The figures in this section can be recreated by downloading What_is_site_response. It requires Matlab.
Site response is the tendency of local geologic structure and property variations to alter the composition of seismic energy. The most significant form of site response to the field of earthquake engineering is site amplification, the phenomenon whereby ground energy is amplified at specific frequencies. The common structure that causes this effect is a sedimentary basin with soft, low density, low velocity overburden overlying hard, high density, high shear wave velocity basement. Some examples of cities built on this structure are San Francisco, Mexico City, Seattle and Boston among many others. Site response poses an issue for infrastructure and, by association, citizens of cities built on these basins due to the tendency of buildings to resonate with the ground. This can have devastating effects as was seen in the 1985 Michoacán earthquake in Mexico City and the 1989 Loma Prieta earthquake in San Francisco. Quantifying basin resonance is essential to develop resilient infrastructure in earthquake-prone areas.
Let’s think about site response in the context of music, specifically the guitar. It doesn’t matter if you’re a campfire guitarist, Jimi Hendrix or have never touched the instrument in your life, the idea of a guitar is the same: you pluck a string which vibrates and makes sound, whose degree of cacophony varies inversely with the hours of practice you have accumulated. There are two main ways to vary the note, or pitch of a guitar: 1) change the tension on the string or 2) shorten a string by placing your finger on a fret. Increasing the tension will raise the pitch and decreasing the tension will lower the pitch. Shortening the string (fretting the guitar up the neck) will raise the pitch, lengthening the string (fretting down the neck) will lower the pitch. Now bear with me here: in the context of site response changing the tension on a string means changing the stiffness of the geologic body, and changing length of the string means changing the thickness of the geologic body. So an increase in geologic stiffness (how hard the material is) corresponds to an increase in frequency and an increase in thickness of the geologic layer corresponds to a decrease in frequency. Now imagine a layer of soft (not stiff) soil over a hard (stiff) rock with earthquake energy coming from below. As with nearly everything in the natural sciences, energy is conserved across the boundary between the two layers. In order for energy to be conserved, the frequency content must change. Since the soil layer is less stiff than the hard layer and since less stiff means lower frequency, lower frequencies are amplified in the soft layer with respect to the hard layer. This is the fundamental theory of site response.
Let’s look at Mexico City as a case study to show site effects. We will use data at two stations from the Mexico City RACM network recorded during the 2017 Puebla earthquake. Mexico City is an ideal site for a case study in basic site response. It is built on the location of three historically shallow lakes, Texcoco, Chalco and Xochimilco. It was here that the Aztecs founded their capital City of Tenochtitlan in 1324. These lakes have since been filled in by windblown volcanic ash and are now characterized by compressible, high plasticity, high water content clays interspersed with horizontal lenses of sand and soil layers. This means that Mexico City is a bowl of hard rock filled with soft soil, a textbook amplifying geologic structure. This amplifying geology made zoning by fundamental site frequency and amplification essential for building code design. The City is separated into three geotechnical zones: hill, transition and lake. The hill zone (Zone I) is composed of rock and hard soil with some sandy deposits or soft clays. The transition zone (Zone 2) is shallower (< 20 m), with greater heterogeneity than the lake zone. The lake zone (Zone III) is split into four subcategories to improve resolution in the building codes. It has soft clays which amplify at frequencies depending on the depth to bottom of the basin (Figure 1).
Figure 1. Mexico City basin with site (CE32) and reference (TP13). The geotechnical zoning is indicated, Zone I is hills, Zone II is transition and Zone III is lake. The map is based off of Çelebi et al. (2017).
In Earthquake engineering practice, we can use two main techniques to look at which frequencies amplify and by how much. The first is to drill a borehole and put an instrument at the bottom and an instrument at the surface and compare the shaking between the two. The second is to put an instrument on some hard rock outside of the basin, called a reference site, and compare it to an instrument at the surface in the basin. This strategy approximates the shaking at the basin but is less direct than the first method. In figure 1, we’ve plotted a site, CE32, which is sitting on soft soil in the basin and a reference site, TP13, which is sitting on hard rock outside the basin and in figure 2, we’ve envision the profile of the subsurface between station TP13 and CE32. We will compare the two and see how frequencies are amplified between the inside and outside of the basin.
