Energy Loss and Partitioning Environmental and Exploration Geophysics II Energy Loss and Partitioning tom.h.wilson tom.wilson@mail.wvu.edu Department of Geology and Geography West Virginia University Morgantown, WV Tom Wilson, Department of Geology and Geography

Objectives for the day Review time-distance plots Discuss energy loss through divergence and absorption mechanisms Present basic definition of reflection coefficient and how it varies with offset or incidence angle Basic definition of transmission coefficient Tom Wilson, Department of Geology and Geography

Can you name the events? Miller et al. 1995 What events can you identify in this shot record? Miller et al. 1995 Tom Wilson, Department of Geology and Geography

Ground roll = noise to the exploration geophysicist Tom Wilson, Department of Geology and Geography

We start off with these noisy looking field records and with some effort get a more geological look at the subsurface Tom Wilson, Department of Geology and Geography

Diffraction (point source) Events Engineers may want to think in terms of ground penetrating radar - Drums will be diffractors, perhaps pipelines and underground storage tanks Geologists may want to think of diffractors as voids or narrow meanedring channels that the seismic line might cross. They could also be associated with the edges of layers that are truncated by faults Draw Diagram Tom Wilson, Department of Geology and Geography

Diffractions, like reflections are hyperbolic in a time-distance plot Diffractions, like reflections are hyperbolic in a time-distance plot. They are usually symmetrical about the apex. Diffractions usually arise from point-like discontinuities and edges ( for example the truncated edges of stratigraphic horizons across a fault). Tom Wilson, Department of Geology and Geography

Note that the apex of the diffraction event is not located at the source location (X=0). The apex is located over the discontinuity. Tom Wilson, Department of Geology and Geography

Use the image point to help you get reflection times Ray Trace Exercise I A direct path Use the image point to help you get reflection times How do you figure out Xmin & tmin for the critical refraction? What is the critical refraction time at longer offsets? Tom Wilson, Department of Geology and Geography

In ray-trace exercises I-III 1) label all plotted curves, 2) label all relevant points and 3) Describe the relationships that you’ve graphed. Tom Wilson, Department of Geology and Geography

Do Exercise II and III using Excel. Ray Trace Exercise II What to you think the influence of higher velocity will be on the moveout? What’s the t0 for these two cases? Do Exercise II and III using Excel. In exercise II comment on the origins of the differences in the two reflection hyperbola? What is their relationship to corresponding direct arrivals. Tom Wilson, Department of Geology and Geography

Although the time intercept remains the same - how do the shapes of the reflection hyperbola differ in these two cases? Tom Wilson, Department of Geology and Geography

Ray Trace Exercise III Hands on Work your way along each path For Exercise III, explain the differences observed in the arrival times of the reflection and diffraction observed in the shot record. Why does the diffraction event drop below the reflection? Tom Wilson, Department of Geology and Geography

In Exercises I-III Your plots should be consistent with expected relationships discussed in class Tom Wilson, Department of Geology and Geography

Questions? Tom Wilson, Department of Geology and Geography

Review - Questions? Tom Wilson, Department of Geology and Geography

First arrival picks and velocity 170 feet offset Can you pick out the direct arrival? Can you pick out the refraction arrivals? How many critical refractions are there? How would you determine the refraction velocity? How would you determine the air wave or direct arrival velocity? 0.15 seconds Tom Wilson, Department of Geology and Geography

Velocities > VA, VB VC, Vairwave Tom Wilson, Department of Geology and Geography

Section 2.3 What happens to the seismic energy generated by the source as it propagates through the subsurface? Basic Concepts- Energy - The ability to do work. It comes in two forms - potential and kinetic Work expended (W) equals the applied force times the distance over which an object is moved. Power is the rate at which work is performed. As a mechanical disturbance or wavefield propagates through the subsurface it moves tiny particles back and forth along it path. Particle displacements are continually changing. Hence, it is more appropriate for us to consider the power, or the rate at which energy is being consumed at any one point along the propagating wavefront. Tom Wilson, Department of Geology and Geography

