Strings and things for locating Earthquakes

When one hears that an earthquake has occurred, one of the first questions is “where was it?” For the general public, this question oftentimes determines the importance of another important question, “how big was it?” Once learning the location and size, some might wonder how this information was determined. We present here an interactive, three-dimensional analog computer that uses a map, strings, and a timedistance scale to find the location of an earthquake (lat, long and depth) based on seismic wave arrival times. We have also developed a set of lesson plans to present the ideas used for locating earthquakes to grade and high school students and the general public. The device is suitable for both permanent mounting in a science museum or can be easily transported to schools or other places. The analog earthquake locator is located in the Public Earthquake Resource Center, PERC, at the Center for Earthquake Research and Information at The University of Memphis and is taken to classrooms for education outreach. Introduction In many cases, the public’s first concern after hearing that an earthquake has occurred is its location. For those who are also interested in how the answer is obtained, we have built a three dimensional analog computer that can be used not only to locate Proceedings of the 2007 Earthquake Engineering Symposium for Young Researchers paper 04 1 Seattle, Washington | August 8-12, 2007 earthquakes but also illustrate many of the ideas associated with this process. We have installed this analog computer in the Public Earthquake Resource Center, PERC, at The University of Memphis, but it can be easily transported to classrooms or other locations. We have also developed a set of instructional materials which we use with the locator in presentations to grade school and high school students. The most popular introductory method for showing how earthquakes are located is based on drawing circles about seismic stations on a map or globe (Figure 1). The circles’ radii are determined by using the difference between the arrival times of the P and S waves. This time difference is converted to a distance that can be measured on the map. Undergraduate Physical Geology laboratory exercises and presentations about earthquakes to grade and high school groups typically use this method with a compass, regional map, and data from at least three seismic stations. By drawing circles with appropriate radii around the seismic stations, the surface location, or epicenter, of the earthquake is found where the three circles intersect at a point. This demonstration “works” perfectly for earthquakes at the surface and synthetic, noiseless P and S wave arrival time data. Most earthquakes, however, do not occur at the surface. In this case, the circles drawn on the map will not intersect at one point. We can fix this problem if we modify the method whereby the radii found from the Ts-Tp arrival times are used to create spheres in three-dimensions (Figure 2), instead of two-dimensional circles on the surface. We now find that there are two points of intersection of the three radii. Both points are equidistant to the earth’s surface. One is located within the earth and the other above the ground surface. The point of intersection beneath the surface is the focal location of the earthquake; hence the distance from the surface to the point of intersection above the Proceedings of the 2007 Earthquake Engineering Symposium for Young Researchers paper 04 2 Seattle, Washington | August 8-12, 2007 earth's surface also denotes the depth of the earthquake. The radii represent the ray paths of the seismic rays traveling through a homogeneous media. The analog earthquake locator forms the intersection of the radii of spheres in three dimensions above the surface of a map, demonstrating not only the epicentral location on the surface directly beneath the intersection, but the depth of the earthquake as well. Analog Computing The term analog is now oftentimes used to mean continuous, as opposed to discrete. While many analog computers use continuously varying quantities to perform their calculations, this is not a requirement for analog computing. The fundamental principal behind analog computing is based on the observation that many seemingly different physical systems can be described mathematically by equations of the same form, differing only in the interpretation of the parameters and variables. If two systems are equivalent mathematically, we can investigate the behavior of one system in terms of the other. Electrical and plumbing circuits are an example of analogous systems. In these two systems the mathematical relationships between the conservation of current or fluid flow, or the voltage and pressure, electrical current and water flow, and resistance to electrical or fluid flow are the same. One could therefore use electric circuits to simulate plumbing systems. Analog electrical computers were a popular method for solving large differential equations before the digital computer revolution. Another example of an analog computer, this time a mechanical one, is the simple slide rule. There are an incredible number of interesting non-electrical analog computing devices, such as the famous Norden bombsight of WW II. Before the advent of the digital computer, the Proceedings of the 2007 Earthquake Engineering Symposium for Young Researchers paper 04 3 Seattle, Washington | August 8-12, 2007 USGS used the circle method to locate teleseisms by finding the intersection of circles on a globe (Figure 3). The Analog Earthquake Locator The analog computer method for locating earthquakes provides students and visitors of the PERC at CERI a hands-on visual presentation to help understand how to determine earthquake focal locations in three-dimensions. It also introduces some of the physical ideas, such as ray paths, associated with this process. The locator was inspired by a device built and used by the late Argentine seismologist F. Volponi, of the Instituto Geofísico Sismológico Zonda (now Instituto Sismológico "Fernando Volponi”) of the Universidad Nacional de San Juan in San Juan, Argentina. S. Sacks suggested the device to Volponi to address the issue of locating intermediate