# 6.6: Summary

- Page ID
- 3551

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Let's review what we learned about earthquakes in this chapter:

## Earthquake Geometry and Process

Generally speaking, it is the movement of faults that leads to earthquakes. There are three basic types of faults; strike slip, normal, and reverse. The focus is the location where the earthquake initiates. The epicenter is the projection of the focus to the surface.

Tectonic forces are moving in opposite directions and cause applied shear stress, \(\sigma_s\).

The equation for frictional strength is

\[\sigma_f=c+\mu\sigma_n\]

If \(\sigma_s>\sigma_f\), then failure (an earthquake) occurs.

We also learned how earthquakes actually happen. In a fault rupture, the earthquake initiates at the focus.

- At 5 seconds, the rupture continues to expand as a crack along the fault plane. When the rupture front reaches the surface, displacements occur along the surface trace, and rocks at the surface begin to rebound from their deformed state.
- At 10 seconds, the rupture front progresses down the fault plane, reducing the stress and allowing the rocks on either side to rebound. Seismic waves continue to be emitted in all directions as the fault propagates.
- At 20 seconds, the rupture has progressed along the entire length of the fault. The fault has reached its maximum displacement, and the earthquake stops.

Seismograms record earthquake waves, and are used for determining:

- Time of an earthquake and distance to epicenter
- Earthquake location
- Magnitude (energy release) of the quake
- Type of fault (normal, reverse, strike slip)
- Distribution of slip on the fault surface
- Structure of the interior of the Earth

## Earthquake Magnitude

In order to measure the size and intensity of earthquakes, we have devised several scales to measure them. The Richter Magnitude scale is one such scale. One issue with measuring earthquakes is that as the waves propagate, the energy is spread out over more area. We discovered that amplitude \(\propto\) the initial energy of the earthquake. The actual equation for determining Richter magnitude is:

\[M_L=\log_{10}A-\log_{10}A_0(\delta)=\log_{10}[\frac{A}{A_0(\delta)}]\]

where \(A\) is the maximum excursion of the Wood–Anderson seismograph and A_{0} depends on the distance of the station from the epicenter (\(\delta\)). The Richter scale is both logarithmic and 'empirical'. Empirical means that it is chosen to fit a range of observation of both very small and very large earthquakes. The fact that the scale is logarithmic indicates that there is an order of magnitude difference between each number.

Another problem we have in measuring earthquakes is how to compare the physical size rupture area from the aftershocks and the surface rupture. To solve this problem, we need to integrate and scale a typical amplitude vs time diagram, and convert it to area, or moment. Moment is literally torque. The famed seismologist Hiroo Kanamori used these ideas to create a new magnitude scale, the Moment magnitude scale.

\[M_w=\frac{2}{3}(log(M_o)-10.7)\]

\(M_o\) integrates seismogram data and relates to physical size of the earthquake. Instead of solving for \(M_w\), we can also solve for \(M_o\).

\[M_o=10^{\frac{3}{2}M_w+10.7}\]

## Location and Focal Mechanisms

To find the simplified location of an earthquake there are four basic steps.

- Identify seismic wave phase arrival times at stations
- Calculate distance between individual stations and the epicenter
- Triangulate-use determined distances for multiple stations to locate epicenter
- Calculate origin time

The equation to find wave distance traveled:

\[x=\Delta t[\frac{v_pv_s}{v_s-v_p}]\]

Using this we were able to find distance, but the earthquake could lie anywhere on a circle centered at that station with a radius=distance.

In order to find the exact location of the earthquake, we must repeat the distance calculation for at least two more stations. This will give us at least three circles with radius=distance, which we can cross reference as in the figure below. To locate the focus/hypocenter, you would need at least four stations to draw spheres in 3D in order to get depth.

To find information about the type of deformation in the source region that generates the seismic waves, we used focal mechanisms. Focal mechanisms show a simple seismic model of faulting. They are also a 'moment tensor' or 'beach ball diagram'. The various motions of the fault are represented on what is known as a beach ball. The regions of compression indicate a first motion of up and are represented by a closed dot. The regions of expansion indicate a first motion of down** **and are represented by an open dot. If there is no detected signal, this is represented by an \(x\). The auxiliary plane and the fault plane are also called nodal planes.

There are three principal axes on a beach ball, the N axis, P axis, and T axis which are all orthogonal to each other. The N axis is the null axis. It is the intersection of the two nodal planes. The P axis is the pressure axis. It bisects the angle between the nodal planes, and has the largest amplitude down first motion. The T axis is the tension axis. It also bisects the angle between the nodal planes, and has the largest amplitude up first motion.

Here are some quick glance takeaways of focal mechanisms can tell you:

- Approximate strike of fault
- What type of faulting occurred
- Overall sense of motion in a region
- Regions of compression and tension

## Earthquake Scaling Laws

There is a quantitative way of looking at the distributions of the occurrence of different magnitude earthquakes, the Gutenberg-Richter frequency-magnitude scaling law. Frequency-magnitude describes the number of occurrences as a function of magnitude.

\[log_{10}(N_c)=-bM+a\]

where

- \(M\) is the magnitude,
- \(N_c\) is the cumulative number of earthquakes that occurred with magnitude \(\geq\) M,
- \(a\) is the total seismicity, and
- \(b\) is the 'b value' which is the scaling value relating the number of large events to small events.

a and b are constants calculated for a specific region and time. The equation is a line (y=mx+b). Linearity and fit of the Gutenberg-Richter law occurs in the mid-magnitude range due to roll-off and a higher magnitude drop off.

The most talked about part of the scaling relationship is the \(b\) value. The b value is typically \(0.8\leq b\leq 1.2\) with \(b \simeq 1\). Each |(M\) you go up by, you go down an order of magnitude of occurrence.

10,000 | 1,000 | 100 | 10 | 1 |

M2 events | M3 | M4 | M5 | M6 |