SURFACE WAVE DISPERSION PRACTICAL (Keith Priestley) This practical deals with surface waves, which are usually the largest amplitude arrivals on the seismogram. The velocity at which surface waves propagate is a function of their frequency, that is they are dispersed, and the dispersion primarily depends on the shear wave velocity structure (Fig. 1). In general, the lowest shear wave velocities are found for near surface material. The velocity increases with depth in the lithosphere then decreases in the asthenosphere. Fundamental mode Rayleigh waves (those dealt with in this practical) are most sensitive to the velocity structure within about 0.4 wavelengths of the surface. Therefore intermediate period fundamental mode Rayleigh waves travel with greater speed than both longer and shorted period waves. Average Rayleigh wave dispersion curves for the Earth are shown in Fig.. If we can measure the dispersion characteristics of surface waves in a region, we can use the dispersion curves to determine the Earth shear wave velocity structure by fitting the observed dispersion curves with theoretical dispersion curves calculated for proposed velocity models. This practical shows how this procedure is accomplished. Group Velocity In the lecture we derived the period equation for a SH wave trapped in a low velocity surface layer. The derivation of the Rayleigh wave period equation for an elastic half space and a liquid layer over an elastic halfspace is attached. The period equation relates the phase velocity c to the period T or the frequency ω. By substituting values of T in the period equation we can solve for c and plot c(t ), the phase velocity curve. For each T along the curve there is a c(t ) which is a solution to the wave equation. The total motion (the seismogram) is the sum of the motion for each T (or ω) scaled by an appropriate amplitude A(ω) factor and shifted by an appropriate initial phase φ(ω), that is u(x, t) = 1 π A(ω) exp {i ωt k(ω)x + φ(ω)} = 1 π A(ω) e iφ This is the fundamental form of a dispersed wave train (for a single mode). We can solve an integral of this form by the method of stationary phase. For large t and x, the only contribution to the integral is for the particular combinations of t and x where dφ = d { {ωt k(ω)x + φ(ω)} = t dk x + dφ } = 0
or t = dk x + dφ = 0 t = x u dφ u = x t + dφ/ We can see this by plotting Ree iφ vs. ω Thus the only contribution to the integral occurs near where dφ/ = 0. This defines the group arrival time t at x and the group velocity u, i.e., where constructive interference leads to the presence of energy of frequency ω on the seismogram. The velocity of this point of constructive interference is x t (for dφ = 0) = dk = group velocity Determing the Group Velocity Curve The long period seismograms are identical to the type used in the other practicals. They were recorded in western North America at the World Wide Standard Seismic station located at Dugway, Utah (40.1950 o N, 11.8133 o W), and are for an earthquake which occurred at 3:11:01.0 on April, 1977 in the Solomon Islands (10.0 o S, 160.65 o E). The epicentral distance is 10448 km and the azimuth of Dugway from the source is 50 o. The onset of the Rayleigh waves are indicated on the trace. Use the vertical component (component #4) to measure the Rayleigh wave dispersion since there can be no interference between the Love and Rayleigh waves on the vertical. Dispersion can be seen as a progressive change in frequency of the waves with time. First you must determine the arrival time of the energy of a given frequency or period. Find the time scale for the measurement by measuring the distance between the minute marks bracketing the wave being considered; do not use one scale for the whole seismogram because the rotation speed of the drum will not be reliably uniform and the paper may have stretched. You can record t and T by a direct measurement off the record. You can measure this more accurately (with some smoothing) by plotting phase number vs. arrival time. The slope of the curve at a point then gives the period of the wave at that time.
Make a table of arrival time and period using either of the methods. The group velocity is then given by u = x t(t ) where we have assumed that the initial source phase as is constant across the frequency band being considered. This method of determining the group velocity will not work if the wave is pulse like or does not have a simple dispersed form. In these cases it is necessary to compute the group velocity from the Fourier phase spectrum. Determining the Velocity Structure Plot your dispersion measurements on the attached group velocity curve. These curves are for a variety of water depths and were computed from the Rayleigh wave period equation for a liquid layer over an elastic half space. Choose the theoretical group velocity curve which best fits your observations and estimate the mean water depth across the Pacific Ocean along the propagation path. The epicenter of this earthquake was south of Guadalcanal and the great circle path to Dugway is almost east-west (azimuth 60 o as seen from Dugway). The east west component (instrument #6) is nearly radial and should yield similar dispersion results as observed on the vertical above. Does the particle motion observed from the vertical and east west show retrograde motion typical of Rayleigh waves? The north south component (instrument #5) is oriented nearly transverse and should therefore record very small Rayleigh wave motion. This is true until about 4:06 but following this time there is a train of waves on the north south similar to those observed on the other two components. Beats also appear on instrument 4 (vertical) at this time. Study the great circle path on a globe and find what effects might be causing this change with time.
