**3. Second vertical derivative**

Many researchers [4, 8, 9] have given a thorough picture of the second derivative method of interpretation of gravity and have shown how this method is in effect for the detection of small irregularities in gravity anomalies and thus useful for supposing minute underground mass distribution that cannot be overlooked by the ordinary method. It is fascinating to note that the method is justified only on data of high accuracy [10].

The SVD of the vertical component of gravity, gz, can be calculated in the spatial domain from the horizontal gradients by using Laplace's equation [9].

$$\frac{\partial^2 \mathbf{g}\_z}{\partial \mathbf{z}^2} = -\left(\frac{\partial^2 \mathbf{g}\_z}{\partial \mathbf{x}^2} + \frac{\partial^2 \mathbf{g}\_z}{\partial \mathbf{y}^2}\right) \tag{11}$$

For an anomaly extended along the y-axis, the SVD can be approximated by the second horizontal derivative of the gravity data along the x-axis in Eq. (13).

$$\frac{\partial^2 \mathbf{g}\_z}{\partial \mathbf{z}^2} \approx -\frac{\partial^2 \mathbf{g}\_z}{\partial \mathbf{x}^2} \tag{12}$$

In the wave number domain or Fourier domain, the SVD is usually calculated by using the following Eq. (14) [11].

$$\frac{\partial^2 \mathbf{g}\_z}{\partial \mathbf{z}^2} = \mathbf{F}^{-1} \left( |\mathbf{k}|^2 \mathbf{G}\_z \right) \text{ with } |\mathbf{k}|^2 = \mathbf{k}\_\mathbf{x}^2 + \mathbf{k}\_\mathbf{y}^2 \tag{13}$$

The second vertical derivatives are the measure of curvature where large curvature is connected to shallow anomalies. It is frequently used to enhance localized subsurface features, that is, weak anomalies due to the sources that are shallow and limited in-depth and lateral extent [10].
