Depth Estimation Using the Tilt Angle of Gravity Field due to the Semi-Infinite Vertical Cylindrical Source

Tilt angle filter is an interpretation method that is used to determine the source borders locations from potential fields data. Moreover, the tilt angle is applied for estimation of the anomaly source depth, such as contact-depth method and tilt-depth method.


Introduction
In most methods of potential fields modelling, the geological structures configuration is exhibited by simple geometrically shaped models. The models may not be geologically realistic, but usually approximate similarity to the gravity anomaly causative masses is sufficient for quantitative and qualitative interpretations and modelling. The geometric shapes of relevant some important geologic structures are similar to vertical cylinder such as igneous plugs, salt domes, and sink holes. The land and airborne gravimetry method are used worldwide for detection of these structures. Determination of the depth of the gravity anomaly sources is one of the main aims of quantitative interpretation of gravity field map. Many methods have been presented for calculating the depth. An excellent review is given in Nettleton [1]. Euler deconvolution gives initial estimates of source location and depth [2,3]. Abdelrahman and El-Araby [4], Abdelrahman et al. [5], and Essa [6] indicated that the window curves method can be used to estimate the depth and shape of the buried structure from gravity data. Abdelrahman et al. [7] and Abdelrahman and Abo-Ezz [8] developed a least-squares variance analysis method for shape and depth determination from gravity data.
The tilt angle has been used as the basis for a variety of methods for edge enhancement of potential field anomalies, such as Cooper [9,10], Ferreira et al. [11], and Verduzco et al. [12]. Salem et al. [13] exhibited a contact-depth estimation approach based on the tilt angle of the magnetic data which was called the tilt-depth method. Cooper [14] proposed the contact-depth method based on the tilt angle for the automatic determination of the location, depth, and dip of contacts from aeromagnetic data. In this paper, an approach for the source depth determination based on the tilt angle of the 2D gravity data and its distance from origin, that is, the coordinate of the anomaly center on the profile, is presented. For shortness, this method is named the tilt-distance-depth (TDD) approach.

Methodology
Verduzco et al. [12] proposed the tilt angle filter. This filter is defined as and are the tilt angle and gravity anomaly, respectively. The gravity anomaly expression produced by a semi-infinite vertical cylinder is given in Abdelrahman et al. [15] as In (2), is the depth, is the position coordinate, is the density contrast, is the universal gravitational constant, and is the radius (Figure 1). For semi-infinite vertical cylinder model, depth to the top ( ) is much smaller than the source length (ℎ); that is, ≪ ℎ ( Figure 1).
The first horizontal and vertical derivatives of are then Inserting (3) into (1), after simplification, yields and consequently = ⋅ tan .
Here, = Δ = − 0 is the distance of the origin of the model from the gravity points ( Figure 1). The coordinate of 0 is correspondent to the coordinate of the maximum gravity value irregardless of its positive or negative sign. The -axis is positive downward.  the TDD approach to noise is investigated in Figure 3. The gravity data in Figure 3(a) was contaminated with random noise with standard deviation of 0.0001 mGal and average 0.0005 mGal (Figure 3(a)). Figure 3(b) shows the tilt angle curve for gravity data in Figure 3  was added to the gravity data set ( Figure 5(a)). Figures 5(b), 5(c), and 5(d) show the tilt angle of the gravity data in Figure 4(a) and the depth estimates. The contiguous solutions with the same depth obtained from the TDD approach using the gravity data corrupted by random-valued noise indicate the exact depth value, as shown in Figures 3(d) and 5(d).

Application to Synthetic Data
The depth estimates of the synthetic models, with and without random noise, for both methods have been given in Table 1.

Field Example
The study region is located at the northwest of Iran. Miocene units in this region include sequences of marl, salt, and chalk. The salt domes in this area are the result of the upward movement of the Neogene evaporative materials. These salt domes are positioned at the shallow depth and have mostly high alloy.   Figure 6 shows the residual gravity anomaly from the exploratory region. The salt dome with negative anomaly is recognizable on the gravity map (SD on the Figure 6). For eliminating the short wavelengths, such as shallow and small anomaly sources and noise, the 10 m upward continuation filter was used (Figure 7). Figures 8(a) and 9(a) show the gravity field variations along profiles A and B, respectively. Figures  8(b) and 8(c) display the tilt angle of the gravity data and depth estimates from the TDD method along profile A and Figures 9(b) and 9(c) exhibit the tilt angle of the gravity data and depth estimates from the TDD method along profile B.
Solutions from the TDD and Euler deconvolution method are marked as red + and blue × symbols. The scattered form of the solutions has been exposed in Figures 8(d) and  9(d).
The clustered and proximate solutions of the TDD procedure have computed the average depth of 25 m for the salt dome. It should be noted that the upward height has been subtracted from the depth estimates.  Moreover, the Modelvision software has been used for the inverse modelling of salt dome for estimating the upper depth of it. A profile of initial model (green rectangular) is depicted in Figure 10. As it can be seen that the graphs show the gravity field of the cylinder model (blue line) and variations of gravity caused by salt dome (black line), which has been measured through profile C shown in Figure 11. The location of cross section of initial model (green circle) has been presented in Figure 11.
In this model, the value of average density of the salt dome is assumed to be 2.2 gr/cm 3 . The final models are displayed through Figures 12 and 13. The estimated depths of upper and lower surface of cylinder model are 28 m and 58 m, respectively. The estimations for the depth of the top of the salt dome by both approaches, that is, the proposed method TDD and inverse modelling, are almost equal. Table 2 summarizes the average depth estimate results of the upper surface of the salt dome using the TDD and  Euler deconvolution methods and inverse modelling by the Modelvision software.

Conclusions
In this paper, a new depth estimation method based on the tilt angle of the gravity data has been presented. In comparison with the diffused solutions of the Euler deconvolution method which show a lower accuracy, the responses of the TDD method have high precision and with increasing the noise, the solutions remain stable. The results demonstrate the efficiency and usefulness of explained procedure. The results of the inverse modelling confirm the acceptable performance the TDD method.