Static Anomalies and Energy Partitioning Environmental and Exploration Geophysics II Static Anomalies and Energy Partitioning tom.h.wilson wilson@geo.wvu.edu Department of Geology and Geography West Virginia University Morgantown, WV

Velocity Analysis Problem Original due dates - and pb 4.8 Due Today, Oct. 24th Due Tuesday, Oct. 29th Velocity Analysis Problem

Recall Energy partitioning

Geophone output is often designed to be proportional to pressure, particle velocity, acceleration or displacement. Land geophone output is typically proportional to particle velocity, while marine geophones record pressure variations.

V Interval Velocity Particle Velocity v

Normal Incidence Raypaths Pi PT PR Boundary Conditions

We can rewrite boundary condition 1 as The subscript P indicates that pressure variations are being considered in this case

From the wave equation, we have that This allows us to rewrite boundary condition II in terms of the pressures, as - By convention, up is negative, thus

Our two boundary conditions become which implies As a matrix equation, we have

Part II Geophone output is often designed to be proportional to pressure, particle velocity, acceleration or displacement. Land geophone output is typically proportional to particle velocity, while marine geophones record pressure variations.

V Interval Velocity Particle Velocity v

Note that Pv =Vv2 Thus Ev2

We have, as expected, a decrease of energy across the interface We have, as expected, a decrease of energy across the interface. Energy is conserved!

Compute and plot two-way interval transit times, two-way total reflection time, layer impedance and boundary reflection coefficients

Density, velocity and impedance plots are usually represented in step-plot form. The values as listed are constant through an interval and marked by abrupt discontinuity across layer boundaries.

Reflection coefficients exist only at boundaries across which velocity and density change, hence their value is everywhere 0 except at these boundaries. Stick Plot

Simplified representation of the source disturbance Subsurface model Simplified representation of the source disturbance

Follow the wavefront through the subsurface and consider how its amplitude changes as a function only of energy partitioning. A. What is the amplitude of the disturbance at point A? B. At point B we have transmission through the interface separating media 1 and 2. At C? We consider only transmission and reflection losses. Geometrical divergence and absorption losses are ignored. Hence PA = 1psi. - hence the amplitude of the wavefront at B is Tp 12 PA.

At C? - At D? -

Consider for a moment- the general n-layer case.

…. Solve for the Ps and vs and then plot

The plot portrays the amplitude of the wavelet at subsurface points A, B, C and D. Input wavelet Provide a general representation of wavelet amplitudes measured at points A - D. Do for both the pressure and velocity measurements

Total loss - incorporating divergence, absorption and reflection/transmission effects. We have considered the above factors individually. All of them act to attenuate seismic waves as they propagate through the earth. Recall that divergence and absorption losses were combined into the following equation Each mechanism acts as a factor that scales the amplitude of the propagating wavefield. So the net effect on amplitude determined by taking the product of all effects on source amplitude AS.

Energy partitioning is a step-like function Energy partitioning is a step-like function. Wave amplitudes will take a jump to higher or lower amplitude across individual interfaces, however, we can consider the effect of transmission through a series of layers having various average values of reflection and transmission coefficient as shown below. Recall that on the decibel scale the relationship between two amplitudes is expressed as where A is in decibels

Total amplitude decay at distance r If average reflection coefficient is not too high (for example 0.05 or 0.1) then the effect is relatively constant over a large range of depths and we can represent transmission reductions by a single scale factor - say T. Total amplitude decay at distance r

These amplitude effects are non-geological in a sense These amplitude effects are non-geological in a sense. Geologists are interested to have accurate information about the reflection coefficients - not only their position, but their value. The above equation indicates that the amplitude of a reflection from a particular reflector will equal The geologist would like to have

ACCH Note amplitude/stratigraphic relationships

Accurate portrayal of reflection coefficients is important in stratigraphic interpretations of seismic data. 10,000 18,500 14,000 19,500 18,500 16,500 14,500 21,000

? ? ? ? “True Amplitude” … with some computer glitches Once again note the amplitude relationships

This seismic display has been “gain corrected” Note that some of the lithology dependant amplitude differences have disappeared.

Note that the amplitudes in the gain corrected trace at right do not accurately portray relative differences in the value of reflection coefficients

Gain incorporates amplitude averaging over short time windows Truer amplitude display - amplitude averaging is undertaken over longer time windows Gain incorporates amplitude averaging over short time windows From Ylmez

Also due next Friday, Nov. 8th Additional Homework - The basic synthetics exercises handed out today will be due next Friday. Look over them and bring questions to class this Thursday. Problem 4.14, Chapter 4 Also due next Friday, Nov. 8th

Background reading Read over the paper I handed out to you last Thursday by Sheriff. A proper understanding of resolution issues is critical to stratigraphic interpretations and also to structural interpretations where the identification of subtle structures, such as faults with small offset may be important. We’ll be studying resolution in forthcoming computer labs and relating resolution limits to your exploration data set.