depth earthquakes beneath San Juan, where the circle method fails due to the 100+ km depth of earthquakes whose epicentral distances for events immediately beneath the network were oftentimes much smaller than their depths. The size of our device is scaled for determining the epicenter and depth for earthquakes within a small part of the New Madrid Seismic Zone, where earthquake depths up to 21 km can be equal to or larger than the epicentral distances. Figure 4 shows a schematic of the locator construction and use. The front of the display contains a set of educational materials and instructional activities, a map showing four seismic stations to be used in the exercises, a time-distance scale with adjustable markers, and several seismograms. The time-distance scale shows both the Ts-Tp times and the corresponding distances traveled by the seismic waves. The distance portion of the time-distance scale is at the same scale as the map and the corresponding Ts-Tp times Proceedings of the 2007 Earthquake Engineering Symposium for Young Researchers paper 04 4 Seattle, Washington | August 8-12, 2007 are found from the P and S wave velocity model. The map is approximately 0.5 m by 0.5 m and covers a 40 km area over the thrust arm of the New Madrid seismic zone. In this region the seismic stations have a spacing of 10 to 30 km and the earthquakes vary from 5 to 21 km in depth. The region includes Reelfoot Lake, located in the northwest corner of TN, and a section of the Mississippi River. Holes are drilled through the map at four seismic stations (shown by stars). A metallic loop is connected to the end of a string that can be pulled out of the hole, away from the surface of the map. The string passes over the time-distance scale where markers show the distance the end of the string has moved. Retracting mechanisms mounted to the back of the locater keep the strings taut and retract the string when the end is moved back towards the hole. In Volponi’s original analog locator, the map was mounted on the top of a table and short sections of pipe were slipped onto the four table legs. The strings were connected to these pipes which served as weights to maintain the tension as the pipes slid up and down the table legs with the change in the position of the other end of the string. The components of the locater are very simple. It consists of a wooden frame and board, a laminated and mounted map, a scale marked in Ts–Tp time and distance, strings, metallic loops, key-ring retractors, and a device for holding the ends of the strings together (Figure 5). The time markings on the scale are based on the equation: D = [(Vp Vs)/(Vp-Vs)](Ts-Tp) D = VT Proceedings of the 2007 Earthquake Engineering Symposium for Young Researchers paper 04 5 Seattle, Washington | August 8-12, 2007 where D is the distance of one ray path, Ts is the S wave arrival time, Tp is the P wave arrival time, Vs is the S wave velocity, and Vp is the P wave velocity. For this implementation of the locater we used a homogeneous half space model for the New Madrid seismic zone with a Vp of 6 km/sec and Vs of 3.5= Vp/(√3) km/sec. The markers are placed on the string associated with each seismic station based on the Ts-Tp time for that station. The position of the marker on the string is also equal to the distance the seismic waves traveled from the focus to the respective seismic station; this distance can be read off the distance scale. By using Ts-Tp times, the locator can be operated several ways. In the first method the markers for the Ts-Tp arrival times are placed at the appropriate positions on the scale for each seismic station, the ends of the strings are pulled together and joined, and this union is moved around in 3-D until the Ts-Tp markers line up on the time scale (Figure 6). Since the markers are initially at the distance the seismic waves traveled to each station, they will line up at zero. In most cases, the union of the ends of the strings must be pulled away from the map for the markers to line-up. When the markers line up, the position of the union represents the earthquake focus and the strings signify the ray paths to each station. The distance of the union of the strings above the map, obtained using a measuring device that has the same scale as the map, gives the depth of the earthquake. The point on the map directly beneath the union is the epicenter. This is the method one would use if the analog earthquake locator was the principal method of locating an earthquake. The next method we present is more didactic but still uses the distance traveled by each of three seismic waves through conversion of the Ts-Tp arrival times to distance Proceedings of the 2007 Earthquake Engineering Symposium for Young Researchers paper 04 6 Seattle, Washington | August 8-12, 2007 through the time-distance scale. Start with all the strings retracted so their ends are at the seismic stations. Next, place the time markers at the appropriate places on the strings using the Ts-Tp scale. Then pull the end of each string away from the station to bring its individual marker to zero. Now lock or hold down all the strings. Finally connect the ends of the strings together to find the solution. Using data from three stations, the solution will be the unique place where all the strings are taut. As with the first method, the location will be above the surface of the map, indicating the depth of the earthquake’s focus and the epicenter beneath the intersection. A variation of the previously discussed method, with additional teaching opportunities, is to start manipulating only one of the locked strings. Demonstrate that this defines a hemisphere above the map by moving the string in all directions while keeping it taut. A similar hemisphere could also be defined below the map. Each string defines its own hemisphere about its seismic station. Now one can hold the ends of two of the strings together and find that this limits the allowed motion of the union of the two strings to following an arc. This arc is half of the circle that is defined by the intersection of two spheres. Each pair of strings defines an arc. Finally, by connecting all three strings together we find that there is a unique place (actually there is also one below the map) where all three strings