Phase Transformation of the Analytic Signal for Enhancing the Resolution of Potential Field Data ^†

Abstract

Enhancement methods based on first-order derivatives are well established tools for gravity and magnetic data processing. Higher-resolution filters have been developed using high-order derivatives, but they are generally more sensitive to noise. Based on a transformation that sharpens the instantaneous phase of the complex analytic signal, which corresponds to the tilt angle map, we obtain an enhancement filter that improves the resolution of the total horizontal gradient (THG) without the need for additional derivative calculations. The steps of the proposed method are as follows: (1) compute the horizontal and vertical derivatives of the data; (2) compute the tilt angle and the analytic signal amplitude; (3) apply a sigmoidal-type transformation to the tilt angle; and (4) reconstruct the THG from the analytic signal amplitude and the transformed tilt angle. The reconstructed THG provides sharper boundary estimates than the true THG, as qualitatively shown with noise-corrupted synthetic data and aeromagnetic data from the Seival copper mining area in Caçapava do Sul, Brazil.

1. Introduction

The complex analytic signal is a theoretical concept that is useful in geophysical data processing. In the context of seismic methods, the analytic signal of a seismic trace can provide information on the reflection strength and the instantaneous phase [1]. For gravity and magnetic data, the complex analytic signal is usually defined from horizontal derivatives of the anomalous field [2,3]. Its amplitude and instantaneous phase, which are known as the analytic signal amplitude [4] and the tilt angle [5], respectively, are well-known enhancement filters.

Porsani and Ursin [6] have recently proposed a transformation that sharpens the instantaneous phase of the analytic signal, improving the temporal and spatial resolution of reflections in seismic traces (see also [7]). We noted that this transformation is effective for potential field maps as well.

In fact, the transformed instantaneous phase turns out to enhance the tilt angle filter. Enhancements of the tilt angle have been previously obtained, e.g., by multiplicative factors [8,9] or when the horizontal tilt angle [10] and the tilt angle are combined [11]. The novelty of our work is to use the transformed tilt angle to reconstruct the components of the complex analytic signal, namely the vertical derivative [12] and the total horizontal gradient [13,14]. The resolution of the latter was significantly improved by the phase transformation, and is comparable to enhancements of the total horizontal gradient based on high-order derivatives, such as the enhanced horizontal derivative [15,16], the tilt angle of the horizontal gradient [17] and nonlinear transformations of this filter [18,19,20,21,22], the enhanced horizontal gradient amplitude [23], and the modified horizontal gradient amplitude [24].

To test the proposed methodology, we consider an example with synthetic anomalies generated by interfering sources and corrupted with Gaussian noise. Moreover, to illustrate its potential for processing potential field data in the context of mineral exploration, we employ aeromagnetic data from a copper mineralization area near Caçapava do Sul, southern Brazil. In both synthetic and field examples, the reconstructed total horizontal gradient provides sharper boundary estimates than the true one.

2. Method

Let us recall that Roest et al. [4] defined the analytic signal for potential field data $f=f(x,y,z)$ as follows (see also [3,25]):

$$\mathbf{AS}=(f_x,f_y,i{f}_z),$$

(1)

whose amplitude and instantaneous phase are

$$A=\sqrt{f_x^2+f_y^2+f_z^2},\theta ={tan}^{-1}{\displaystyle \frac{f_z}{f_h}}\left(f_h=\sqrt{f_x^2+f_y^2}\right).$$

(2)

The analytic signal may be written in terms of A and $\theta$ as

$$\mathbf{AS}=A\left({\widehat{\nabla }}_{h}fcos\theta +isin\theta \mathbf{k}\right),{\widehat{\nabla }}_{h}f={\displaystyle \frac{1}{f_h}}{\nabla }_{h}f,$$

(3)

where ${\nabla }_{h}f=f_x\mathbf{i}+f_y\mathbf{j}$ is the horizontal gradient and $\mathbf{i},\mathbf{j},\mathbf{k}$ are unit vectors in the positive x-, y-, and z-directions, respectively. Regarding the analytic signal components,

$$\left\{\begin{array}{ccc}\hfill f_z& =& Asin\left(\theta \right),\hfill \\ \hfill {\nabla }_{h}f& =& A{\widehat{\nabla }}_{h}fcos\left(\theta \right).\hfill \end{array}\right.$$

