An earthquake is the result of a sudden displacement across a fault that releases stresses that have accumulated in the crust of the earth.

An earthquake is the result of a sudden displacement across a fault that releases stresses that have accumulated in the crust of the earth.

Measuring an Earthquake s Size Magnitude and Moment Each can be measured from seismogram - but Moment can be related to physical description of earthquake source.

Seismic Moment Mo = υlwd

And there is an empirical relationship between Magnitude and Seismic Moment Log Mo = 15M + 16.1 (dyn-cm) that can also be derived from physical principles 4

Earthquake Statistics and Behavior of Faults When one looks at seismicity over broad regions - there are some general observations that ALWAYS seem to be true

If one plots a catalog of earthquakes from any region of the world like this (where N=number of earthquakes > Mag) One will always observe There are many more small earthquakes than large The slope of this line (ratio of small to large earthquakes) is invariably equal to ~1

This is referred to as the Gutenberg-Richter or Ishimoto- Iida Relationship b-value N N Where n = number of events M= Magnitude or

N Where Nn = number of events M= Magnitude or Log N = a - bm Where N is number of earthquakes >= M

Log N Given an area with a seismic network (e.g California) Seismograph network stations operating in California in 1986.

Value for seismic hazard analysis Log(N)

ISC catalog 1900-2010

Predicting Earthquake Recurrence with Earthquake Statistics... >=M9 each 1/.0001 years = 10,000 years BUT, it does not tell us that magnitude 8 or 9 earthquakes will occur or where the earthquakes will occur, other than in broad region of the catalog... >=M8 each 1/.001 years = 1000 years for that, the historical record and geology is often needed...

The GR relationship can be written in terms of Seismic Mo which is valuable because fault data can be used to predict rates of seismicity.

The question exists whether or not seismicity on a particular fault also satisfies a Gutenberg Richter distribution?

Endmember models of fault behavior we ll first assume this 2nd model

size of an expected earthquake is function of the fault length - and can empirically be estimated slip rate = Ug/T

empirical regressions of earthquake size to fault length for historical continental earthquakes Wesnousky 2008

similar from Wells and Coppersmith

!!! But what if a fault in addition to the largest earthquakes that occur on it also has many smaller earthquakes that satisfy the Gutenberg-Richter distribution?! Intuitively, these smaller events will take up slip during the time between the biggest earthquakes and in turn make it take longer for the fault to accumulate sufficient slip to produce that biggest earthquake.! Or, saying another way, the total moment released between the occurrence of the maximum event will be greater than the moment solely of the big quake.! Expressing the Gutenberg-Richter distribution in Terms of Seismic Moment is first step in addressing these questions!!

Mo = µlwd And relationship between magnitude and seismic moment is Log Mo = 1.5 M + 16.1 (dyn-cm) Or generically Log Mo = dm + c can write magnitude in terms of seismic Moment M = (Log Mo c)/d And can substitute into Gutenberg Richter relationship Log N = a bm = a (b/d)(log Mo c) = a + bc/d (b/d)log Mo And with definition of Log N = 10 (a + bc/d (b/d)log Mo)

N = 10 (a + bc/d (b/d)log Mo And because Log e 10 =2.3 and A b-c = A b /A c can rewrite as N = exp 2.3(a + bc/d) Mo b/d N(Mo) = ηmo B where η=exp 2.3(a + bc/d) and B = b/d Which is the Gutenberg-Richter magnitude frequency distribution written in terms of seismic moment or the Moment-Frequency Distribution Which provides capability to address question of what is rate of seismicity on a fault if it behaves according to the Gutenberg-Richter distribution rather than than the Characteristic or Max-Moment model whereby all fault slip occurs in largest earthquake.

N(Mo) = ηmo B where η=exp 2.3(a + bc/d) and B = b/d Question: what is total seismic moment release (ΣMo) resulting from all earthquakes during the repeat time T max of the largest event Mo Max earthquake on a fault. Define moment-density distribution by taking derivative w.r.t. Mo dn(mo)/dmo = -η max B Mo (B+1) η max is defined such that N(Mo max ) = 1 = η max Mo B And now one can integrate the moment density distribution from the limits of smallest expected earthquake to largest to calculate the total seismic moment release during the return time of one Tmax earthquake

dn(mo)/dmo = -η max B Mo (B+1) η max is defined such that N(Mo max ) = 1 = η max Mo B

both fault models in a nutshell we can apply them Characteristic or Max-Mo Model B-value model o T = Mo/Mo return time of Biggest Quake return time of smaller Quake

Here is example of effect of small earthquakes - this example assumes max event similar to that expected on San Andreas and a slip rate chosen to result in 100 year repeat time for Max-Mo calculation.. impact is most significant when considering seismic hazard

Geologic data holds record of prehistoric earthquakes. Historical Record is short - which is problematic Problem is to extract the record for use in seismic hazard analysis Requires knowledge of which fault model is correct Japan as example Active faults mapped - slip rates estimated for all Long historical record

B-value model Max-mo model-solid symbs this particular test suggests Max-mo or Characteristic earthquake model is better

But what does it mean when one considers the Gutenberg-Richter distribution? In essence, it shows that the Gutenberg-Richter distribution of earthquakes that is always observed over regions need not apply to individual faults. Individual faults need not display a Gutenberg Richter distribution of seismicity over the time period between large earthquakes and analogously, the Gutenberg Richter distribution results from the combined effect of the population of fault lengths in the earths crust and the slip rates of each

Another way to look at the problem- what is the magnitude frequency distribution on a single fault

We know that the b-value is about 0.9 for all of S. Cal

Paleoearthquake and fault slip rate data exist for all these faults. They provide an independent measure of the repeat time between large earthquakes on each fault which can be compared to the predictions of earthquake statistics along each fault.

The seismicity along each fault within each box can be counted and used to predict the average return time of the largest earthquakes. Estimates of the expected size (slip or moment) and average recurrence time of large earthquakes have been estimated from Geology for these faults So we can compare what geology says to what seismology says..

Main Message of Lecture 1. Seismicity over broad areas satisfies Gutenberg Richter Distribution. 2. Slope of Gutenberg-Richter Distribution always near value of 1 3. Seismicity on individual Faults appears NOT to satisfy Gutenberg-Richter distribution. 4. Seismic Moment is measure of earthquake size that may

Papers that discuss principles of lecture in more detail. Wesnousky, S. G., C. H. Scholz, and K. Shimazaki (1983), Earthquake Frequency Distribution and the Mech Wesnousky, S.G. (1990), Seismicity as a Function of Cumulative Geologic Offset: Some Observations from S Wesnousky, S. G., The Gutenberg-Richter or Characteristic Earthquake Distribution, Which is it? (1994) Bul Wesnousky, S. G. Crustal Deformation Processes and the Stability of the Gutenberg-Richter Relationship, Bu