# Earthquake Rates and Probabilities in the United States: A PSHA Perspective

## Introduction

Many people want to know if there is going to be a sizable earthquake within some
relatively small distance of their home, office or school within a relatively
short time interval. Earthquake science is making strides in this direction.
Whereas specific information about the relatively immediate future is desired,
whether it can be provided with much reliability is debatable. However, numerous
models exist whose purpose is to make estimates of earthquake probabilities.
Probabilistic seismic hazard assessment requires estimates of *mean* earthquake
frequencies in order to produce estimates of ground shaking probabilities. These mean
frequencies are less specific, that is, they refer to expected values for rate of
earthquakes averaged over many time intervals rather than rate associated with the
immediate future or any other specific time interval.

PSHA models that scientists have developed for the U.S.A, and for which PSHA maps and other products have been distributed from the USGS-Golden office, are used at this Website to compute earthquake rates and probabilities within a given distance and a given time interval of any U.S. site. Either the site’s Zip Code or its geographic coordinates may be entered to obtain the map.

## Some Details

Earthquake rates are converted into expected numbers of earthquakes in a *random* time-span T by multiplication:

(1)

where E[N(S)] is the symbol for expected number of earthquakes from source S in time T. The mean rate of this source is l(S). At this web site the time unit is years, so the rate parameter unit is “number of earthquakes per year.” The expected number of earthquakes is converted into a probability of one or more earthquakes by the formula,

Pr[1 or more earthquakes in time T] (2)

where the summation ( symbol) is over earthquake sources that are within a prescribed distance R of the site. The estimate of equation (2) uses the Poisson probability model. The Poisson model of earthquake occurrences is at the heart of standard PSHA at USGS Golden. The Poisson probability is a time-independent probability. That is, its determination is independent of recent history of earthquakes in the region, such as the time and location of the last large earthquake.

Most reasonable people would agree that any attempt to estimate the probability of a future earthquake should take into account all pertinent information about earthquake processes. The Poisson model, by its time-independent earthquake-occurrence assumption, is not necessarily the best model for such determinations. However, it is the model that is used for many seismic-hazard assessments and it can be used as a benchmark for comparisons with other more sophisticated earthquake-generation models.

To better understand what is meant by time-independent earthquake occurrences,
we compare estimates using the Poisson model with another model which assumes
the same annual rate of earthquakes. Suppose the rate of earthquakes which
have magnitude **M** greater than 5 within R km of the site is 1 per year.
In the Poisson model, the probability of one or more earthquake in a 1-year
period is 1-exp(-1)=0.63, approximately. Suppose, on the other hand, that
the earthquake-production model shows a long history of exactly one earthquake
greater than **M**5 within R km of the site occurring once per calendar year
(i.e., the earthquakes seem to keep tract of leap years). Then, supposing the
earthquake has already occurred in the current year, and supposing the prior
pattern continues, the probability is zero that it will occur for the rest of
the year. Or, supposing the earthquake has not yet occurred, the probability
is one that it will occur before January 1. This model yields a very different
probability assessment than that of the Poisson model, which gives a blanket
probability estimate of 0.63 of one or more per year, regardless of the recent
or long-term history, even though both models assume the same mean rate of earthquakes.

The Poisson probability is a function of exposure time, t. For the above example, with rate parameter of 1, Pr[t]=1-exp(-t). For example, the probability of one or more such earthquakes in a three-month interval is 1-exp(-3/12)=1-exp(-0.25)0.22. Thus, if today is October 1, we might be comparing our time-independent probability of 0.22 with either a probability of 1 (if it hasn't occurred yet) or a probability of 0 (if it has) for this particular time-dependent model. Note, also, that the Poisson model has positive probability of more than one earthquake in a short time interval, such as one year, whereas the alternate model has zero probability of two or more per calendar year. Time-dependent models may have other constraints based on physical models and/or historical data, such as a minimum waiting time between earthquakes that exceed a given size in the region. The Poisson model has no such constraint.