(4)

The maps of A and $\theta$ in Equation (2) are very popular enhancement filters in potential field data processing, being known as the analytic signal amplitude (ASA) and the tilt angle (TA), respectively. The former is known for being independent of the magnetization direction for tabular sources (but not in general for 3D sources [26]), while the latter highlights the centers of the sources when applied to gravity or reduced-to-the pole (RTP) magnetic data and equalizes anomalies from shallow and deep sources [5,17,27]. Moreover, $f_h$ is the horizontal gradient magnitude, or total horizontal gradient (THG), which is helpful in delineating source boundaries in the cases of gravity or RTP magnetic data [28].

Similarly to [6], we transform the instantaneous phase as follows:

$$\tilde{\theta }={\displaystyle \frac{1}{2}}\pi {\displaystyle \frac{p^{\alpha }-q^{\alpha }}{p^{\alpha }+q^{\alpha }}},p=1+{\displaystyle \frac{2\theta }{\pi }},q=1-{\displaystyle \frac{2\theta }{\pi }},$$

(5)

where the parameter $\alpha >1$ controls the degree of the phase transformation. As shown in Figure 1a, it is a sigmoid-type transformation where the higher the value of $\alpha$, the sharper the transformed instantaneous phase $\tilde{\theta }$ is. If $\alpha =1$, then $\tilde{\theta }=\theta$. The cosine and sine of $\tilde{\theta }$, which are employed to reconstruct the analytic signal components in Equation (6), are shown in Figure 1b and Figure 1c, respectively. It is remarkable that the transformed cosine develops a peak at is maximum; this property will enhance the resolution of THG near its peaks.

As in Equation (4), we reconstruct the vertical derivative $f_z$ and the horizontal gradient ${\nabla }_{h}f$ as follows:

$$\left\{\begin{array}{ccc}\hfill {\tilde{f}}_z& =& Asin\left(\tilde{\theta }\right),\hfill \\ \hfill {\tilde{\nabla }}_{h}f& =& A{\widehat{\nabla }}_{h}fcos\left(\tilde{\theta }\right).\hfill \end{array}\right.$$

(6)

Moreover, as $|{\widehat{\nabla }}_{h}f|=1$, it readily follows that the reconstructed THG (rTHG) is

$${\tilde{f}}_h=\left|{\tilde{\nabla }}_{h}f\right|=Acos\left(\tilde{\theta }\right).$$

(7)

In summary, we compute the rTHG from a potential field data f and a phase transformation parameter $\alpha$ using the following steps:

• compute the horizontal and vertical derivatives of the data, $f_x$, $f_y$, and $f_z$;
• compute TA and ASA, Equation (3);
• compute the transformed instantaneous phase $\tilde{\theta }$ from TA, Equation (5);
• reconstruct the THG from ASA and the transformed TA, Equation (7).

We compute the x- and y-derivatives with centered finite differences in the spatial domain [14], while the vertical derivative (VDR) is computed in the wavenumber domain ([29], p. 326). As we aim to emphasize noise sensitivity, we do not employ any regularization or filtering in our calculations.

3. Results and Discussion

3.1. Synthetic Data

We illustrate the proposed approach with a synthetic example from [30]. The theoretical model consists of two magnetic prisms defined by the parameters shown in

Table 1. Parameters of the magnetic model of two prismatic sources.

Parameter	P1	P2
Easting coordinates of center (km)	120	90
Northing coordinates of center (km)	120	90
Width (km)	50	60
Length (km)	50	60
Depth of top (km)	7	9
Depth of bottom (km)	9	13
Magnetization (A/m)	1.5	2

. Figure 2a shows the total-field anomaly with declination $D=-13.6^{\circ }$, inclination $I=-37.2^{\circ }$, and grid spacing $\Delta x=\Delta y=$ 1 km, contaminated with Gaussian random noise with a standard deviation corresponding to 1% of the maximum absolute value of the anomaly. Afterwards, we apply the RTP transformation to these data (Figure 2b).

