# Presentation: 598485

##  Theoretical Seismology 1: Sources

-      Elastic Rebound
-      Fault Slip  Double-couple Force
- ・ Seismic Moment Tensor
- ・ Models of Earthquake Faults
- ・ Earthquake Size
-       Magnitudes
-       Seismic Moment
-       Energy

-  Theoretical Seismology 1: Sources

## What is an Earthquake ?

-     Fault mechanisms
-      The Shaking
-          Wave propagation
-           Structures

**Notes:**
We can think of an earthquake as two components, the source and the shaking. Most people are not seismologists and tend to think about shaking when they talk about earthquakes; however, many seismologists are more interested in the what is happening at the source – the fault mechanisms - than in the waves this generates. 

## Giuseppe Mercalli

- <date/time>

- (1850-1914)

- John Milne
- (1850-1913)

- Chang Heng ‘Seismometer’ AD132

- Sassa Seismometer (~1935), Abuyama, Kyoto Univ.

**Notes:**
People have always been interested in earthquakes and have attempted to build instruments to record the ground shaking (with varying levels of success) for thousands of years.

##  What is the cause of

-         Earthquakes ?

- ・ Associated with faults
-    (source or cause?)
- ・ Associated with magma?

- (Most) Earthquakes are
- fault movements

**Notes:**
However, it’s only relatively recently that movement on faults were identified as the cause of most earthquakes – until the early 1900’s faults were thought of as an affect of ground shaking rather then the cause. Many myths have been recorded over the years relating to the cause of earthquakes, one of the more famous is an old Japanese idea that the earth sits on the back of a giant catfish, with a god holding the catfish still and that earthquakes happen when the god looses concentration and the catfish flicks his tail.

## Breaking of Chopstick

- ・ Build-up of stress (strain energy)
- ・ Difficult to predict time and place
- ・ Breaks at weakest point
- ・ Hear precursors
- ・ Sound of breaking same as seismic waves

**Notes:**
However we now know that most earthquakes happen on faults and the process that causes them is similar to what happens if you bend a chopstick until it breaks. 

## Elastic Rebound Theory

-   Reid (1910)

- (Data in 1851-65, 1874-92, 1906)

- 8.5 feet offset in San Andreas fault
- from 1906 earthquake. Marin County

**Notes:**
The first person to come up with the model that fault movement caused earthquakes, was Henry Reid in 1910 – after studying the famous 1906 San Francisco earthquake. He camd up with elastic rebound theory – the idea that strain slowly builds up for many years (may be as much as ~100 or ~200 years for areas with irregular large earthquakes) and then, when the strain is greater than the fault strength it is rapidly released in a few seconds as the fault slips to remove the strain, creating the earthquake.

## San Francisco Earthquake

- April 18, 1906
- Mw 7.7-7.9
- 470 km rupture of
- San Andreas fault

**Notes:**
This model was generated after studying the affects of the 1906 earthquake in San Francisco

## Slide 8

**Notes:**
this is an exaggerated model of what happens

## Equivalent Body Forces

- Single Force

- Dipole

- Couple
- (Single Couple)

- Double Couple

**Notes:**
So after making the connection between fault movement and earthquakes, the next question is what are (and how do we represent) the force on the fault? This caused a considerable amount of discussion in the early and mid 1900s. 

The simplest forces to visualize are single forces, the next simple are dipoles – which are a pair of force separated by a distance. More complicated are couples, these are offset forces, which generate a torque (i.e. rotational force). These forces can also be combined, for example at the bottom of the page we have a double couple.

## Single Couple versus Double Couple

- Single Couple

- Double Couple

- ・ P polarity pattern same
- ・ S polarity pattern different
- ・ Single Couple ‘resembles’ fault slip

**Notes:**
The P-wave energy radiated from earthquakes was studied in the early 1900s and it was found to have a pattern that does not fit with the forces on the fault being single forces or dipoles. But for a long time there was a debate over whether single or double couples better represented the fault. This was solved by Maruyama in 1963 who showed that the S wave energy supported a double couple source. This is also intuitive, as most earthquake are not associated with rotations, and a single couple would be associated with a torque, whereas the torques in the double couple cancel each other out.

## Moment tensor: dipoles and couples

- (LW p.343; AR p.50)

- 9 components
- Symmetric matrix so 6 independent

**Notes:**
The most complete way to describe the forces on the fault is through the moment tensor. This is the complete series of 3 dipoles (each at 90 degrees) and 6 couples (each at 90 degrees). 

## Moment Tensor for an Explosion

**Notes:**
This is a very versatile way of modeling the earthquake, it doesn’t even require a double couple source mechanism – for example an explosion would be represented by three positive forces on the 3 dipoles.