are taut and this single place, where all the strings intersect, is the focus and hypocentral location of the earthquake. This point is the intersection of the three spheres and the three arcs of circles we round with individual strings and pairs of strings. If we have more data (more seismic stations with strings strung to them) we will again find a place where all the strings are taut, but only if the data are perfect. If the data are not perfect, we will always be able to keep at least three strings taut at any time. Proceedings of the 2007 Earthquake Engineering Symposium for Young Researchers paper 04 7 Seattle, Washington | August 8-12, 2007 By changing the time-distance scale, the analog computer can also be used to locate earthquakes using P wave arrival times only. The time-distance scale is changed from Ts-Tp times and the corresponding distances to a scale marked in P wave travel times and the corresponding distances P waves travel. Unlike the case for using Ts-Tp, where the velocity used to scale time to distance does not represent a physical velocity, the velocity used to generate the new scale is the actual P wave velocity. In this method one does not know the distance to the hypocenter beforehand so one cannot lock the strings at a fixed distance or know where they should line up. The direction of the time and distance marks on the scale now run the other direction and one places the time markers based on their relative P wave arrival times. The last arrival defines zero time and distance. The earlier arrivals are marked at their relative arrival times/distances ahead of the last arrival. The ends of the string are again joined and the union moved around until the markers line up. In contrast to the case of using Ts-Tp, the markers will now line up at some arbitrary position. The distance from the station with the latest P arrival time to the earthquake is read directly from the distance portion of the scale and the distances to the other stations can be determined by the relative arrival times and corresponding distances. For this method, P arrivals from at least four seismic stations are required to estimate the four parameters – latitude, longitude, depth, and origin time (There are actually six data measurements in the Ts-Tp method as we need both a P and S wave arrival data at each of the three stations. In addition in the Ts-Tp method, we are estimating one less parameter since the origin time is determined from the known distance to each station.). The concepts associated with using P wave arrival times only Proceedings of the 2007 Earthquake Engineering Symposium for Young Researchers paper 04 8 Seattle, Washington | August 8-12, 2007 are more advanced than those associated with the Ts-Tp method and would be suited for undergraduate or graduate class presentations. College level classes may benefit from demonstrations of other concepts that are more advanced than those presented with the museum display. The analog locator can be used to illustrate the idea of a “best fit” when one has more than the minimum number of data and one introduces noise or errors into the arrival time measurements. This can be demonstrated using four sets of Ts-Tp times. For perfect data, the four markers will all line up at zero. For data with errors, the user will have to find a “best” arrangement of the markers near zero. The analog locator can also be used to show how the determination of origin time and depth are coupled when one has only P wave arrival times. As the union is moved vertically over the epicenter, there is very little relative movement between the markers, so it is difficult to decide when they are best lined up. The device is also well suited to illustrate problems arising when the earthquake to be located is well outside the network. In this case, with either type data (Ts-Tp or P), one can move the union perpendicular to the line connecting the union to the stations a considerable amount with very little variation in the position of the markers. Explanatory and Educational Materials About 2000 people visit the PERC yearly and the average visitor is between the ages of 8 and 15. Those who visit the PERC learn about paleoseismology, plate tectonics, geology, earthquake hazards – especially in the New Madrid area, and how CERI records earthquakes in the New Madrid Zone. The PERC has several activities involving seismology, but no display dedicated to the subject of locating earthquakes. Our physical Proceedings of the 2007 Earthquake Engineering Symposium for Young Researchers paper 04 9 Seattle, Washington | August 8-12, 2007 analog model provides a hands-on, interesting and fun way to obtain a basic understanding of the process of locating earthquakes. Explanatory materials are displayed on the locator to give an introduction to the process of locating earthquakes. Given the diverse target audience of the PERC and the relatively complex nature of earthquake location, importance was placed on selecting development material that was both rich in content and suitable for both adults and children. The goal was to create a model that would appeal to the average visitor (8-15) and to visitors that we hope to do a better job of attracting (15-24). The introductory materials introduce the ideas behind locating earthquakes using the travel time of waves by analogy to the common method of estimating the distance to a lightning bolt by counting the seconds between the flash and the thunder. Several example seismograms are shown with the P and S arrivals marked. These seismograms are used to provide the data for the hands-on exercise. Finally there is a short discussion with a simple explanation of seismic waves, their velocities, the idea of ray paths, and the relation of the elements of the analog model to the actual physics of the problem. We have prepared additional materials that teachers can use to prepare their class before the visit and that teachers and students can take with them when they leave. Importance was placed on tying our subject matter to the State of Tennessee science standards. Earth and space science, physical science, geography, and science as inquiry are the main connections to the curriculum. Academic “bullets” (and grade levels) addressed in the display include: