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5.4: Summary

  • Page ID
    3544
  • Let's review what we learned about seismology in this chapter:

    Basics of Wave Propagation

    The basic ways we measure waves are through \(\lambda\) (the wavelength in meters) and \(f\) (the frequency in Hz). They have the relation to velocty: 

    \[v=f\lambda\;[\frac{m}{s}]\]

    One category of seismic waves are body waves. Body waves are P waves and S waves. P waves act like an accordion, and move parallel to the propagation direction. S waves can have two components of motion, vertical and horizontal. Most S waves usually have both components, S-vert and S-horiz, which can be polarized. The other category of seismic waves are surface waves, which decay with depth. There are two main surface waves, Love waves and Rayleigh waves. Love waves have particle motion like that of the S-H component. Rayleigh waves have the components of a P wave and a S-V wave. The particle motion is retrograde. 

    We also reviewed some constants we learned earlier that will be helpful in determining P and S wave velocity. 

    E-Young's Modulus (elongation dilation)

    \(\mu\)-Poisson's Ratio (compensation deformation)

    G-shear strain

    \(\kappa\)-volume change

    In seismic calculations, you can use one set (E and \(\mu\)) or the other (G and \(\kappa\)).

    For P-waves:

    \[v_p=\sqrt{\frac{\kappa+\frac{4}{3}G}{\rho}}\;or\;\sqrt{\frac{E}{\rho}\frac{(1-\nu)}{(1-2\nu)(1+\nu)}}\]

    For S-waves, we can calculate velocity as:

    \[v_s=\sqrt{\frac{G}{\rho}}\;or\;\sqrt{\frac{E}{\rho}\frac{(1-2\nu)}{(2(1+\nu))}}\]

    Using Fermat's Principle and the fact that \(\theta_i=\theta_r\), we derived Snell's Law:

    \[\frac{sin\theta_i}{v_1}=\frac{sin\theta_r}{v_2}\]

    In special cases such as critical refraction,

    \[\theta_{ic}=sin^{-1}(\frac{v_1}{v_2})\]

    Seismic Refraction (Single Layer)

    One of the most important uses of seismic waves is for seismic surveys. There are two types of seismic surveys:

    1. Refraction ie 'critical refraction' which is useful for determining velocities of layers
    2. Reflection, which is useful for determining layers and structure

    \[t=\frac{x}{v_1}\]

    The above equation is the direct ray travel/arrival time. The equation is a line whose slope is \(\frac{1}{v_1}\) and intercept=0. Using the vertical incidence of a reflected ray, we can derive an equation to calculate the time the ray took to reach the receiver:

    \[t=\frac{2h}{v_1}\]

    There are important values that can be seen when the refracted waves are graphed such as the cross over distance, xco is when the tdir=thead(rfr)

    \[x_{co}=2h(\frac{v_2+v_1}{v_2-v_1})^{\frac{1}{2}}\]

    In this section, we were essentially learning how to interpret/extract information from a seismic refraction survey. This allows us to determine the subsurface structure.

    How do you determine subsurface structure?

    • Identify direct wave arrivals
      • Measure slope→m1
      • Calculate v1(layer 1)→\(v_1=\frac{1}{m_1}\)
    • Identify headwave arrivals
      • Measure slope, m, intercept, to
      • Calculate v2(layer 2)→\(v_2=\frac{1}{m_2}\)
    • Use to, v1, and v2 to calculate h
      • \(t_o=\frac{2h(v^2_2-v^2_1)^{\frac{1}{2}}}{v_1v_2}\)
      • \(h=\frac{t_ov_1v_2}{2(v^2_2-v^2_1)^{\frac{1}{2}}}\)
    Seismic Reflection (Single Layer)

    Reflection surveys follow the same basic principles as refraction surveys, using reflected waves instead of refracted waves. We can use reflected waves to get h and v1, but we cannot get v2 unless we have a reflection from a deeper layer. The time the wave traveled down and back up to the receiver (2d) can be represented as:

    \[t=\frac{\sqrt{x^2+4h^2}}{v_1}\]

    The intercept time is:

    t(x=0)=to=\(\frac{2h}{v_1}\)

    For a vertically reflected wave, if you know v1, you can get h.

    \[h=\frac{v_1t_o}{2}\]

    Reflected waves graph as hyperbolas instead of lines, where a=to and b=v1to. The slope at large x=\(\frac{a}{b}=\frac{t_o}{v_1t_o}=\frac{1}{v_1}\).

    \[1=\frac{t^2}{t_o^2}-\frac{x^2}{v_1^2t_o^2}\]

    We analyze data using a method called Normal Move-out (NMO). 

    \[T_{NMO}=t(x)-t_o\]

    The equation for TNMO gives us the "shape of the hyperbola". By correcting for NMO, we are looking at data close to the first few recievers, ie those that have taken shorter paths and have not 'moved-out'.

    \[v_1\approx\frac{x}{\sqrt{2t_oT_{NMO}}}\]

    When you are actually analyzing a travel time graph, there are several steps you can take to get information on the subsurface.

    1.  Measure (or read off) to, t(x), and x
    2. If x has a small value, calculate TNMO=t(x)-to
    3. Calculate \(v_1\approx\frac{x}{\sqrt{2t_oT_{NMO}}}\)
    4. Calculate \(h=\frac{v_1t_o}{2}\)