Description of earthquake aftershock sequences using prototype point patterns

Transcription

1 Describing aftershocks using prototype point patterns.1 Description of earthquake aftershock sequences using prototype point patterns Frederic Paik Schoenberg 1 and Katie Tranbarger 1 Abstract The behavior of a typical aftershock sequence from the global Harvard earthquake catalog is characterized using prototype point patterns. These prototypes are used not only to summarize the data but also to identify outliers and to classify regions into groups of earthquakes ehibiting similar aftershock behavior. Key words: aftershocks, earthquakes, point process, prototypes. 1 Department of Statistics, 8142 Math-Science Building, University of California, Los Angeles,

2 Describing aftershocks using prototype point patterns.2 1 Introduction. a) ZZQ: Describe purpose characterization of earthquake aftershock patterns. Cite similar works. 2 Data ZZQ: describe the data, and the Harvard catalog. Mainshocks: m w 7.5 to 8.0, depth < 70km. Limited to those without shallow events with m w 7.5 within 2 yrs or m w 8.0 within preceding 4 yrs. Aftershocks: m w 5.5, depth < 70km days after mainshocks, distance < 100 km. Molchan, G., Kronrod, T., and Panza, G. F. (1996,1997). Kagan (2003, 2004). There are 49 earthquake sequences in all, each containing an average of 5.47 aftershocks. The times of the aftershocks in each sequence are shown in linear and logarithmic scales in Figures 1 and 2, respectively. 3 Point Process Distances and Prototypes A natural definition of a prototype is the point pattern Y minimizing the sum n d(x i, Y ), (1) i=1 where d is some distance function, that is d(x, Y ) is the distance between two point patterns X and Y. Many choices of d are possible. One option, used successfully in the description of

3 Describing aftershocks using prototype point patterns.3 aftershock sequences sequence number days since mainshock Figure 1: Times of aftershocks, linear scale neuron firings by Victor and Purpura (1997), is to define d(x, Y ) as the minimal cost needed to transform point pattern X into pattern Y using a series of elementary operations. These elementary operations including adding a point to X, which is given some cost p a, deleting a point from X, which given cost p d, and moving a point of X by some amount of time, which is given a cost of p m. Note that p a must equal p d in order for d to be a symmetric distance function, since otherwise d(x, Y ) d(y, X). Provided p a = p d and p a, p m 0, d is a well-defined distance metric (see Victor and Purpura, 1997). Relate to earth mover s distance and likelihood distance. ZZQ Some problems of how to determine the distance between two point patterns and how to

4 Describing aftershocks using prototype point patterns.4 aftershock sequences, logarithmic time scale sequence number days since mainshock Figure 2: Times of aftershocks, log scale find the prototype for a collection of point patterns are discussed in the Appendi. We list here a few key facts facilitating the solution to these problems. The first is that the distance between two point patterns X and Y is simply given by the sum of the penalties associated with adding or deleting points, plus the sum J j=1 p m (j) y (j), where { (1),..., (J) } and {y (1),..., y (J) } are the sorted remaining points of X and Y, respectively. Hence the problem of determining which point of X gets moved to which point of Y is not an issue, and finding the distance between X and Y simply amounts to deciding which points are added or deleted and which are moved. The second fact worth mentioning is that, for a given collection of point patterns, each point of the prototype must always be a point in one of the point

5 Describing aftershocks using prototype point patterns.5 patterns in the collection. Therefore one way of finding the prototype is by searching over all combinations of points in the collection. As noted in Victor and Purpura (1997), one of the main difficulties inherent in prototype analysis is that of determining the penalties for adding, deleting, and moving points. Since only the relative values of these parameters are relevant, we hereafter fi the moving penalty p m at unity. The ratio of the adding/deleting penalty to p m can have a large impact on the resulting prototypes. In general, the prototype contains a point t if an aftershock is present at time t (or a nearby time) in a fraction of at least p a /(p a + p d ) of the mainshocks in the dataset. Indeed, if a given earthquake has no aftershock in the interval (t p a, t + p a ), then the occurrence of a point in the prototype at time t results in an increase by an amount p a in the distance from the point pattern associated with this earthquake to the prototype. On the other hand, for each earthquake with an aftershock near time t, the lack of a point near time t in the prototype causes the distance from the point pattern to the prototype to increase by the amount p d. The prototype will contain a point at t if it is more economical to do so, i.e. if n 0 p a n 1 p d, where n 0 and n 1 are the numbers of earthquakes with and without aftershocks near time t, respectively. Hence if p a is small, then the prototype will basically contain only those points occurring in at least p a /(p a + p d ) of the point patterns in the dataset. The fact above can be helpful in determining appropriate values of p a and p d. If p a = p d, then the above ratio is 1/2, but it may be of interest to include points in the prototype even though those points may occur in less than half the point patterns in the dataset. If one desires that the prototype Y should represent all aftershocks occurring in at least some