The rate of change of work in unit time. One horsepower equals 550 foot-pound force /second 745 Newton meters/second 1 watt is one joule per second This force is the force exerted by the seismic wave at specific points along the propagating wavefront. Tom Wilson, Department of Geology and Geography

and since force is pressure (p) x area (A), we have The power generated by the source is PS. This power is distributed over the total area of the wavefront A so that We are more interested to find out what is going on in a local or small part of the wavefield. * Note we are trying to quantify the effect of wavefront spreading at this point and are ignoring heat losses. Tom Wilson, Department of Geology and Geography

Over what surface area is the energy generated by the source distributed? Tom Wilson, Department of Geology and Geography

Area of a hemisphere is 2R2 - hence - This is the power per unit area being dissipated along the wavefront at a distance R from the source. (Recall p is pressure, v is the particle velocity, PS magnitude of the pressure disturbance generated at the source and R is the radius of the wavefront at any given time. Tom Wilson, Department of Geology and Geography

In the acoustic wave equation a quantity Z which is called the acoustic impedance. Z = V, where  is the density of the medium and V is the interval velocity or velocity of the seismic wave in that medium. Z is a fundamental quantity that describes reflective properties of the medium. We also find that the pressure exerted at a point along the wavefront equals Zv or Vv, where v is the particle velocity - the velocity that individual particles in the disturbed medium move back and forth about their equilibrium position. Tom Wilson, Department of Geology and Geography

Particle velocity versus interval velocity Tom Wilson, Department of Geology and Geography

Combining some of these ideas, we find that v, the particle displacements vary inversely with the distance traveled by the wavefield R. Tom Wilson, Department of Geology and Geography

We are interested in the particle velocity variation with distance since the response of the geophone is proportional to particle (or in this case) ground displacement. So we have basically characterized how the geophone response will vary as a function of distance from the source. The energy created by the source is distributed over an ever expanding wavefront, so that the amount of energy available at any one point continually decreases with distance traveled. This effect is referred to as spherical divergence. But in fact, the divergence is geometrical rather than spherical since the wavefront will be refracted along its path and its overall geometry at great distances will not be spherical in shape. Tom Wilson, Department of Geology and Geography

The effect can differ with wave type The effect can differ with wave type. For example, a refracted wave will be confined largely to a cylindrical volume as the energy spreads out in all directions along the interface between two intervals. z R The surface area along the leading edge of the wavefront is just 2Rz. Tom Wilson, Department of Geology and Geography

Hence (see earlier discussion for the hemisphere) the rate at which source energy is being expended (power) per unit area on the wavefront is Remember Zv2 is just Following similar lines of reasoning as before, we see that particle velocity Tom Wilson, Department of Geology and Geography

The dissipation of energy in the wavefront decreases much less rapidly with distance traveled than does the hemispherical wavefront. This effect is relevant to the propagation of waves in coal seams and other relatively low-velocity intervals where the waves are trapped or confined. This effect also helps answer the question of why whales are able to communicate over such large distances using trapped waves in the ocean SOFAR channel. Visit http://www.beyonddiscovery.org/content/view.page.asp?I=224 for some info on the SOFAR channel. Tom Wilson, Department of Geology and Geography

Sound Surveillance System Detected a 1 pound TNT explosion 2000 miles away (Bahmas to Africa). See also - http://www.beyonddiscovery.org/Includes/Dialogs/ExternalLink.asp?ID=3101&URL= Tom Wilson, Department of Geology and Geography

Divergence losses decrease particle velocity or the amplitude of a seismic wave recorded by the geophone - i.e. the amplitude of one of the wiggles observed in our seismic records. Unless we correct for this effect – turn up the volume – the signal will become too weak for us to hear. Tom Wilson, Department of Geology and Geography

Geophone Output Since amplitude (geophone response) is proportional to the square root of the pressure, we can rewrite our divergence expression as Thus the amplitude at distance r from the source (Ar) equals the amplitude at the source (AS) divided by the distance traveled (r). This is not the only process that acts to decrease the amplitude of the seismic wave. Tom Wilson, Department of Geology and Geography

