# Wavelet Entropy: A New Tool for Edge Detection of Potential Field Data

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Wavelet Decomposition

#### 2.2. Wavelet Energy

_{x}, L

_{y}) can be given as:

_{x}), t varies from 1 to (q/L

_{y}), p and q are the sizes of the detail coefficient matrices, and N represents the total number of wavelet coefficients in level k. The mean wavelet energy was computed for all windows at a specific level of k using Equation (6). We tested different window sizes to calculate the WSE and found that it gave a good resolution of the sources with a window length of 1 × 1 at the first level. The resolution decreased with more decomposition levels due to the variable windowing effect.

#### 2.3. Wavelet Space Entropy

- First, 2D potential field data with dimensions of M × N were decomposed into different wavelet decomposition levels (k = 1 to D) to obtain approximation coefficients $({W}_{\varphi}\left({j}_{0},p,q\right)),$ and detail (horizontal, vertical, and diagonal) coefficients $({W}_{\psi}^{H,V,D}\left(j,p,q\right))$ using Equations (A1)–(A10). The analyzing wavelet used was Daubechies of order one (db1) at the first level.
- The horizontal and vertical detail coefficients $\left({W}_{\psi}^{H}\left(j,p,q\right)\mathrm{and}{W}_{\psi}^{V}\left(j,p,q\right)\right)$ were combined and normalized to calculate the mean, total, and relative wavelet energies using a window of length L
_{x}× L_{y}at a specific level using Equations (6)–(8). We avoided the utility of the diagonal detail coefficients $\left({W}_{\psi}^{D}\left(j,p,q\right)\right)$ due to their poor resolution and noisy information. - The normalized wavelet space entropy was calculated with the relative wavelet energy using Equation (9).

## 3. Synthetic Cases

#### 3.1. Performance Analysis

#### 3.1.1. Comparison with Conventional Methods

#### 3.1.2. Effect of Noise

## 4. Application to Bishop Model

## 5. Source Boundaries in the Delhi Fold Belt Region

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Multiresolution Analysis Using DWT

