‘Hilbert Transform’ Remains a Valuable Tool

Geological interpretation of seismic data is commonly done by analyzing patterns of seismic amplitude, phase and frequency in map and section views across a prospect area. Although many seismic attributes have been utilized to emphasize geologic targets and to define critical rock and fluid properties, these three simple attributes – amplitude, phase and frequency – remain the mainstay of geological interpretation of seismic data.

Any procedure that extracts and displays any of these seismic parameters in a convenient and understandable manner is an invaluable interpretation tool.

A little more than 30 years ago, M.T. Taner and Robert E. Sheriff introduced the concept of using the Hilbert transform to calculate seismic amplitude, phase and frequency instantaneously – meaning a value for each parameter is calculated at each time sample of a seismic trace.

That Hilbert transform approach now forms the basis by which almost all amplitude, phase and frequency attributes are calculated by today’s seismic interpretation software

The Complex Seismic Trace

The action of the Hilbert transform is to convert a seismic trace x(t) into what first appears to be a mysterious complex seismic trace z(t) as shown on figure 1.

In this context, the term “complex” is used in its mathematical sense, meaning it refers to a number that has a real part and an imaginary part. The term does not imply that the data are difficult to understand.

This complex trace consists of the real seismic trace x(t) and an imaginary seismic trace y(t) that is the Hilbert transform of x(t).

On figure 1 these two traces are shown in a three-dimensional data space (x, y, t), where t is seismic time, x is the real-data plane, and y is the imaginary-data plane. The actual seismic trace is confined to the real-data plane; the Hilbert transform trace is restricted to the imaginary-data plane.

These two traces combine to form a complex trace z(t), which appears as a helix that spirals around the time axis.

The projection of complex trace z(t) onto the real plane is the actual seismic trace x(t); the projection of z(t) onto the imaginary plane is the Hilbert transform trace y(t).

At any coordinate on the time axis, a vector a(t) can be calculated that extends perpendicularly away from the time axis to intercept the helical complex trace z(t) as shown on figure 2. The length of this vector is the amplitude of the complex trace at that particular instant in time – hence the term “instantaneous amplitude.”

The amplitude value is calculated using the equation for a(t) shown on the figure.

The orientation angle Φ(t) that defines where vector a(t) is pointing (figure 2) is defined as the seismic phase at time coordinate t – hence the term “instantaneous phase.” Numerically, the phase angle is calculated using the middle equation listed on figure 2.

As time progresses, vector a(t) moves down the time axis, constantly rotating about the time axis as it maintains contact with the spiraling helical trace z(t).

Mathematically, frequency can be defined as the rate of change of phase. This fundamental definition allows instantaneous frequency ω(t) to be calculated from the time derivative of the phase function as shown by the bottom equation on figure 2.


The calculation of these three interpretation attributes – amplitude, phase and frequency – are illustrated on figures 3 and 4. Application of the three equations listed on figure 2 yields first the instantaneous amplitude for one seismic trace x1(t) (figure 3), and then instantaneous phase and frequency are shown on figure 4 for a different seismic trace x2(t).

Note that the instantaneous frequency function is occasionally negative – a concept that has great interpretation value, as has been discussed in a previous article (April 2008 Geophysical Corner).

For those of you who click on a menu choice to create a seismic attribute as you interpret seismic data, you now see what goes on behind the screen to create that attribute.

Comments (0)

 

Geophysical Corner

The Geophysical Corner is a regular column in the EXPLORER that features geophysical case studies, techniques and application to the petroleum industry.

VIEW COLUMN ARCHIVES

Image Gallery

Next month – Part two

In the two-part series on the role of reference surfaces in seismic interpretation: Defining reflection events and their polarities.

See Also: Book

Desktop /Portals/0/images/_site/AAPG-newlogo-vertical-morepadding.jpg?width=50&h=50&mode=crop&anchor=middlecenter&quality=90amp;encoder=freeimage&progressive=true 4479 Book

See Also: Bulletin Article

This article concentrates on the question, Which parameters govern recovery factor (RF) behavior in channelized turbidite reservoirs? The objective is to provide guidelines for the static and dynamic modeling of coarse reservoir-scale models by providing a ranking of the investigated geologic and reservoir engineering parameters based on their relative impact on RF. Once high-importance (H) parameters are understood, then one can incorporate them into static and dynamic models by placing them explicitly into the geologic model. Alternatively, one can choose to represent their effects using effective properties (e.g., pseudorelative permeabilities). More than 1700 flow simulations were performed on geologically realistic three-dimensional sector models at outcrop-scale resolution. Waterflooding, gas injection, and depletion scenarios were simulated for each geologic realization. Geologic and reservoir engineering parameters are grouped based on their impact on RF into H, intermediate-importance (M), and low-importance (L) categories. The results show that, in turbidite channel reservoirs, dynamic performance is governed by architectural parameters such as channel width, net-to-gross, and degree of amalgamation, and parameters that describe the distribution of shale drapes, particularly along the base of channel elements. The conclusions of our study are restricted to light oils and relatively high-permeability channelized turbidite reservoirs. The knowledge developed in our extensive simulation study enables the development of a geologically consistent and efficient dynamic modeling approach. We briefly describe a methodology for generating effective properties at multiple geologic scales, incorporating the effect of channel architecture and reservoir connectivity into fast simulation models.
Desktop /Portals/0/PackFlashItemImages/WebReady/the-impact-of-fine-scale-turbidite-channel-architecture.jpg?width=50&h=50&mode=crop&anchor=middlecenter&quality=90amp;encoder=freeimage&progressive=true 3664 Bulletin Article

See Also: CD DVD

Desktop /Portals/0/images/_site/AAPG-newlogo-vertical-morepadding.jpg?width=50&h=50&mode=crop&anchor=middlecenter&quality=90amp;encoder=freeimage&progressive=true 4398 CD-DVD

See Also: Learn! Blog

Listen to Dr. Ronald Nelson as he shares his knowledge and insights on a practical approach to defining reservoir fluid and pressure related natural fracture generation and fracture property alteration in conventional and unconventional reservoirs.

Desktop /Portals/0/PackFlashItemImages/WebReady/ec-fec-rock-fluid-interactions-and-natural-fracture-development-and-alteration.jpg?width=50&h=50&mode=crop&anchor=middlecenter&quality=90amp;encoder=freeimage&progressive=true 12283 Learn! Blog

See Also: Workshop

This workshop brings together new technologies, practices, and procedures that can be applied to new and mature fields in order to revitalize them and increase / optimize recovery. The presentations will focus on case studies, research findings, and field applications for new and existing plays, including Texas, Gulf Coast region, and Latin America.
Desktop /Portals/0/PackFlashItemImages/WebReady/gtw-revitalizing-reservoicrs-texas-gulf-coast-latin-12dec-2015-hero.jpg?width=50&h=50&mode=crop&anchor=middlecenter&quality=90amp;encoder=freeimage&progressive=true 21310 Workshop