How many of you remember how to solve such an equation? Absorption When we set a spring in motion, the spring oscillations gradually diminish over time and the weight will cease to move. In the same manner, we expect that as s seismic wave propagates through the subsurface, energy will be consumed through the process of friction and there will be conversion of mechanical energy to heat energy. We guess the following - there will be a certain loss of amplitude dA as the wave travels a distance dr and that loss will be proportional to the initial amplitude A. i.e. How many of you remember how to solve such an equation? Tom Wilson, Department of Geology and Geography

 is a constant referred to as the attenuation factor In order to solve for A as a function of distance traveled (r) we will have to integrate this expression - In the following discussion,let Tom Wilson, Department of Geology and Geography

Tom Wilson, Department of Geology and Geography

Mathematical Relationship Graphical Representation Tom Wilson, Department of Geology and Geography

 - the attenuation factor is also a function of additional terms -  is wavelength, and Q is the absorption constant 1/Q is the amount of energy dissipated in one wavelength () - that is the amount of mechanical energy lost to friction or heat. Tom Wilson, Department of Geology and Geography

 is also a function of interval velocity, period and frequency Tom Wilson, Department of Geology and Geography

 is just the reciprocal of the frequency so we can also write this relationship as Tom Wilson, Department of Geology and Geography

Smaller Q translates into higher energy loss or amplitude decay. Tom Wilson, Department of Geology and Geography

increase f and decrease A Run through brief discussion of db scale note that logato the x = x log a in this case a is e and log base 10 of e is approximately 0.43 and 20 (0.43) = 8.68 Higher frequencies are attenuated to a much greater degree than are lower frequencies. Tom Wilson, Department of Geology and Geography

When we combine divergence and absorption we get the following amplitude decay relationship The combined effect is rapid amplitude decay as the seismic wavefront propagates into the surrounding medium. We begin to appreciate the requirement for high source amplitude and good source-ground coupling to successfully image distant reflective intervals. Tom Wilson, Department of Geology and Geography

But we are not through - energy continues to be dissipated through partitioning - i.e. only some of the energy (or amplitude) incident on a reflecting surface will be reflected back to the surface, the rest of it continues downward is search of other reflectors. The fraction of the incident amplitude of the seismic waves that is reflected back to the surface from any given interface is defined by the reflection coefficient (R) across the boundary between layers of differing velocity and density. Tom Wilson, Department of Geology and Geography

Z1 and Z2 are the impedances of the bounding layers. Tom Wilson, Department of Geology and Geography

http://www. crewes. org/ResearchLinks/ExplorerPrograms/ReflEx/REcrewes http://www.crewes.org/ResearchLinks/ExplorerPrograms/ReflEx/REcrewes.htm Tom Wilson, Department of Geology and Geography

You can also have a look at AVO.xls linked on the class site Tom Wilson, Department of Geology and Geography

AVA response predicted for shallow strata in the Central Appalachians, Marshall Co., WV Tom Wilson, Department of Geology and Geography

The transmitted wave amplitude T is Tom Wilson, Department of Geology and Geography

Consider the following problem At a distance of 100 m from a source, the amplitude of a P-wave is 0.1000 mm, and at a distance of 150 m the amplitude diminishes to 0.0665 mm. What is the absorption coefficient of the rock through which the wave is traveling? (From Robinson and Coruh, 1988) Any Questions? Tom Wilson, Department of Geology and Geography

Assignments Please read through Chapter 3, pages 65 through 95; Chapter 4, pages 149 to 164. Hand in problems 2.3 and 2.6 (Today) Questions about problems 2.7 and 2.12? Hand those in on Wednesday. Start working Exercises I-III and bring questions to class on Wednesday. Any questions about the attenuation problem? We’ll probably end up discussing this some more in class on Wednesday. This will be due next Monday. Tom Wilson, Department of Geology and Geography