## ⇒

- Double Couple
-     Fault - Slip

- Moment Tensor for Fault Slip

- North

**Notes:**
The double couple force associated with most earthquakes is represented as a force on two of the couples. 

## Types of faults

- Thrust (Reverse) fault

- Normal
- fault

- Strike-slip fault

**Notes:**
It is also useful (and common) to characterize earthquakes using the geometry of the fault and the slip direction. The slide shows the three end member faults. 
Normal faults are associated with tectonic tension and movement on the fault results in extension, the faults are often steep (at least in the near surface), commonly dipping at ~60 degrees. 
Reverse faults and thrust faults are associated with compressional tectonics and movement on the fault results in shortening. Reverse faults are steeply dipping and thrust faults shallow dipping. The movement on normal and reverse/thrust faults is sometimes referred to as dip-slip, distinguishing them from the third type of fault.
Strike-slip fault are formed in areas with shearing forces, one side of the fault slides past the other without causing either extension or shortening.

## Strike-Slip Faults  

- Strike-Slip Faults

- Left-lateral

- Right-lateral

**Notes:**
Strike-slip faults can be described as either left-lateral, where the far side of the fault moves to the left – or right-lateral, when the far side of the fault moves to the right.

## 1940 Imperial Valley, California (Ms 7.1)

- <date/time>

**Notes:**
You can see from the off-set trees in this orchard that the fault here had recent right-lateral movement. However, normally the fault moves deep in the crust and there is not surface expression of the rupture, so we need to look at the seismic wave to try and work out what happened at the fault

## P-wave first motions

- This type more likely to produce large tsunamis

**Notes:**
The figure on the left hand side is a representation of a fault, either of the blue lines could be the fault, the blue arrows show the double couple forces on the fault. The two possible fault planes separate regions where the ground would move towards a distant station (+) and regions where the ground would move away (-). One of the features of earthquakes generated by coupled forces is that the pattern of positive and negative first motion (also called compression and dilation) is the same for the two possible fault orientations (at 90 degrees to each other). This distribution of compression and dilation can be seen in the polarity of the recorded P-waves (i.e. whether the first motion on the seismogram is positive/upwards, or negative/downwards).

The figures on the right are focal mechanism diagrams (also called beachballs), which area  graphical way of representing the orientation of the two possible fault planes and the nature of the slip on the fault. These are discussed in a later session.

## Harvard/NEIC Moment Tensor Solutions

**Notes:**
Harvard University and the USGS National Earthquake Information Center (NEIC) both calculate the moment tensor and give a pictorial representation of the fault and publish this information freely on the internet within a few hours of a large earthquake. The slide shows the moment tensor information and beachball diagram published by Harvard following the Great Sumatra earthquake in Dec 2004. The data give details on the data used (here the global GSN stations, 300 sec period surface waves); the centroid location – which is in effect the center of the rupture area; time of the earthquake; the duration of the rupture (which is underestimated in this case because the waves are insensitive to a rupture that last longer then their period, but it is very unusal for the rupture to last longer than can be recorded by the surface waves); the 6 independent components of the moment tensor and the scale; then the scale and orientation of the tension, pressure and neutral axes; and finally seismic moment and the strike and dip of the nodal planes for the best double couple fault plane solution (which is what the graphical beachball represents).

## Single-force earthquakes volcanic eruptions and landslides

- Mount St. Helens, USA

- Kanamori et al. 1984

**Notes:**
There are some ways of generating earthquakes that are not double couple sources. The eruption of Mount St Helens in 1980 involved a large explosion as the side of the volcano gave way, this generated seismic waves that are best modeled as a source generated by a single horizontal force (associated with the explosion). Also landslides and magma movement within volcanoes can generate small earthquakes that are not double couple sources.

## Circular Crack – Sato and Hirasawa, 1973

**Notes:**
So far we have really been talking about earthquake as though they are generated from a single point; however, we know that ruptures have a finite size, so seismologists have spent much time trying to image/model the finite nature of ruptures. This circular crack model is one of the early attempts and is probably OK for small earthquakes (less than Magnitude 5).

## Haskell Line Source

- Dislocation Source

- Haskell, 1964

- sumatra

- Sumatra earthquake            Ishii et al., 2005

**Notes:**
The Haskell line source is basically a 1D model and appears to do better for large earthquakes, the rupture propagates along the fault with only a small section slipping at any one time. Models of the Sumatra earthquake calculated from the recorded seismograms show that something similar to this happened in this earthquake, the rupture propagated northward for ~500 sec, with different section rupturing at any give time.

