The Volume Derivative process calculates the first or second derivative of a trace using the finite difference method.
Create a Volume Derivative process
- In the Control Panel, open the Process tab.
- At the tab header, click the Add icon, and select New Process.
- Scroll down and double-click on Volume Derivative.
- Type a name for the process, and click OK.
Calculate the derivative of a trace
- In the Details Panel, select a Volume.
Note: There are several options of derivative method to choose from with different trade-offs. Below is a comparison of their frequency responses for a dt of 4ms.
Derivative method: Select a derivative method to use. Each of the derivative methods is described below:
- Fourier: Computes the derivative in the frequency domain to any order. This is the most accurate method available, however, the operator is very long and may appear cosmetically ringy, even though it is correct.
Butterworth: Computes the derivative in the frequency domain to any order, just like the Fourier method, but with an additional low-pass Butterworth filter. The low-pass filter considerably shortens the operator and gives control of the high-frequency attenuation. This is the default method.
- Low-pass 3dB frequency/wavenumber: 3dB down frequency/wavenumber of the low-pass filter.
- Low-pass slope (db/Oct): Slope of the low-pass filter.
Taylor: Designs and applies a derivative filter of a given length and order based on the Taylor series. Longer filters provide a broader bandwidth output, but the filter always rolls off to zero at Nyquist. This method helps avoid ringing in the output at the loss of some higher frequencies.
- Filter length: The length of the filter. Must be odd. A length of 3 results in the "leapfrog" method.
Lanczos: Designs and applies the first derivative of a Lanczos interpolator. The Lanczos filter can be short or long and has the ability to control the rejection of higher frequencies. The disadvantage is that it has ripples around the Maximum frequency and has the potential to alias.
- Maximum frequency/wavenumber: The low-pass frequency/wavenumber.
- Filter length (cycles): The length of the filter expressed in cycles of the Maximum frequency.
- Leapfrog (legacy): A simple first or second-order leapfrog derivative. This method exists for compatibility with older versions of Insight. It has decreasing accuracy with increasing frequency above about 16% of Nyquist. Unlike the other methods, it tolerates NaNs in the middle of traces and halves the sample interval for TWT volumes.
Order: The order of the derivative. Orders above 1st are limited to certain methods.
- First derivative: Calculate the first derivative of the trace.
- Second derivative: Calculate the second derivative of the trace.
- Adjoint: Whether to apply the adjoint of the operator.
As a result of this process, a new volume is available in the Volume tab.
Note: The derivatives are calculated using a central finite difference method.
- The first and last values of the first derivative are calculated using the forward and backward finite difference respectively.
- The first and last values of the second derivative will be null (NaN).