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There are a small number of extremely large earthquakes with magnitude greater than 7 each year, for example approximately 13 in 2000, and 5 in January of this year.
Here is the code which generates this applet:
// Sandpile.java
import comphys.graphics.*;
import comphys.Easy;
import java.awt.*;
import java.awt.event.*;
public class Sandpile extends Animation {
int L = 20; // length of pile
int grains; // total grains added
int lost; // grains lost off edge
int[] height; // height
int[] slope; // local slope
int[] change; // change in height in toppling event
int criticalSlope = 1; // topples if slope > critSlope
boolean random; // add grain at random position
boolean unstable; // pile is unstable
boolean[] move; // move unstable grains to neighbor sites
int t; // time step
int[] N; // accumulate number of topplings
int[] E; // accumulate avalanche sizes
int maxHeight; // for plotting
void initial () {
if (height == null || height.length < L) {
height = new int[L];
slope = new int[L];
change = new int[L];
move = new boolean[L];
N = new int[L];
E = new int[L];
}
for (int i = 0; i < L; i++) {
height[i] = slope[i] = change[i] = 0;
move[i] = false;
}
for (int i = 0; i < N.length; i++)
N[i] = 0;
for (int i = 0; i < E.length; i++)
E[i] = 0;
maxHeight = L;
unstable = false;
grains = lost = 0;
t = 0;
}
int topplings;
int unstableSites;
void checkStability () {
unstable = false;
for (int i = 0; i < L; i++) {
if (slope[i] > criticalSlope) {
unstable = move[i] = true;
++unstableSites;
} else {
move[i] = false;
}
}
if (unstable) {
++topplings;
} else {
if (unstableSites > E.length - 1) {
int len = E.length;
int[] temp = new int[unstableSites + 10];
for (int i = 0; i < len; i++)
temp[i] = E[i];
E = temp;
}
++E[unstableSites];
unstableSites = 0;
if (topplings > N.length - 1) {
int len = N.length;
int[] temp = new int[topplings + 10];
for (int i = 0; i < len; i++)
temp[i] = N[i];
N = temp;
}
++N[topplings];
topplings = 0;
++t;
}
}
void timeStep () {
for (int i = 0; i < L; i++) {
height[i] += change[i];
change[i] = 0;
if (height[i] > maxHeight)
maxHeight = height[i];
}
for (int i = 0; i < L - 1; i++)
slope[i] = height[i] - height[i + 1];
slope[L - 1] = height[L - 1];
checkStability();
if (unstable) {
for (int i = 0; i < L; i++) {
if (move[i]) {
change[i] -= criticalSlope;
unstableSites += criticalSlope;
for (int j = 1; j <= criticalSlope; j++) {
if (i + j < L)
change[i + j] += 1;
else
++lost;
}
}
}
} else { // add grain
int site = 0;
if (random)
site = (int) (L * Math.random());
change[site] = 1;
++grains;
}
}
class Pile extends Plot {
Pile () {
setSize(300, 300);
}
public void paint () {
clear();
setWindow(0, maxHeight + 1, 0, maxHeight + 1);
setColor("yellow");
for (int i = 0; i < L; i++) {
for (int j = 0; j < outer.height[i]; j++) {
if (j == outer.height[i] + change[i])
break;
boxArea(i + 0.1, i + 0.9, j + 0.1, j + 0.9);
}
}
setColor("red");
for (int i = 0; i < L; i++) {
for (int j = outer.height[i]; j < outer.height[i] + change[i]; j++)
boxArea(i + 0.1, i + 0.9, j + 0.1, j + 0.9);
}
}
}
class PowerLaw extends Plot {
PowerLaw () {
setSize(300, 300);
}
// power law plot and exponent
int maxIndex;
double[] logData;
double exponent;
void analyze (int[] data) {
if (logData == null || logData.length < data.length)
logData = new double[data.length];
maxIndex = 0;