6 Describing aftershocks using prototype point patterns.6 proportion p of the aftershock sequences in the dataset, then one may set the ratio of p a to p d to p/(1 p). The prototype may still be defined as before, i.e. as the point pattern Y minimizing the sum i d(x i, Y ); the fact that the function d is no longer symmetric and thus not a proper distance function is immaterial. In addition to the determination of prototypes, the (possibly non-symmetric) distance function d can also be used for identification of outliers or clusters. We propose classifying certain individual earthquake sequences as outliers if their distance from the prototype is unusually large. Further, subgroups of earthquake sequences can be classified into clusters, based on the principle of minimizing the total distance from the earthquake sequences to their respective cluster prototypes. 4 Results. Figure 3 shows the times of events in the prototypes for the datasets of linear and logarithmic aftershock times described in Section 2, with symmetric distance function specified by p a = p d = 6. In a rough sense, one may interpret these prototypes as representing the times of the collection of aftershocks occurring in the majority of aftershock sequences. Figure 4 shows a histogram and boplot of the distances from each of the point patterns to the prototype with symmetric distance function. One sees that there are two outliers, with abnormally large distances to the prototype of over 8000, while by contrast, most of the other aftershock sequences have a distance to the prototype of 4000 or less. Collectively the mean distance is 3000 and the standard deviation is approimately ZZQ: put in eact numbers here.

7 Describing aftershocks using prototype point patterns.7 Prototype aftershock sequence, linear scale days since mainshock Prototype, logarithmic time scale days since mainshock Figure 3: Temporal prototype aftershock sequence, linear and log scale One problem with the prototype in Figure 3 and its capacity to adequately represent a typical aftershock sequence in this dataset is that the prototype has only four points, whereas the average point pattern in the Harvard dataset considered here contains nearly 5.5 aftershocks. ZZQ indicate the median and SD. The prototype contains slightly atypically few aftershocks. Figure 5 shows the five aftershock sequence closest to the prototype, in terms of the distance described in Section 3. Several of the sequences, particularly the Honshu and Papua New Guinea sequences. Figure 6 shows the five outlier sequences, i.e. those with greatest distance from the prototype. Not surprisingly, these include sequences containing

8 Describing aftershocks using prototype point patterns.8 Frequency distance from prototype distance from prototype Figure 4: Histogram and boplot of distances from prototype many aftershocks, such as the New Ireland 2000 and Kermadec Island 1986 sequences. As noted in Section 3, the distance function used to define a prototype may contain penalties not only for changing the times of points, but also for changing their magnitudes. Using sliding penalties for both time and magnitude, one obtains a temporal-magnitude prototype for the 49 aftershock sequences; this two-dimensional prototype is depicted in Figure 7. One sees the characteristic decline in moment magnitude of the aftershocks as the temporal distance from the mainshock increases, in accord with the Omori law. It has been argued recently (ZZQ: embellish this) that earthquake activity leading up to

9 Describing aftershocks using prototype point patterns.9 5 sequences closest to prototype oo o o Prototype Honshu 1978 mw 7.62, d=166 Papua N.G mw 7.58, d=261 Costa Rica 1991 mw 7.61, d=706 Mariana Isl mw 7.74, d=714 Honshu 1983 mw 7.71, d= days since mainshock Figure 5: Five sequences closest to prototype an earthquake should mirror its aftershock activity; this idea is embodied in recent reports of so-called accelerated moment release (AMR). Prototypes can be used to shed some light on this, in typifying a typical fore-shock sequence which can then be compared to a typical aftershock sequence. Figure 8 shows the times of foreshocks with m w 5.5 for two years prior to each mainshock described above, with the same spatial constraints as used for the aftershocks as described in Section 2. One sees that for the vast majority of the major earthquakes, there were no foreshocks at all in the two years prior, and in fact the prototype for the foreshock sequences contains no points whatsoever.