Figure 2. Cartoon profile between TP13 and CE32 exhibiting the basic structure of a sedimentary basin.
Let’s begin by comparing the 2017 Puebla earthquake time series at CE32 to the time series at TP13 of the same event. A “time series” is a record of ground motion with amplitude of shaking on the y-axis and time on the x-axis. Imagine a buoy floating in the sea. You could put a sensor in the buoy which measures how much it moves up and down, left and right and forward and backward. This would be the “times series motion” for that buoy and each direction is referred to as a “component”. A seismometer, the instrument that measures ground motion, does the same thing as our buoy example only instead of “left-right” and “forward-backward” a seismometer is generally placed such that its horizontal components are oriented north-south and east-west.
Figure 3. The time series for CE32 (in the left column) and TP13 (in the right column) with each component of motion. The units are in acceleration (m/s^2).
Let’s take a look at these time series. The soft site, CE32 is in the left column and the hard reference site, TP13 is in the right column. Two things stick out, 1) the amplitude and duration is greater at CE32 than at TP13 and 2) there is, on average, more low frequency energy in the CE32 time series than in the TP13 time series. To come to this conclusion yourself, look at the distance between peaks. The distance between each peak in the CE32 time series is greater than the distance between each peak in the TP13 time series. The CE32 time series is more spread out in time the TP13 time series. Both time series have low amplitude vertical components, so we can draw a tentative conclusion that the horizontal component is more significant for earthquake engineering purposes.
We are talking about frequencies amplifying though and thus far have only qualitatively from the time series said that frequencies are lower at CE32 than at TP13. How do we quantify the amplification of a frequency? The answer is the Fourier transform, an invaluable and theoretically beautiful tool in the sciences. The premise under which Joseph Fourier operated is that all continuous signals can be broken down into an infinite sum of sin waves of varying frequency and amplitude. In our context, we are working with discrete time series data, thus, we decompose it into a sum of finite frequencies using the “Fast Fourier Transform” or “FFT” a computer implementation which is often ranked as a “top ten most significant algorithm of the 20th century”. Basically what it does is take the time series in figure 3 and turns it into a set of frequencies of differing amplitudes. Imagine again strumming a chord on a 6-string guitar. If you recorded the chord and took the FFT, you would get a plot with 6 spikes corresponding to the 6 frequencies being strummed with each spike with an amplitude corresponding to how loudly it was played.
Let’s look at the frequency content of the times series from the Puebla earthquake at Mexico City stations CE32 and TP13 in figure 3. In the north south component at station CE32 in the top left position of figure 4, there are 3 distinct spikes in the frequency content at around 0.25, 0.75 and 1.25 hz. In the corresponding north-south component at station TP13 in the top right position of figure 4, there are also some spikes but they are of lower amplitude. The same is true for the east-west components of each station with some small differences. The vertical component of both CE32 and TP13 are each low amplitude and similar looking to one another, thus we reinforce our former hypothesis that the vertical component is not as significant as the horizontal component in earthquake engineering studies.
Figure 4. Frequency content of the 2017 Puebla earthquake at Mexico City soft station CE32 (left column) and hard reference station TP13 (right column).
If you remember from our original definition of site response, we are looking for amplification at specific frequencies. Analyzing figure 4, we were able to see that certain frequencies at CE32 have higher amplitude than the corresponding frequency at TP13, but how much? Well, what is an amplification? An amplification is really just a ratio of one thing to another, so to determine the amplification of CE32 with reference to TP13, we simply divide the frequency content of CE32 by the frequency content of TP13. This is called a spectral ratio and is used often and in earthquake engineering as well as many other engineering disciplines. In the case of this example we are using a “simple spectral ratio”, which is the ratio of a soft site to a hard reference site outside of the basin. Other examples of spectral ratios are the “borehole spectral ratio” which is the ratio of a surface instrument to a borehole instrument and a “horizontal to vertical spectral ratio” which is the ratio of the horizontal component of motion to the vertical component of motion.
Figure 5. Spectral ratios of soft station CE32 with reference to hard station TP13 in Mexico City for the 2017 Puebla earthquake. These represent the frequency amplification of each of the three components during the event.