Rayleigh Waves in a Homogeneous Half space In order to have surface waves it is necessary for the apparent velocity to be less than the P and S velocities, i.e. c < β < α. If this were not the case the SV energy would radiate into the half-space and the seismic energy would not be trapped in the upper layers. φ = Ae i(ωt kx x+k x γ αz) ψ = Be i(ωt kxx+k xγ β z) For surface waves the exponentials in z must have negative, real exponents so we chose the signs of the square roots γ α, γ β such that γ α = i 1 c x α γ β = i 1 c x β and c x = ω k x (1) Using the stress free, free surface boundary condition we find the following x system of homogeneous linear equations c x β α 1 A + ck A + β 1 c k β B = 0 B = 0 () For a nontrivial solution for these two equations in two unknowns, the determinant of the system must be zero, that is c x β = 4 β 1 α 1 We can solve this numerically for a given model (specific values of α, β). There will be four roots, one of which will be zero, and only one of the three remaining roots will be consistent with our surface wave condition (c < β < α). For the case of a Poisson solid (λ = µ, therefore α β = 3) we can further simplify the above equation to obtain c x β 6 4 β 6 8c x β 4 + 56c x 3β 3 = 0 3
This is a cubic in c x/β whose roots are 0, 4, and ± / 3. c x/β = 0 has no physical significance and the only root which is consistent with the surface wave condition is / 3. Therefore c x β = 0.8453 = c x = 0.9194β Thus, the apparent velocity of the Rayleigh wave in a half-space which is a homogeneous Poisson solid is independent of frequency and only slightly less than the shear velocity. To find the displacement, substitute () in (1) and find the displacements from the potentials. Taking the real part of the exponentials and using the numerical values of c x /β and c x /α for a Poisson solid u x = Ak x sin(ωt k x x)(exp( 0.85k x z) 0.58 exp( 0.39k x z)) u z = Ak x cos(ωt k x x)( 0.85 exp( 0.85k x z) + 1.47 exp( 0.39k x z)) At the surface these become u x = 0.4Ak x sin(ωt k x x) u z = 0.6Ak x cos(ωt k x x) If we look at the motion of a particle at a fixed x while a monochromatic wave passes, the x and z displacements which are out of phase combine to give retrograde motion along an ellipse. The maximum vertical displacement is about 1.5 times the maximum horizontal motion at the surface for a Poisson solid. Since the decay of displacement with depth is controlled by factors like exp( k x z) = exp( πz/λ), long wavelength Rayleigh waves penetrate deeper than shorter wavelength Rayleigh waves. Therefore, long wavelengths provide information about deeper structure than higher frequency waves.
Dispersion of Rayleigh Waves Simple model of a liquid layer over a half-space. B.C. 1. Z = H Stress Free. Z = 0 Continuity of Vertical Stress Continuity of Vertical Displacement There are no shear waves in the fluid and the displacements must satisfy the scalar (P-wave) equation. φ = A L e i(ωt kx x+k xγ L z) where γ L = α L 1 Both P and S waves propagate in the solid. φ = Ae i(ωt kx x+k xγ αz) ψ = Be i(ωt kx x+k xγ β z) In the half-space the exponentials in z must have negative, real exponents so we chose the signs of the square roots γ α, γ β such that γ α = i 1 c x α γ β = i 1 c x β We must satisfy the boundary conditions: σ zz = 0 at z = H gives and c x = ω k x
B = A L e iγ LH continuity of u z at z = 0 gives continuity of σ zz at z = 0 gives γ α A + kb = iγ L A L e iγ LH cos γ L H A(ω k β ) + Bkγ β β = i ρ 1 ρ ω A L e iγ LH sin γ L H For a non-trivial solution to exist for this set of equations the determinant of the coefficients must vanish. This gives the period equation tan H ω α L k = (c x /α L 1) 1/ ρβ 4 (1 c x /α ) 1/ 4(1 c x 1/ αρ 1 c 4 x α ) (1 c x 1/ β ) ( c x β ) Now the velocity c x is a function of frequency, that is, the propagation is dispersed. If H 0 or the frequency 0 or the period of the waves becomes very long, the above equation reduces to the half-space equation. The phase and group velocities curves are (where the frequency is scaled by N = Hω/α L, the time taken for the p-wave to cross the fluid layer) The eigenfunctions for the u z component of displacement is also shown. For this case (α, β > c x > α L ) waves propagate within the water layer but the motion exponentially decays in the half-space. Copyright c 1993 Keith Priestley