Monthly Archives: October 2014

Class Summary – 15 Oct – Earthquake Descriptive Statistics

I began with a presentation that reviews some basic earthquake spatial and temporal patterns, and introduced terms like foreshock, mainshock, aftershock.

Students worked on an in-class activate related to the 2011 Tohoku earthquake. They performed a Gutenberg-Richter analysis of the aftershock magnitudes, an Omori’s Law analysis of the timing of the aftershocks, and used the spatial distribution of the aftershocks to estimate the area of the rupture, and then with the seismic moment, they estimated the mean-slip in the earthquake.

The Gutenberg-Richter Relation: $$log_{10}[N(M)] = a – b M$$ where \(N(M)\) is the number of earthquakes with a magnitude greater than \(M\), and \(a\) and \(b\) are constants. The productivity \(a\), depends on the duration of the catalog, and is equal to the logarithm of the number of earthquakes with \(M≥0\). The “b-value” is the negative of the slope and is usually a parameter of geologic/tectonic interest. Typically, \(b\sim1\), but the number will vary depending on the type of earthquake catalog being analyzed or earthquake sequence being investigated.

Omori’s “Law” describes the typical temporal evolution of an aftershock sequence, $$n(t)=\frac{K}{(c+t)^P}$$ which is the equation of a hyperbolic decrease in the number of aftershocks following an earthquake. The values of \(K\),  \(c\), and  \(P\) and \(t\) represents time. What does hyperbolic indicate? The number of aftershocks in a fixed time interval (think day) decreases very fast early in the sequence, but then decays slowly much later. There are many aftershocks early on, the number of aftershocks decreases quickly, but a few hang around for a long time. The values of  \(K\) and  \(c\) depend on the size of the mainshock (the amount of strain energy release). Typically,  \(P\sim 1\). Not all sequences follow this form, but even then, Omori’s Law is a good reference to use to compare and contrast earthquake-sequence temporal histories. Note, Omori’s Law does not say anything about when the largest aftershock occurs, just the numbers of aftershocks.

Bath’s law is relatively simple – in a typical earthquake sequence, the magnitude of the largest aftershock is roughly 1.2 magnitude units below the magnitude of the mainshock. Not all sequences are typical, but this is the usual pattern.

In-class activity: Students started an analysis of aftershock data from the Great 2011 Tohoku Earthquake.

Class Summary – 13 October – Earthquake Size – Seismic Moment

Students completed the in-class activity on estimating the Lg magnitude of the 2011 Virginia earthquake.

Then I completed the presentation on earthquake size, discussing a more physically-based metric, the seismic moment. Seismic moment is a quantity that combines the area of the rupture and the amount of fault offset with a measure of the strength of the rocks – the shear modulus.

Seismic Moment = (Shear Modulus) x (Rupture Area) x (Fault Offset)

Usually we measure the moment directly from seismograms, since the size of the very long-period waves generated by an earthquake is proportional to the seismic moment. The physical units of seismic moment are force x distance, such as newton-meter or dyne-centimeter. These are the same units as energy, but we use the explicit N-m or dyne-cm forms to distinguish the physical character of the quantity as a moment.

For scientific studies, the moment is the preferred measure we use to compare earthquake size since it has fewer limitations than the magnitudes, which often reach a maximum value (we call that magnitude saturation).

To compare seismic moment with magnitude we use a formula constructed by Hiroo Kanamori of the California Institute of Technology:

Mw = log10(Seismic Moment)/1.5 – 10.7

where the units of the moment are in dyne-cm. We call Mw the moment magnitude. Note that moment is often reported in N-m (newton-meters), to convert to dyne-cm, multiply the N-m value by 107.

Class Summary –10 Oct – Magnitude

I began a lecture on earthquake size; we will finish next class. Get the notes and read the appropriate sections in the class texts if you need more background. Some of the material online is very good on these subjects. Detail can be found in wikipedia, for example.

The earliest metrics of earthquake size were based on their impact upon societies. The suffering, social impact, and economic impacts are recorded historically for many earthquakes. Since the 1700’s we have tried to be more systematic about measuring earthquake size using the scale of damage to human-made structures. In the late 1800’s the development of the seismometer led to classic magnitude scales (developed in the 1930’s) and satellite geodesy has led to additional measures based on ground deformation.

