Spectral Decomposition Resolution: Do We Really Need Frequency Resolution?

The Transformation: Time to Frequency Domain

Fourier Transformation (FT) is arguably the most widely recognized and utilized transform, particularly within the domains of electrical engineering and signal processing. Its popularity stems from its ability to convert signals between time and frequency domains, providing a powerful tool for analyzing and interpreting various types of data. However, the landscape of transformation techniques is vast and diverse, encompassing numerous other transforms that are regularly employed by engineers and mathematicians. some of the transforms are:

• Hilbert transform
• Short-Time Fourier transform (STFT)
• Radon Transform,
• Wavelet transform, (WT)
• etc.

Each of these transformation techniques has its own unique area of application, bringing with it distinct advantages and disadvantages that make it suitable for specific types of data and analysis needs.

In most practical applications, signals start out as time-domain signals, meaning they show how a signal changes over time. While this gives us a picture of the signal’s behavior, it often doesn’t reveal the most important details. These crucial insights are usually hidden in the signal’s frequency content. By converting the signal from the time domain to the frequency domain, we can uncover these hidden features. This transformation helps us identify patterns, periodicities, and other significant details that aren't obvious in the time-domain form. Understanding and interpreting the frequency content of signals is therefore a key part of effective signal processing and analysis.

The Thin Bed Interference

The amplitude spectrum of a thin-bed layer shows a response that includes the reflectivity overprint. By examining this spectrum, we can estimate the actual temporal thickness of the layer. This is done by analyzing the interference patterns within the spectrum, which are related to the layer's reflectivity. These patterns give us clues about the thickness and the elastic properties of the rock. Essentially, when seismic waves interact with the rock layers, they create these interference patterns. By studying these patterns, geophysicists can learn about the physical characteristics of the subsurface. This ability to determine the thickness and properties of rock layers is crucial for applications like resource exploration and geological research, making the analysis of amplitude spectra a valuable tool in geophysics.

The tuning effect is closely linked to the thickness of a layer, and it becomes challenging to resolve layers thinner than 10 milliseconds. When the reflections from the top and bottom of the layer interfere destructively, the signal gets lost, creating what's known as an amplitude notch. In the amplitude spectrum, the interference caused by thin beds shows up as V-shaped amplitude notches. These notches appear at frequencies that are equal to the inverse of the layer's time thickness (1/time-thickness). Essentially, this means that the thinner the layer, the higher the frequency at which these notches occur. This phenomenon is crucial for understanding and interpreting the data, as it helps identify and characterize thin layers that might otherwise be undetectable.

The Time (Vertical) and Frequency Resolution

In spectral decomposition, resolution can be divided into two types: time (vertical) resolution and frequency resolution. On one hand, we aim for better separation in the time (vertical) domain to distinguish different layers more clearly. On the other hand, we also want to see thin bed interference more distinctly. The greater the amplitude difference in each color channel, the more contrast we get in the RGB blending volume, making it easier to identify and analyze different features.

It's important to note that higher frequency resolution, particularly with methods like Basis Pursuit, provides better contrast and feature delineation. This enhanced contrast allows us to see the finer details and variations within the data, which is crucial for accurate analysis. This is why achieving higher frequency resolution is essential; it helps us to get a clearer and more detailed picture, ultimately leading to better interpretation and understanding of the subsurface features.

Frequency Selection For RGB Blending: Highlighting The Thin Bed Interference

The frequency for each RGB blending color channel can be chosen using a Spectral Decomposition Analysis tool. This selection is based on the differences in the spectrum at the same frequency for each color channel. Essentially, the greater the amplitude difference in one color channel, the more distinct the color variation in the RGB blending volume will be.

However, real data is often complex, making it challenging to analyze the spectrum directly. To address this, frequencies for each RGB channel can be more effectively selected by visualizing and analyzing the data in slice or map views. This approach allows for a clearer understanding of the spectral differences and helps in making more accurate frequency selections for each color channel.

Some features might be clearer in one frequency, and some others might be clearer in other frequency

Since human eye has limitation on separating certain color (i.e. features), discontinuity / edge detection on each color channel volume may delineate the particular feature or lithology better compared to the conventional RGB blending.

Horizon Edge Stacking can be used to identify edges or features in the input amplitude volume of each color channel.

The Horizon Edge Stacking results then used as RGB blending input to get better visual separation of the features of lithology.

Changing the frequency combination could help the interpretation.

Note the better feature delineation on RGB Blending using The Horizon Edge Stacking results.

stay tuned for more articles from me.

p.s. HampsonRussell and JasonWorkbench are trademark of GeoSoftware