. In general, various methods can be used to obtain τ, such as autocorrelation function [39], experience [40], and mutual information function [41]. However, compared with the above two methods, the third method is expected to process non-linear and non-stationary data. For the selection of m in this paper, false nearest neighbors (FNN) [42] are introduced to obtain m, where m must satisfy m > 2h + 1 (Takens theory) [43] and h represents the real dimension of attractors.

#### 2.2.1. Mutual Information Function

_{i}) over s

_{i}as follows:

_{n}(τ) is employed as time delay τ.

#### 2.2.2. False Nearest Neighbors

_{i}= {u

_{i}, u

_{i}

_{+}

_{τ}… u

_{i}

_{+(m−1)}

_{τ}} by x

_{i}

^{0}= {u

_{i}

^{0}, u

_{i}

_{+}

_{τ}

^{0}… u

_{i}

_{+(m−1)}

_{τ}

^{0}} within a certain distance in m dimensions is denoted. Then, the square Euclidian distance is as follows:

_{1}(i, m) is given as follows:

_{1}(i, m) ≥ 10, the neighbor is identified as false; in contrast, m is the embedding dimension.

#### 2.3. Discrete Wavelet Transform

_{0}is the scale. t is the temporal shift and t

_{0}is initial condition. W

_{f}(m, n) denotes the wave coefficients, which satisfy the following expression:

_{0}and t

_{0}are correlated with them [30]. Thus, the inverse wavelet transform is obtained as follows:

#### 2.4. SVD Decomposition

_{1}

^{T}, x

_{2}

^{T}, … x

_{n}

^{T}]

^{T}, where X∈R

^{m}

^{× n}. The following formulation of the SVD method [47] is:

^{m}

^{× n}represents a diagonal matrix, O is the zero matrix, σ

_{1}≥ σ

_{2}≥ …≥ σ

_{p}≥ 0, and p = min(m, n), which are the singular values of X in descending order. U is an m × m matrix, of which the columns are orthonormal. V is an n × n matrix, of which the rows are orthonormal.

_{s}

^{i}and X

_{s}represent the effective signal section and σ

_{n}

^{i}and X

_{n}are the noise section. The two-norm $f={\left({\Vert {\displaystyle \sum \overline{{X}_{s}}}\Vert}_{2}\right)}_{min}$ is employed to estimate the signal and the reconstruction matrix of the signal is expressed as follows:

- Overlap data to segments to ensure the continuity of the data;
- Calculate the MDE for each data segment, for which the value below the threshold is the noisy data segment;
- Apply phase space reconstruction to calculate the number of wavelet decomposition level in noisy data segments;
- Perform discrete wavelet transform and discrete wavelet inverse transform for multiple components;
- Decompose the components using iterative SVD to obtain the de-noised section;
- Reconstruct the MT de-noising signal with the useful data segments given in Step 1 and the de-noising data segments given in Step 5, where the average value of the two segments is adopted for the overlapping data.

## 3. Synthetic Cases

_{2}is a 2-norm function, and S/N denotes the level of added noise. The value of 0 dB indicates that the intensity of the added noise is much stronger than the useful signal, which means that the noise level is high. On the contrary, 40 dB represents that the intensity of added noise is low. When S/N is 40 dB, the intensity of the noise is only one percent of the useful signal, meaning that it has little influence on the useful signal.

#### 3.1. Entropy Analysis

#### 3.2. Parameter Calculation

#### 3.3. Performance Evaluation

## 4. Implementation for MT Field Data

_{yx}in site 340 and ρ

_{xy}in site 280, the de-noising curves of two methods for ρ

_{xy}in site 340 and ρ

_{yx}in site 280 are overlapping, which further verifies the stability and effectiveness of the proposed method.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Noise-free MT data, (

**b**) data contaminated by square wave noise, (

**c**) data contaminated by charge–discharge triangular wave noise, (

**d**) data contaminated by impulse noise, (

**e**) data contaminated by various noise. (

**f**–

**j**) is the frequency spectrum corresponding to (

**a**–

**e**). Noise-free MT data were collected in Linze Province with a sampling rate of 15 Hz. All the noise was randomly simulated.

**Figure 3.**Entropy value of noise-free data and noisy data at different data segments. (

**a**) Noise-free MT data, (

**b**) data contaminated by square noise, (

**c**) data contaminated by charge–discharge triangular wave, (

**d**) data contaminated by impulse noise, (

**e**) data contaminated by various noise. The green dashed line is the baseline of MDE, and the blue dashed line is the baseline of MSE. The red solid line shows the MDE, and the black solid line shows the MSE.

**Figure 4.**The choice of the best wavelet decomposition level made through repeated tests. (

**a**) shows SNR and MSE of the whole noisy signal at different wavelet decomposition levels, (

**b**) shows NCC and E of the whole noisy signal at different wavelet decomposition levels. Red solid line indicates SNR, black solid line indicates MSE, red dashed line indicates NCC, and black dashed line indicates E.