Magnitude is based on observations of the size of the ground shaking produced by an earthquake. The simple idea is that larger earthquakes produce larger vibrations. To measure the ground vibrations precisely, we use seismometers. Seismometers are sensors used to measure ground motion; seismographs are instruments used to record the motions as a function of time; and seismograms are plots of the motion versus time.

Magnitude is usually computed using two observations from each seismogram – the largest amplitude and the period of the motion. We will talk about period and frequency later.  The magnitude of the earthquake can be measured using a number of different parts of a seismogram, and thus we have a number of different types of magnitudes.

Richter originally used the logarithmic (base 10) difference between the observed largest ground motion and a reference value $$M_L = log_{10} A – log_{10}~A_0~,$$where \(A\) now represents the amplitude of the ground motion. The subscript in \(M_L\) stands for “local”, and we call this measure, local magnitude because originally this measure was local to southern California, where Richter was working. The above formula works fine if the earthquakes are the same distance away. However,  we must also account the fact that vibrations observed farther from an earthquake are naturally smaller. Richter recognized that some type of distance-based correction would be necessary if seismometers at all distances were to produce consistent estimates for the same earthquake.

Having even a repeatable, easy-to-measure estimate of earthquake size is very valuable if you want to compare earthquakes. Richter recognized this quickly and in collaboration with a colleague at CalTech, Gutenberg, he developed other magnitude measures that were appropriate for waves recorded at greater distances from the earthquake.

In-class activity: Students started a calculation of the magnitude of the 2011 Virginia Earthquake.

Mid-Term Exam Notes

The mid-term exam on 17 October will consist of

  • A section of multiple-choice questions that will assess your appreciation of certain facts and your understanding of certain processes associated with earthquakes that were covered in the readings and/or the class lectures and activities.
  • A section that assesses your data analysis skills. I will provide data and ask you to perform tasks similar to some of those that you performed as part of the in-class activities.
  • A section of questions that you will answer with a concise ‘essay’.

I will load some practice questions for the multiple choice into ANGEL sheet this weekend so that you can review the readings and the material for the multiple choice assessment. You should study the in-class activities to review how you performed the various data analyses. I will provide equations as needed with one exception, I expect that you know that $$ time = \frac{distance}{speed}~.$$

The following is the list of potential essay questions. These questions should be answered with five-to-ten good sentences. I strongly encourage you to include a hand-drawn  cartoon or simple graphic, a timeline, etc. to help you explain the topic. You must support your answers with specific information from class notes or the readings. I will not ask all of these, but I may ask any of them.

  1. What should you do before, during, and after an earthquake (try here)?
  2. What is science and what is the relationship between science and mathematics?
  3. Summarize the history of the Universe and the origin of the elements that constitute everything we know.
  4. What is Plate Tectonics and what relationship does it have to earthquake occurrence (don’t forget to mention heat transfer)?
  5. What is an earthquake?
  6. Discuss the various measures of earthquake size and how they developed (historically) and how they are related (human impact, intensity, magnitude, moment).

Class Summary – 08 Oct – Measuring Ground Deformation – Seismograms

Earthquakes are associated with the deformation of the lithosphere, so it makes sense that the way to observe them is to measure ground deformation. We use several different technologies to make these measures. We can classify them into two overlapping types of measurements based on the “frequency” or “period” of the signals. Slow deformations are generally measured using geodetic methods such as GPS, InSAR, or survey methods. Quick deformations are waves that are usually observed with seismometers.

A seismogram is a plot of the motion of the ground with time. The motion may be ground displacement or its derivative, ground velocity, or its derivative, ground acceleration.

Modern seismograms are digital samples of the position of the ground, much like an mp3 file is a sample of the sound associated with an audio recording.

Seismometers operate 24 hours a day 7 days a week around the globe and many return their data to centralized, openly available web sites from which scientists (or anyone with an internet connection) can acquire the signals. You can browse a list of some of the near-realtime, openly available seismometers at http://www.iris.edu/mda/_REALTIME or see them on a map at http://www.iris.washington.edu/gmap/_REALTIME.

Students completed an in-class activity on reading seismograms.

Class Summary – 03 Oct – Plate Motion and Quiz

Students completed an in-class activity related to the motion of the Pacific Plate over the Hawaiian hot spot. Using the ages of volcanic rocks found on the islands and seamounts that form the Hawaiian-Emperor Seamount Chain, students estimated the speed of plate motion (relative to the hot spot).

Then we “played” a team-based quiz game where I asked review questions and students answered.