## Complicated Slip Distributions

-      -

- 1999 Chi-Chi, Taiwan Earthquake

**Notes:**
However, any simple model is not going to capture the full complexity of rupture, which is normally very variable even within the area that slipped. This figure shows the amount of slip on each part of the fault for the 1999 Chi-Chi earthquake in Taiwan, it is clear that some areas of the fault slip far more than others. This is very important when thinking about tsunami generation – tsunamis are generated by displacing a large volume of water, with variable slip the seabed deformation will be complicated and so some areas above the rupture may generate a tsunami while others do not.

## Earthquake Size – Magnitude

- M = log A – log A0

- Richter, 1958

- Charles Richter
- 1900-1985

- log of amplitude

- Distance correction

**Notes:**
Magnitude is a single value that characterizes the earthquake and is one of the primary pieces of information used by scientists, the media and the public when talking about earthquakes. The first magnitude scale was developed by Richter, who worked out that you could correct for the amplitude change with distance from the earthquake and fit a single curve through the data, characterizing the size of the earthquake. The Richter-scale was developed for southern California and for data recorded on a wood-anderson seismometer (never used any more), so it isn’t really applicable to earthquake recorded today – the modern equivalent is the local magnitude scale ML – but it’s hard to stop the news media from referring to the Richter Scale, even though this isn’t what the seismologists are using.

## Types of Magnitude Scales

-                                                          surface waves
- Mj       JMA (Japan Meteorol. Agency)    regional S and           5-10 sec
-                                                           surface waves
- mb    Body wave magnitude    	         teleseismic P waves    1-5 sec
- Ms     Surface wave magnitude 	         teleseismic surface      20 sec
-                                                             waves
- Mw       Moment magnitude                   teleseismic surface     > 200 sec
-                                                              waves
- Mwp     P-wave moment magnitude        teleseismic P waves    10-60 sec
- Mm      Mantle magnitude                      teleseismic surface     > 200 sec
-                                                                                               waves

- Period Range

**Notes:**
There are many magnitude scales, all are logarithmic, but each is calculated using different types of seismic wave and different constants. Some of the more common scales are listed here. Which is the most appropriate depends on what will be done with the data, for example ML is quite popular with engineers because it calculates the magnitude using a period of wave that is similar to the resonance frequency of most buildings, so it’s quite well related to the amount of damage caused by the earthquake. 

## Relationship between different types of magnitudes

**Notes:**
This slide shows that magnitude calculated using different scales compared to the seismic moment – the moment magnitude scale is in theory the true measure of the seismic moment, the other scales under estimate the magnitude for very large earthquakes. This underestimation is related to the period of the waves used to calculated the magnitude.

## Earthquake size - Seismic Moment

- Area (A)

- Slip (S)

- 15 km

- 10

- 0

- M4    M5     M6

- 5

**Notes:**
Seismic moment has been the standard measure for earthquake magnitude for the last ~20 years. It is a physically meaningful number, unlike the magnitudes scales, related to the size of the fault that slips, the amount of slip and the rigidity of the fault.

On the right the colored blocks illustrate the approximate fault size for different magnitude earthquakes.  Note that the fault size does not increase by 10 for each magnitude step, but instead increases by 2-3 times.

## Seismic moments and fault areas

- of some famous earthquakes

**Notes:**
This slide shows the fault area (and the seismic moment and moment magnitude) for some famous damaging earthquakes. Chile 1960 is the largest ever recorded earthquake, Alaska 1964 was also larger than Sumatra 2005, but the Sumatra event had the longest ever recorded rupture length (it was just a narrower rupture than the other two). Note however, that many of the damaging earthquakes do not have particularly large seismic moments – 5000 people died in Kobe, even though it was only a M=6.9 event.

## Types of Magnitude Scales

-                                                          surface waves
- Mj       JMA (Japan Meteorol. Agency)    regional S and           5-10 sec
-                                                           surface waves
- mb    Body wave magnitude    	         teleseismic P waves    1-5 sec
- Ms     Surface wave magnitude 	         teleseismic surface      20 sec
-                                                             waves
- Mw       Moment magnitude                    teleseismic surface     > 200 sec
-                                                              waves
- Mwp     P-wave moment magnitude         teleseismic P waves    10-60 sec
- Mm      Mantle magnitude                       teleseismic surface     > 200 sec
-                                                                                               waves

- Period Range

**Notes:**
Moment magnitude is the most commonly reported, because is relates the size of the earthquake to the seismic moment. The problem with moment magnitude is that it takes a few 10’s of minutes to calculate as it uses long period surface waves recorded at large distances. This is no problem for academics, but for tsunami warnings and other hazard mitigation uses, this time can be a big problem. 