The original TA and transformed TA of the RTP data are shown in Figure 3. Both maps are dominated by noise around the sources, but the transformed TA (Figure 3b) is sharper.

Figure 4 shows the original and transformed horizontal ($f_h$ and ${\tilde{f}}_h$) and vertical ($f_z$ and ${\tilde{f}}_z$) derivatives of the noisy data. The VDR is not significantly changed by the phase transformation, but a sharper map is obtained for THG, Figure 4b, which can be compared with the transformation with different values of the parameter $\alpha$ (Figure 5). In particular, Figure 5b represents a substantial improvement of the $f_h$ map, both in terms of resolution and signal-to-noise ratio.

For the sake of comparison, we present enhanced maps obtained by two high-order methods, the enhanced horizontal derivative (EHD) and the modified horizontal gradient amplitude (MHGA). We have chosen these approaches as they have the same purpose as rTHG, namely the improvement of the THG method. We implemented the EHD as follows:

$$\mathrm{EHD}=\sqrt{{\varphi }_x^2+{\varphi }_z^2},\varphi =\sum _{i=0}^m{w}_i{\displaystyle \frac{{\partial }^{i}f}{\partial z^i}}.$$

(8)

We use $w_i=k^i$ as in [16], where k equals to the grid spacing [30], and consider the order $m=4$. The MHGA is defined as

$$\mathrm{MHGA}={\displaystyle \frac{|R+1|-|R-1|}{2}},R={\displaystyle \frac{f_{hz}}{\sqrt{f_{hx}^2+f_{hy}^2}}}-{\displaystyle \frac{\pi }{3}},$$

(9)

where $f_{hx}$, $f_{hy}$, and $f_{hz}$ are the derivatives of THG in the x-, y- and z- directions, respectively [24].

The maps of EHD and MHGA, shown in Figure 6, are comparable to the rTHG maps, especially those in Figure 4b and Figure 5b. The higher order of these filters leads to more accurate edges than rTHG, but they are more sensitive to noise. In fact, they depend on stable vertical differentiation [15,31] of prior data filtering [32,33,34] to reduce the noise sensitivity.

To have an accuracy measure of a given enhancement map, we generate a binary map from its peaks and compute the RMS error of this map with respect to the binary map containing the true edges (Figure 7a).

Similarly to [35], we select the peaks at grid points $(x_{i,j},y_{i,j})$ where the enhancement map value $z_{i,j}$ satisfies at least two of the following four conditions,

$$\begin{array}{cc}{z}_{i,j}>z_{i-1,j}\mathrm{and}{z}_{i,j}>z_{i+1,j},& z_{i,j}>z_{i-1,j+1}\mathrm{and}{z}_{i,j}>z_{i+1,j-1},\\ z_{i,j}>z_{i,j-1}\mathrm{and}{z}_{i,j}>z_{i,j+1},& z_{i,j}>z_{i+1,j+1}\mathrm{and}{z}_{i,j}>z_{i-1,j-1},\end{array}$$

(10)

In addition, $z_{i,j}$ must be greater than 25% of the global maximum. The selected peaks of rTHG with $\alpha =8$, EHD, and MHGA are shown in Figure 7b–d, and their RMS errors are 0.0640, 0.1794, and 0.2096, respectively. An acceptable value of the RMS error should be less than the error of a blank (null) binary map, which is 0.0712.

We study the dependence of the RMS error on the data noise level in Figure 8. The data are contaminated with Gaussian random noise with standard deviation varying from 0% to 10% of the maximum absolute value of the anomaly. Figure 8a compares the rTHG with $\alpha =8$ with EHD and MHGA. The RMS errors of the latter two are lower than the one of rTHG in the absence of noise, but they quickly increase beyond the reference value 0.0712 as the noise increases, while the RMS error of rTHG is less sensitive to noise. Figure 8b illustrates the dependence of the RMS error on the phase transformation parameter $\alpha$, showing that $\alpha >2$ provides qualitatively accurate results.

3.2. Field Data

3.2.1. Magnetometric Database

The proposed approach has also been tested with aeromagnetic anomalies from the Seival mining area, in Caçapava do Sul, Brazil, which were the object of study in [36]. The primary objective of that study was to investigate the geophysical signature of copper mineralization within volcanic rocks of the Hilário Formation, situated in the Lavras do Sul and Caçapava do Sul regions of Rio Grande do Sul state, Brazil (Figure 9a). The analysis focused on understanding the subsurface distribution and geometry of mineralized bodies through 3D inversion of aeromagnetic data.

3.2.2. Geological Settings

The geological setting and lithologies are situated in the southern portion of the Mantiqueira Province [38,39], commonly referred to as the Southern Rio Grande Shield (SRGS). The SRGS encompasses geological units that form part of the Dom Feliciano Belt, which was established during the Neoproterozoic. It is represented by the São Gabriel and Tijucas terranes, as well as the Pelotas Batholith. Paleoproterozoic units are also present, including the Taquarembó and Tijucas terranes [37] (Figure 9b).

São Gabriel Terrane, which hosts the study area, is notable as an accretionary prism, characterized by petrotectonic associations typical of passive margin and back-arc environments. It includes ophiolites, volcanic–sedimentary sequences, and plutonic magmatic arcs [37]. Zircon ages, determined using TIMS and SHRIMP methods, range from 735 to 680 Ma [40,41,42].

Additionally, the region hosts a suite of sedimentary and volcanogenic units formed in elongated tectonic basins, dated to the late Neoproterozoic and early Paleozoic. These basin deposits consist of thick successions of conglomerates, sandstones, and shales, developed in continental, coastal, and marine environments. They are frequently associated with volcanogenic rocks of alkaline affinity. These sequences, known as the Camaquã Basin, overlie the Precambrian basement and are predominantly composed of siliciclastic and volcanogenic rocks [39,43].

3.2.3. The Seival Copper Mining Area

The mining area is located in a region that gained prominence in the early 20th century due to the Seival Mine, which played a significant role in copper production in southern Brazil. Currently, the deposit is considered economically exhausted due to the high concentration of oxidized copper. The region also contains records of several formerly exploited, now inactive mines, as well as mineral occurrences known as Quero-Quero, Lagoa do Jacaré, and Vila do Torrão [44] (Figure 10).

Reischel [46] and Lopes [47] provided detailed descriptions of the inactive mines:

• Barita Mine: The largest of the mines, consisting of a lenticular body oriented NE. It contains chalcocite associated with fault zones, with estimated reserves of approximately 64,000 tons and a copper grade of 1.7%;
• João Dahne Mine: A small body also oriented NE, but with low copper grades;
• Morcego Mine: Characterized by disseminated sulfides (chalcocite and malachite), with NS and NE orientations, located near fracture zones in andesite;
• Meio Mine: Displays irregular concentrations of malachite, tectonically controlled by a fracture pattern in andesitic tuff, with NS–NE orientations and a steep SW dip;
• Cruzeta Mine: Defined by the presence of chalcocite in veins along fractures in porphyritic andesite, with NW orientations and subvertical dip;
• Alcides Mine: Composed of disseminated sulfides (chalcocite) within volcanic breccias, with barite as the gangue mineral.

3.2.4. Airborne Data—Main Characteristics

The aerogeophysical database used in this study, compiled by [36], consists of aeromagnetic and gamma-ray spectrometric data acquired during the Primavera Project, carried out in 2010. Data acquisition and initial processing were performed by LASA Prospecções S.A. The project area is located in the southernmost region of Brazil, covering the municipalities of Caçapava do Sul and Lavras do Sul, with a total survey coverage of approximately 12,681 km². The technical parameters applied in the aerogeophysical survey were as follows:

• Flight lines direction: N14W;
• Flight lines spacing: 200 m;
• Flight average altitude: 100 m.

The RTP total-field anomaly was interpolated to a rectangular grid with spacing $\Delta x=\Delta y=50$ m by a least-squares interpolation using finite elements with B-splines [48] and is shown in Figure 11, where the copper mines and occurrences are numbered according to

Table 2. Cooper mines and occurrences in the Seival mining area.