int numValues = 0;
double maxValue = 0;
double xAv = 0, yAv = 0, xxAv = 0, xyAv = 0;
for (int i = 0; i < data.length; i++) {
double value = 0;
if (data[i] > 0) {
maxIndex = i;
value = Math.log(data[i]);
if (value > maxValue)
maxValue = value;
xAv += i;
yAv += value;
xxAv += i * (double) i;
xyAv += i * value;
++numValues;
} else {
logData[i] = 0;
}
logData[i] = value;
}
exponent = 0;
if (numValues > 0) {
for (int i = 0; i <= maxIndex; i++) {
logData[i] /= maxValue;
}
xAv /= numValues;
yAv /= numValues;
xxAv /= numValues;
xyAv /= numValues;
exponent = (xyAv - xAv * yAv) / (xxAv - xAv * xAv);
}
}
public void paint () {
clear();
setWindow(-0.1, 1.1, -0.15, 2.25);
// topplings
setColor("green");
analyze(N);
if (maxIndex == 0)
return;
for (int i = 0; i <= maxIndex; i++) {
double x1 = i / (maxIndex + 1.0);
double x2 = (i + 1) / (maxIndex + 1.0);
double y1 = 1.1;
double y2 = 1.1 + logData[i];
boxArea(x1, x2, y1, y2);
}
plotStringLeft("power = " + Easy.format(exponent, 3), 1, 2.0);
setColor("black");
plotString("log(N)", 0, 2.15);
plotLine(0, 1.1, 1, 1.1);
plotLine(0, 1.1, 0, 2.1);
double d = 0.5 / (maxIndex + 1);
plotStringCenter("0", d, 1);
plotStringCenter("Topplings", 0.5, 1);
plotStringCenter("" + maxIndex, 1 - d, 1);
// unstable sites
setColor("blue");
analyze(E);
if (maxIndex == 0)
return;
for (int i = 0; i <= maxIndex; i++) {
double x1 = i / (maxIndex + 1.0);
double x2 = (i + 1) / (maxIndex + 1.0);
double y1 = 0;
double y2 = logData[i];
boxArea(x1, x2, y1, y2);
}
plotStringLeft("power = " + Easy.format(exponent, 3), 1, 0.9);
setColor("black");
d = 0.5 / (maxIndex + 1);
plotLine(0, 0, 1, 0);
plotLine(0, 0, 0, 1);
plotStringCenter("0", d, -0.1);
plotStringCenter("Unstable sites", 0.5, -0.1);
plotStringCenter("" + maxIndex, 1 - d, -0.1);
}
}
class Output extends Plot {
Output () {
setSize(610, 30);
}
public void paint () {
clear();
setWindow(0, 100, 0, 3);
setColor("blue");
boxArea(0, 100, 0, 3);
setColor("white");
plotString("Time step " + t, 5, 1);
plotStringCenter("Grains added " + grains, 30, 1);
plotStringCenter("Grains lost " + lost, 50, 1);
plotString("Topplings " + topplings, 65, 1);
plotString("Unstable sites " + unstableSites, 80, 1);
}
}
Pile pile;
PowerLaw powerLaw;
Output output;
Reader LReader, criticalSlopeReader, skipReader;
Checkbox randomBox;
int skip;
public void init () {
initial();
add(pile = new Pile());
add(powerLaw = new PowerLaw());
add(output = new Output());
add(LReader = new Reader("L = ", L));
add(criticalSlopeReader = new Reader("Critical slope", criticalSlope));
add(skipReader = new Reader("Skip steps", skip));
add(randomBox = new Checkbox("Add random", random));
randomBox.addItemListener(new ItemListener() {
public void itemStateChanged(ItemEvent ie) {
random = randomBox.getState();
}
});
addControlPanel();
}
public void step () {
for (int s = 0; s < skip; s++)
timeStep();
timeStep();
pile.repaint();
powerLaw.repaint();
output.repaint();
}
public void reset () {
L = LReader.readInt();
criticalSlope = criticalSlopeReader.readInt();
skip = skipReader.readInt();
initial();
pile.repaint();
powerLaw.repaint();
output.repaint();
}
Sandpile outer;
public Sandpile () {
super();
outer = this;
}
public static void main (String[] args) {
Sandpile sandpile = new Sandpile();
sandpile.frame("One-dimensional sandpile", 650, 500);
}
}
Questions or comments:
phygons@acsu.buffalo.edu