## Types of Magnitude Scales

-                                                          surface waves
- Mj       JMA (Japan Meteorol. Agency)    regional S and           5-10 sec
-                                                           surface waves
- mb    Body wave magnitude    	         teleseismic P waves    1-5 sec
- Ms     Surface wave magnitude 	         teleseismic surface      20 sec
-                                                             waves
- Mw       Moment magnitude                    teleseismic surface     > 200 sec
-                                                              waves
- Mwp     P-wave moment magnitude         teleseismic P waves    10-60 sec
- Mm      Mantle magnitude                       teleseismic surface     > 200 sec
-                                                                                               waves

- Period Range

**Notes:**
So the tsunami warning centers generally use Mwp, Mj or Mm – particularly Mwp which is an attempt to calculate the moment magnitude from just the first few cycles of the P wave, this allows the magnitude estimate to be made very quickly (within a few minutes) which may be very important for saving lives. However, the Mwp scale doesn’t work very well for very large earthquakes, it tends to underestimate events with magnitudes greater than 8.

## Magnitudes for Tsunami Warnings

- ・ Want to know the moment (fault area and size)
-     but takes a long time (hours) to collect surface wave
-     or free oscillation data
- ・ Magnitude from P waves (mb) is fast but
-     underestimates moment
-   ⇒ If have time (hours),
-               determine Mm from mantle waves
-     ⇒  For quick magnitude (seconds to minutes),
-               determine Mwp from P waves

## Mm       Mantle Magnitude 

- Mm = log10(X()) + Cd + Cs – 3.9

- Distance Correction

- Source Correction

- Spectral Amplitude

- ・ amplitude measured in frequency domain
- ・ surface waves with periods > 200 sec

**Notes:**
The equation for the mantle magnitude scale is similar to the Richter scale, but a little more complicated as it specifies a specific amplitude and corrections.

## ∫uz(t)dt  ∝ Mo

- ∫uz(t)dt  ∝ Mo

**Notes:**
Mwp is based on the integration of the displacement (the double integration of the velocity trace), which gives the maximum displacement. This is then proportional to the seismic moment.

## Mo = Max |∫uz(t)dt| 43r/Fp

- ・ Quick magnitude from P wave
- ・ Uses relatively long-period body waves (10-60 sec)
- ・ Some problems for M>8.0

- Mo = Max |∫uz(t)dt| 43r/Fp
- Mw = (log10Mo – 7.1)/1.5

**Notes:**
This shows the equation used to derive seismic moment from the twice integrated P, note in inclusion of a distance correction (r). The moment is converted to moment magnitude (Mw) using the lower equation. 

## Magnitudes for the Sumatra Earthquake

- mb	7.0               1 sec P wave                  131 stations
- mblg  	6.7               1 sec Lg waves                   6 stations
- Mwp   	8.0 – 8.5        60 sec P waves
- Ms    	8.5 - 8.8        20 sec surface waves       118 stations
- Mw   	8.9 - 9.0       300 sec surface waves
- Mw   	9.1 - 9.3     3000 sec free oscillations

**Notes:**
The first PTWC bulletin for the Dec 26, 2004 tsunami (11 mins after the earthquake) announced an Mwp=8.0, after approximately one hour this was increased to Mw=8.5 after including surface wave data. Harvard CMT made a special calculation of the Mw using longer than usual surface waves (300 sec compared to the normal 135 sec period) and published Mw=9.0 within 5 hours of the event. Days later the Mw was increased to 9.1-9.3 using very long period normal mode data.

## Fault Areas of 

- 1944 Tonankai
- Mw 8.1

- 1946 Nankai
-  Mw 8.1

- 1995 Kobe
- Mw 6.9

- Fault Areas of
- Damaging Earthquakes

-        Deaths
- 1944     1223
- 1946     1330
- 1995     5310

**Notes:**
Earthquakes that kill and cause damage are not necessarily the largest quakes, for example, these 2 great earthquakes in the 1940s killed fewer people than the far smaller Kobe 1995 event. The major difference was that Kobe happened extremely close to the city, while the outher two happened offshore – how close the earthquake is has as much affect the magnitude of the event.

## Seismic Radiated Energy  

- Seismic Radiated Energy

- Kanamori, 1977

**Notes:**
Some comparisons of the amount of energy released in big earthquakes to the energy involved in other sources.

## Things to Remember

- 1. Earthquake sources are a double couple force system
-     which is equivalent to Fault Slip
- 2. The moment tensor describes the Force System
-     for earthquakes and can be used to determine
-     the geometry of the faulting
- 3. Earthquake ruptures begin from a point (hypocenter)
-     and spread out over the fault plane
- 4. The size of an earthquakes can be described by
-     magnitudes, moment, and energy.
-     Mm and Mwp are types of magnitudes used for
-     tsunami warning systems