**Figure 5.**Data detected by MDE and the noise time–frequency spectrum extracted by the proposed method. (

**a**) shows the noisy data selected by MDE in the signal contaminated by square wave noise, (

**e**) indicates the square wave noise extracted by the proposed method. (

**b**,

**f**) are the results of the signal contaminated by charge–discharge triangular wave. (

**c**,

**g**) denote the results of data contaminated by impulse noise. (

**d**,

**h**) represent the results of data contaminated by various noise types. (

**i**–

**l**) are the frequency spectrum corresponding to (

**e**–

**h**).

**Figure 6.**Time–frequency domain results after application of the proposed method. (

**a**–

**j**) shows the original and de-noised time series of noise-free MT data, data contaminated by square noise, data contaminated by charge–discharge triangular waves, data contaminated by impulse noise, data contaminated by various types of noise. (

**k**–

**o**) is the frequency spectrum corresponding to (

**f**–

**j**). Red boxes indicate the differences between the de-nosing signal and original signal at 500 data point. Blue boxes indicate the differences between the de-nosing signal and original signal at 2900 data point.

**Figure 7.**The de-noising results obtained after the use of different methods. Black lines represent the original signal as a reference, blue lines indicate the de-noising results of SVD, red lines denote the processing results of WT, purple lines show the results improved by the proposed method, completely covered by black lines. (

**a**–

**c**) show the time series of data contaminated by square noise after the use of SVD, WT, and the proposed method. (

**d**–

**f**) show the time series of data contaminated by impulse noise after the above different methods have been used. (

**g**–

**i**) show the time series of data contaminated by charge–discharge triangular waves after the above different methods have been used. (

**j**–

**l**) shows the time series of data contaminated by various types of noise after the above different methods have been used.

**Figure 8.**Frequency domain results after the use of the different methods. (

**a**–

**c**) show the spectrum of data contaminated by square noise after the use of SVD, WF, and the proposed method. (

**d**–

**f**) show the spectrum of data contaminated by impulse noise after the above different methods have been used. (

**g**–

**i**) show the spectrum of data contaminated by charge–discharge triangular waves after the above different methods have been used. (

**j**–

**l**) show the spectrum of data contaminated by various types of noise after the above different methods have been used.

**Figure 9.**(

**a**) Recovery error (E), (

**b**) NCC, (

**c**) MSE, and (

**d**) SNR of the signals recovered by different methods at different S/Ns. Black line indicates wavelet transform, red line denotes SVD decomposition, and green line shows the proposed method.

**Figure 10.**Time series segments of the real sites 380 and 360 in Qilian area with a sampling rate of 15 Hz. (

**a**) Raw signal of Site 380, (

**b**) noise extracted by the proposed method, and (

**c**) signal de-noised by the proposed method. (

**d**) Raw signal of site 360, (

**e**) noise extracted by the proposed method, and (

**f**) signal de-noised by the proposed method.

**Figure 11.**Time series segment of site 380 after the use of different methods. (

**a**) Raw signal of site 380, (

**b**) noise de-noised by SVD, (

**c**) signal de-noised by WT, (

**d**) noise de-noised by the proposed method, and (

**e**–

**h**) the frequency spectrum corresponding to (

**a**–

**d**).

**Figure 12.**Apparent resistivity phase curves of field sites. The red curves represent the results obtained using raw data. The green curves denote the results obtained using the data de-noised using the proposed method. The blue curves stand for the results obtained using the data de-noised by the robust method.

Different Signals | MAE | MSE | MFE | MDE |
---|---|---|---|---|

Noise-free signal | 0.2531 | 1.6493 | 0.1582 | 2.7568 |

With square wave | 0.4519 | 0.9279 | 0.1369 | 1.1521 |

With triangle wave | 0.3458 | 1.3142 | 0.1174 | 2.0091 |

With impulse noise | 0.3379 | 1.5034 | 0.1041 | 2.3361 |

With various noise | 0.4323 | 0.9013 | 0.1124 | 1.1055 |

**Table 2.**The wavelet decomposition level calculated by the proposed method for the different noisy data segments in the different signals (signals with different types of noise contain different numbers of noisy data segments).

Signals with Different Noise | Noisy Segment1 | Noisy Segment2 | Noisy Segment3 | Noisy Segment4 | Noisy Segment5 | Noisy Segment6 | Noisy Segment7 | Noisy Segment8 |
---|---|---|---|---|---|---|---|---|

Square wave | 4 | 5 | 4 | 5 | 4 | 5 | - | - |

Triangle wave | 5 | 5 | 4 | 5 | 4 | 4 | 4 | 4 |

Impulse noise | 4 | 5 | 4 | 4 | 4 | 5 | 4 | 4 |

Various noise | 4 | 5 | 4 | 4 | 4 | 4 | 4 | 4 |

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**MDPI and ACS Style**

Zhou, R.; Han, J.; Guo, Z.; Li, T.
De-Noising of Magnetotelluric Signals by Discrete Wavelet Transform and SVD Decomposition. *Remote Sens.* **2021**, *13*, 4932.
https://doi.org/10.3390/rs13234932

**AMA Style**

Zhou R, Han J, Guo Z, Li T.
De-Noising of Magnetotelluric Signals by Discrete Wavelet Transform and SVD Decomposition. *Remote Sensing*. 2021; 13(23):4932.
https://doi.org/10.3390/rs13234932

**Chicago/Turabian Style**

Zhou, Rui, Jiangtao Han, Zhenyu Guo, and Tonglin Li.
2021. "De-Noising of Magnetotelluric Signals by Discrete Wavelet Transform and SVD Decomposition" *Remote Sensing* 13, no. 23: 4932.
https://doi.org/10.3390/rs13234932