Number	Mine/Occurrence
1	Quero-quero
2	Barita
3	João Dahne
4	Meio
5	Seival Geral
6	Morcego
7	Cruzeta
8	Lagoa do Jacaré
9	Vila do Torrão
10	Alcides

.

3.2.5. Phase Transformation of the Analytic Signal

The original TA and THG maps of the aeromagnetic data are shown in Figure 12a,c, whereas the corresponding reconstructed maps using $\alpha =4$ are shown in Figure 12b,d. Figure 13a,b show the rTHG maps with $\alpha =2, 8$, as in the synthetic data, while Figure 13c,d show the EHD and MHGA maps (Equations (8) and (9), respectively).

The MHGA map (Figure 13d) is the sharpest one, but the anomalies have low lateral continuity and there are some high-frequency anomalies. The EHD map (Figure 13c) shows higher continuity at the peaks but is more affected by noise. On the other hand, the rTHG maps (Figure 12d and Figure 13c,d) were not significantly affected by high-frequency content and showed high continuity, making the boundaries highlighted by standard THG clearer (Figure 12c).

Regarding the copper mineralization sites, the transformed tilt angle map (Figure 12b) was able to delimit most of them in the same positive anomalous area, whereas the remaining ones are in adjacent positive anomalies. Furthermore, they are concentrated at the edges of the anomalies as shown in the rTHG with $\alpha =8$ (Figure 13b), where the lateral continuity of the edges is a great aid to visualization. The edges of the high-order filters (Figure 13c,d) are also correlated with the mineralization occurrences. In particular, Cruzeta Mine (number 7) is more clearly linked to an edge than in the rTHG map of Figure 13b.

3.2.6. Analysis and Comparison with Geological and Structural Data

The results obtained from the phase transformation of the analytic signal (Section 3.2.5) were integrated with direct field data, including geological information and the main structures described in [36]. The filters rTHG with $\alpha =4$ (Figure 12d), rTHG with $\alpha$ = 8 (Figure 13b), and MHGA (Figure 13d) enabled the comparison of the interpreted discontinuities and alignments. Furthermore, the method provides a significant contribution to the analysis and interpretation, enhancing the detection of anomalies’ limits when compared with the TA and ASA maps reported in [36] (Figure 14a).

When comparing the Euler results with the filters proposed herein, it becomes evident that the main occurrences are concentrated along the borders of the strong anomalies, corroborating the genetic mineralization model, which proposes that the main occurrences develop in hydrothermal systems where magnetization is subtle due to alteration processes (Figure 14b). This interpretation is supported by the results overlaid on the 3D magnetization vector inversion (MVI), which was applied using the same airborne database. Figure 14c presents three main generated profiles (A–B, C–D, and E–F), where limits of the reconstructed TA (Figure 12b) are clearly coincident with the limits observed in the MVI model. This approach assists in delineating structural boundaries at depth, as well as outlining mines 2, 4, 5, 6, and 9, proving effective when integrated with 3D models and Euler solutions for the detailed characterization of subsurface features.

The phase transformation of the analytic signal in an area with a recognized presence of mineralization (inactive mines), where structural controls play a major role in the distribution of mineral occurrences, supports previous research in the region that emphasized the importance of NE-trending faults as primary controls on copper mineralization and hydrothermal alteration. Moreover, this method provided a significant tool for analysis and interpretation, enhancing the delineation of anomaly magnitudes and limits.

4. Conclusions

We adapted a non-linear transformation of the phase of the analytic signal, which was originally proposed to seismic data, to potential-field data, and our results indicate that reconstructed THG obtained by the proposed transformation is a qualitative enhancement filter that is able to highlight anomaly boundaries, without using high-order derivatives. For field data, the known copper mineralization areas were highly correlated with the edges of rTHG.

Even though the rTHG can have very sharp boundaries, depending on the phase parameter $\alpha$, we must keep in mind that it remains a first-order method; thus, higher-order approaches can delineate the edges more accurately, as shown in the synthetic results. The actual advantage of rTHG is its lower sensitivity to noise; thus, it constitutes an attractive choice when data smoothing or stabilized differentiation cannot be employed.