10 Describing aftershocks using prototype point patterns.10 5 outlier sequences oo o o Prototype New Ireland 2000 mw 7.5 Kermadec Isl mw 7.7 Molucca Psg mw 7.5 Minahasa 1990 mw 7.6 Sumatera 2000 mw days since mainshock Figure 6: Five sequences furthest from prototype 5 Division of earthquake sequences into clusters Using the distance function associated with these penalty parameters, one may subdivide the earthquake sequences into clusters, so that within each cluster, each point pattern is optimally close to the prototype of its cluster. That is, we seek to divide the n = 49 point patterns into k clusters so that the sum of the distances from the sequences to their cluster prototypes is minimized. Obviously, when each aftershock sequence is its own cluster this sum is zero, so a parsimonious choice of k is required. Figure 8 shows the sum of these distances as a function of

11 Describing aftershocks using prototype point patterns.11 Prototype aftershock sequence with magnitudes magnitude days since mainshock Figure 7: Prototype sequence, with magnitude and temporal sliding penalties included in the distance function k. ZZQ 6 Discussion. ZZQ.

12 Describing aftershocks using prototype point patterns.12 preshock sequences sequence number days before mainshock 7 Appendi. Figure 8: Foreshock sequences Suppose X and Y are two temporal point patterns on [0, T ] such that X contains points 1,..., m and Y contains points y 1,..., y n, with m n. Viewing X(t) = ma{i : i t} and Y (t) = ma{i : y i t} as processes, we may investigate conventional distances between the two point patterns as in the following result. Theorem 1. Proof. T 0 m X(t) Y (t) dt = i y i + n (T y i ). (2) i=1 i=m+1

13 Describing aftershocks using prototype point patterns.13 Let a i = ma{ i, y i }, for i = 1,..., m. Let a i = T for i = m+1, m+2,..., n, and a 0 = 0. Similarly let b i = min{ i, y i }, for i = 1,..., m. Let b i = y i for i = m + 1, m + 2,..., n, and b 0 = 0. Let c i,j = y j if a i = i, and c i,j = j if a i = y i. Thus c i,i = b i, for i = 1,..., m. (In the case where i = y i = a i, the value of c i,j is irrelevant.) With this notation, we may write T 0 X(t) Y (t) dt = = = = m a i+1 i=0 a i n i=1 m i=1 m i=1 X(t) Y (t) dt j i:c i,j a i (a i ma{c i,j, a i 1 }) [(a i a i 1 ) + (a i 1 a i 2 ) (a i k c i,i )] + (a i b i ) + n i=m+1 n i=m+1 (T b i ). (T c i,i ). Theorem 2. Suppose m = n, and that p a = p d =, i.e. no points may be added or deleted. Then the total distance between X and Y is given by n i y i. Proof. (ZZQ: See Katie s writeup.) Theorem 3. For any collection of point patterns, X (1), X (2),..., X (n), where X (i) is a point pattern consisting of the points X (i) 1,..., X (i), there eists a prototype P consisting eclusively of points p i such that p i = X (i) j for some i, j. Proof. (ZZQ: See Katie s writeup.) k i=1 An immediate consequence of Theorems 1 and 2 is that if the two sequences X and Y have the same length, i.e. if m = n, then the integrated absolute difference between the two processes X and Y is proportional to the total sliding distance between the two point

14 Describing aftershocks using prototype point patterns.14 patterns, and the proportionality constant is simply the sliding penalty, p m. Hence if p m is very small compared to the addition/deletion penalty, so that addition and deletion is essentially not permitted, then the total distance between two point patterns of equal length is simply p m times the integrated absolute difference between the two processes.