## References

- Fairhead, J.D.; Salem, A.; Cascone, L.; Hammill, M.; Masterton, S.; Samson, E. New developments of the magnetic tilt-depth method to improve structural mapping of sedimentary basins. Geophys. Prospect.
**2011**, 59, 1072–1086. [Google Scholar] [CrossRef] - Sun, Y.; Yang, W.; Zeng, X.; Zhang, Z. Edge enhancement of potential field data using spectral moments. Geophysics
**2016**, 81, G1–G11. [Google Scholar] [CrossRef] - Dwivedi, D.; Chamoli, A. Source Edge Detection of Potential Field Data Using Wavelet Decomposition. Pure Appl. Geophys.
**2021**, 178, 919–938. [Google Scholar] [CrossRef] - Evjen, H.M. The place of the vertical gradient in gravitational interpretations. Geophysics
**1936**, 1, 127–136. [Google Scholar] [CrossRef] - Arısoy, M.Ö.; Dikmen, Ü. Edge enhancement of magnetic data using fractional-order-derivative filters. Geophysics
**2015**, 80, J7–J17. [Google Scholar] [CrossRef] - Cordell, L.; Grauch, V.J.S. Mapping Basement Magnetization Zones from Aeromagnetic Data in the San Juan Basin, New Mexico. In The Utility of Regional Gravity and Magnetic Anomaly Maps; Society of Exploration Geophysicists: Houston, TX, USA, 1985; pp. 181–197. [Google Scholar]
- Blakely, R.J.; Simpson, R.W. Approximating edges of source bodies from magnetic or gravity anomalies. Geophysics
**1986**, 51, 1494–1498. [Google Scholar] [CrossRef] - Hidalgo-Gato, M.C.; Barbosa, V.C.F. Edge detection of potential-field sources using scale-space monogenic signal: Fundamental principles. Geophysics
**2015**, 80, J27–J36. [Google Scholar] [CrossRef] - Miller, H.G.; Singh, V. Potential field tilt—a new concept for location of potential field sources. J. Appl. Geophys.
**1994**, 32, 213–217. [Google Scholar] [CrossRef] - Verduzco, B.; Fairhead, J.D.; Green, C.M.; MacKenzie, C. New insights into magnetic derivatives for structural mapping. Lead. Edge
**2004**, 23, 116–119. [Google Scholar] [CrossRef] - Ferreira, F.J.F.; de Souza, J.; de B. e. S. Bongiolo, A.; de Castro, L.G. Enhancement of the total horizontal gradient of magnetic anomalies using the tilt angle. Geophysics
**2013**, 78, 1MJ-Z75. [Google Scholar] [CrossRef] - Santos, D.F.; Silva, J.B.C.; Barbosa, V.C.F.; Braga, L.F.S. Deep-pass — An aeromagnetic data filter to enhance deep features in marginal basins. Geophysics
**2012**, 77, J15–J22. [Google Scholar] [CrossRef] - Wijns, C.; Perez, C.; Kowalczyk, P. Theta map: Edge detection in magnetic data. Geophysics
**2005**, 70, L39–L43. [Google Scholar] [CrossRef] - Nasuti, Y.; Nasuti, A.; Moghadas, D. STDR: A Novel Approach for Enhancing and Edge Detection of Potential Field Data. Pure Appl. Geophys.
**2019**, 176, 827–841. [Google Scholar] [CrossRef] - Pham, L.T.; Eldosouky, A.M.; Oksum, E.; Saada, S.A. A new high resolution filter for source edge detection of potential field data. Geocarto Int.
**2022**, 37, 3051–3068. [Google Scholar] [CrossRef] - Moreau, F.; Gibert, D.; Holschneider, M.; Saracco, G. Wavelet analysis of potential fields. Inverse Probl.
**1997**, 13, 165–178. [Google Scholar] [CrossRef][Green Version] - Chamoli, A.; Srivastava, R.P.; Dimri, V.P. Source depth characterization of potential field data of Bay of Bengal by continuous wavelet transform. Indian J. Mar. Sci.
**2006**, 35, 195–204. [Google Scholar] - Chamoli, A.; Pandey, A.K.; Dimri, V.P.; Banerjee, P. Crustal configuration of the Northwest Himalaya based on modeling of gravity data. Pure Appl. Geophys.
**2011**, 168, 827–844. [Google Scholar] [CrossRef] - Torrence, C.; Compo, G.P. A Practical Guide to Wavelet Analysis. Bull. Am. Meteorol. Soc.
**1998**, 79, 61–78. [Google Scholar] [CrossRef] - Lockwood, O.G.; Kanamori, H. Wavelet analysis of the seismograms of the 2004 Sumatra-Andaman earthquake and its application to tsunami early warning. Geochem. Geophys. Geosystems
**2006**, 7. [Google Scholar] [CrossRef] - Chamoli, A.; Swaroopa Rani, V.; Srivastava, K.; Srinagesh, D.; Dimri, V.P. Wavelet analysis of the seismograms for tsunami warning. Nonlinear Process. Geophys.
**2010**, 17, 569–574. [Google Scholar] [CrossRef][Green Version] - Chamoli, A.; Ram Bansal, A.; Dimri, V.P. Wavelet and rescaled range approach for the Hurst coefficient for short and long time series. Comput. Geosci.
**2007**, 33, 83–93. [Google Scholar] [CrossRef] - Leise, T.L.; Harrington, M.E. Wavelet-based time series analysis of Circadian rhythms. J. Biol. Rhythms
**2011**, 26, 454–463. [Google Scholar] [CrossRef] [PubMed] - Telesca, L.; Lovallo, M.; Chamoli, A.; Dimri, V.P.; Srivastava, K. Fisher-Shannon analysis of seismograms of tsunamigenic and non-tsunamigenic earthquakes. Phys. A Stat. Mech. Its Appl.
**2013**, 392, 3424–3429. [Google Scholar] [CrossRef] - Telesca, L.; Chamoli, A.; Lovallo, M.; Stabile, T.A. Investigating the Tsunamigenic Potential of Earthquakes from Analysis of the Informational and Multifractal Properties of Seismograms. Pure Appl. Geophys.
**2015**, 172, 1933–1943. [Google Scholar] [CrossRef] - Telesca, L.; Lovallo, M. Fisher-shannon analysis of wind records. Int. J. Energy Stat.
**2013**, 01, 281–290. [Google Scholar] [CrossRef] - AghaKouchak, A. Entropy-copula in hydrology and climatology. J. Hydrometeorol.
**2014**, 15. [Google Scholar] [CrossRef][Green Version] - Nicolis, O.; Mateu, J. 2D Anisotropic Wavelet Entropy with an Application to Earthquakes in Chile. Entropy
**2015**, 17, 4155–4172. [Google Scholar] [CrossRef][Green Version] - da Silva, S.L.E.F.; Corso, G. Microseismic event detection in noisy environments with instantaneous spectral Shannon entropy. Phys. Rev. E
**2022**, 106, 014133. [Google Scholar] [CrossRef] [PubMed] - Rosso, O.A.; Blanco, S.; Yordanova, J.; Kolev, V.; Figliola, A.; Schürmann, M.; Başar, E. Wavelet entropy: A new tool for analysis of short duration brain electrical signals. J. Neurosci. Methods
**2001**, 105, 65–75. [Google Scholar] [CrossRef] - Liang, Z.; Chen, S. Edge detection method based on wavelet space entropy and Canny algorithm. In Proceedings of the 2011 4th IEEE International Conference on Broadband Network and Multimedia Technology, Shenzhen, China, 28–30 October 2011; pp. 245–249. [Google Scholar]
- Mallat, S.G. A Theory for Multiresolution Signal Decomposition: The Wavelet Representation. IEEE Trans. Pattern Anal. Mach. Intell.
**1989**, 11, 674–693. [Google Scholar] [CrossRef][Green Version] - Shannon, C.E. A Mathematical Theory of Communication. Bell Syst. Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef][Green Version] - Rosso, O.A.; Martin, M.T.; Figliola, A.; Keller, K.; Plastino, A. EEG analysis using wavelet-based information tools. J. Neurosci. Methods
**2006**, 153, 163–182. [Google Scholar] [CrossRef] [PubMed] - Pérez, D.G.; Zunino, L.; Garavaglia, M.; Rosso, O.A. Wavelet entropy and fractional Brownian motion time series. Phys. A Stat. Mech. Its Appl.
**2006**, 365, 282–288. [Google Scholar] [CrossRef][Green Version] - Tao, Y.; Scully, T.; Perera, A.G.; Lambert, A.; Chahl, J. A low redundancy wavelet entropy edge detection algorithm. J. Imaging
**2021**, 7, 188. [Google Scholar] [CrossRef] [PubMed] - Bhattacharyya, B.K. Magnetic anomalies due to prism-shaped bodies with arbitrary polarization. Geophysics
**1964**, 29, 517–531. [Google Scholar] [CrossRef] - Williams, S.; Derek Fairhead, J.; Flanagan, G. Realistic models of basement topography for depth to magnetic basement testing. SEG Tech. Progr. Expand. Abstr.
**2002**, 21. [Google Scholar] [CrossRef] - Salem, A.; Williams, S.; Fairhead, D.; Smith, R.; Ravat, D. Interpretation of magnetic data using tilt-angle derivatives. Geophysics
**2008**, 73, 14JF-Z11. [Google Scholar] [CrossRef] - Salem, A.; Green, C.; Cheyney, S.; Fairhead, J.D.; Aboud, E.; Campbell, S. Mapping the depth to magnetic basement using inversion of pseudogravity: Application to the Bishop model and the Stord Basin, northern North Sea. Interpretation
**2014**, 2, T69–T78. [Google Scholar] [CrossRef][Green Version] - Dwivedi, D.; Chamoli, A.; Pandey, A.K. Crustal structure and lateral variations in Moho beneath the Delhi fold belt, NW India: Insight from gravity data modeling and inversion. Phys. Earth Planet. Inter.
**2019**, 297, 106317. [Google Scholar] [CrossRef] - Dwivedi, D.; Chamoli, A. Seismotectonics and lineament fabric of Delhi fold belt region, India. J. Earth Syst. Sci.
**2022**, 131, 74. [Google Scholar] [CrossRef] - Geological Survey of India and National Geophysical Research Institute. GSI-NGRI; “Gravity Map Series of India”; Geological Survey of India and National Geophysical Research Institute: Hyderabad, India, 2006. [Google Scholar]

**Figure 2.**(

**a**) Total magnetic anomaly generated due to 3D synthetic model (M1). The black rectangular lines show the actual upper edges of the prismatic sources. (

**b**) Variations in the normalized wavelet space entropy of the magnetic anomaly. Note the sharp changes over the source edges.

**Figure 3.**Comparison of different techniques to identify source edges for model M1. (

**a**) ASA, (

**b**) THDR, (

**c**) TDR, (

**d**) THETA. Three profiles (L1, L2, and L3) were used for error analysis (Figure 4).

**Figure 4.**Comparison of the techniques for profiles (

**a**) L1, (

**b**) L2, and (

**c**) L3 for model M1 and (

**d**) the RMS errors for the different methods. In profile L2, over the two close edges of prisms S1 and S2, all the methods failed to resolve these due to high interference effects. Vertical grey lines indicate the actual source boundaries.

**Figure 5.**(

**a**) Total magnetic anomaly generated due to 3D synthetic model (M2), (

**b**) THDR, and (

**c**) WSE maps. Comparison of THDR and WSE along the profiles of (

**d**) HL1 and (

**e**) HL2. The RMS errors for THDR and WSE were 44.6 and 44.2. Vertical grey lines indicate the actual source boundaries.

**Figure 6.**Normalized WSE after the addition of Gaussian noise with a mean of zero and standard deviations of (

**a**) 5 nT, (

**b**) 7 nT, (

**c**) 8 nT, and (

**d**) 10 nT to model M1.

**Figure 7.**Comparison of THDR and WSE along profile L1 (Figure 6) for prismatic model M1 with Gaussian noise with standard deviations of (

**a**) 5 nT and (

**b**) 6 nT. The RMS errors for THDR and WSE were 55.4 and 49.2. Vertical grey lines indicate the actual source boundaries.

**Figure 8.**Bishop model. (

**a**) Variations in depth of basement, (

**b**) magnetic susceptibility variations in basement, and (

**c**) total magnetic anomaly (after [3]).

**Figure 9.**The edges identified by normalized wavelet space entropy for the magnetic anomaly of the Bishop model (Figure 8c).

**Figure 10.**Geological map and the gravity anomaly of the Delhi fold belt and surrounding region (after [3,42]). Note the geological structures: the Delhi fold belt (DFB), the axis of the Delhi–Sargodha Ridge (DSR), the Mahendragarh–Dehradun Subsurface fault (MDSSF), the Great Boundary Fault (GBF), and the Kaliguman and Delwara lineaments.

**Figure 11.**The edges identified using normalized wavelet space entropy of the Bouguer anomaly of the Delhi fold belt (Figure 10).

Model | Prismatic Source | Length (m) | Width (m) | Extent (m) | Total Magnetization (A/m) | Susceptibility (SI) | Z Top (m) | Inclination of Prism (Degree) |
---|---|---|---|---|---|---|---|---|

M1 | S1 | 2400 | 400 | 700 | 2.6 | +0.05 | 300 | 0 |

S2 | 400 | 2400 | 700 | 2.6 | +0.05 | 400 | 45 | |

S3 | 400 | 2400 | 700 | 2.6 | +0.05 | 300 | −45 | |

S4 | 2400 | 2400 | 400 | 2.6 | +0.05 | 400 | 0 | |

S5 | 1000 | 1000 | 200 | 2.6 | +0.05 | 200 | 0 | |

S6 | 300 | 300 | 200 | 2.6 | +0.05 | 100 | 0 |

**Table 2.**The synthetic case after changing the parameters and geometries of the prismatic source of the model (M2).

Model | Prismatic Source | Length (m) | Width (m) | Extent (m) | Total Magnetization (A/m) | Susceptibility (SI) | Z Top (m) | Inclination of Prism (degree) |
---|---|---|---|---|---|---|---|---|

M2 | S1 | 2600 | 600 | 700 | 2.1 | +0.04 | 220 | 0 |

S2 | 200 | 2200 | 700 | −1.8 | −0.04 | 220 | 45 | |

S3 | 200 | 2200 | 700 | 2.1 | +0.04 | 250 | −45 | |

S4 | 2000 | 2000 | 400 | −2.3 | −0.05 | 320 | 30 | |

S5 | 800 | 800 | 200 | 2.6 | +0.05 | 150 | 0 | |

S6 | 200 | 200 | 200 | 3.2 | +0.06 | 100 | 0 |

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Dwivedi, D.; Chamoli, A.; Rana, S.K. Wavelet Entropy: A New Tool for Edge Detection of Potential Field Data. *Entropy* **2023**, *25*, 240.
https://doi.org/10.3390/e25020240

**AMA Style**

Dwivedi D, Chamoli A, Rana SK. Wavelet Entropy: A New Tool for Edge Detection of Potential Field Data. *Entropy*. 2023; 25(2):240.
https://doi.org/10.3390/e25020240

**Chicago/Turabian Style**

Dwivedi, Divyanshu, Ashutosh Chamoli, and Sandip Kumar Rana. 2023. "Wavelet Entropy: A New Tool for Edge Detection of Potential Field Data" *Entropy* 25, no. 2: 240.
https://doi.org/10.3390/e25020240