## The rest of this article is a more detailed discussion, maybe of limited interest to most readers.

A great problem with time-dependent assessments is that there is generally no long sequence of regular-occurring earthquakes in the published archives of earthquake history. Thus, most time-dependent models require some amount of faith that the physics of earthquake generation has been adequately modeled. The time-independent model is modest about how it uses past earthquake occurrence information, and is less dependent on physical models. This feature of the Poisson model may seem helpful for those who have little faith in current models, but may seem too limited for those who have some faith in one or more current models of earthquake generation.

## Uncertainty about Earthquakes that is Contained in the PSHA Model

There are three primary sources of uncertainty about hazardous earthquakes: (1), their size, (2) their location, and (3) their rate of occurrence. The USGS-Golden PSHA model of earthquake sources can be partitioned into three broad categories of sources: (1), background or gridded seismicity, (2) earthquakes on identified Quaternary faults, and (3) earthquakes associated with oceanic plate subduction. The manner of incorporating source uncertainties varies with these three broad categories, and varies geographically as well. When we present maps of earthquake probabilities for earthquakes having a specified range of magnitudes and within a specified maximum distance of the site, we could present multiple maps each of which uses a specific subset of earthquake size, location, and recurrence rate. Technically, these multiple maps are the consequence of logic-tree branch analysis. However, we wish to present a “best guess” model, and to do this we present maps that combine the alternate models by weighing them by our measure of confidence in them, and adding the weighted rates.

Even though we do not present effects of alternate models of earthquake recurrence in
these maps, it is helpful to know something about source variability that is in the
PSHA model. The 2002 update report for the conterminous U.S. (Frankel et al, 2002)
describes the source variability. Similar treatments of source variability are found
in the PSHA of Alaska (Wesson *et al*, 1999) and in the PSHA of Puerto Rico and
the U.S. Virgin Islands (PRVI) (Mueller *et al*, 2003).

There is variability in source location and size in the model. To produce a mean-hazard
map, we do not need to model uncertainty in mean recurrence rate (or interval). For New
Madrid Seismic Zone main shocks, we model variability in location by putting the source
on one of three fault traces (Frankel *et al*, 2002) and we model variability in
magnitude by defining four possible magnitudes for the NMSZ main shock, ranging from
**M**7.3 to **M**8. However, for all NMSZ source models, the mean recurrence
interval is fixed at 500 years. Similarly, for the Charleston South Carolina main shock,
there are four magnitudes, ranging from **M**6.8 to **M**7.5, and there are two
models of source location, a broad zone and a narrow zone (Frankel *et al*, 2002).
The mean recurrence interval is fixed at 550 years for these Charleston-region main shocks.
For WUS faults, the fault locations are assumed known, although there is considerable
uncertainty in segmentation, i.e., which segments or combinations of segments might
rupture in a future earthquake. Uncertainty of fault-source magnitude is included in
the PSHA model. The mean slip-rate is assumed known for many WUS faults, and for a
given magnitude, recurrence rate is fixed, and correlates inversely with magnitude.

Much seismic hazard is from background or gridded sources. Background seismicity is not associated with identified Quaternary faults. For modeling purposes, gridded-source locations are fixed at source-cell locations. For gridded sources with magnitude greater than 6, the source is assigned a finite length from a Wells and Coppersmith relation. The center of the fault is placed on the source-cell location and its strike may be random or fixed at a specified strike angle. Randomness of fault strike yields uncertainty in source distance to any given site. Gridded-source rate uncertainty is captured into the mean estimate by determining rates of earthquakes from different catalogues and averaging these rate estimates.

Subduction sources have variability in magnitude, **M**8.3 or **M**9, and location,
described in Frankel *et al.* (2002). The recurrence interval is fixed in the PSHA
model, 500 years for **M**9 and about 130 years for **M**8.3. Many other details
of source variability are discussed in Frankel *et al.* (2002). For Alaska and for
PRVI, there are some differences in detail, but generally the same variability in sources
that affect seismic hazard in that state and territory, respectively, as in the PSHA for
the conterminous USA. Earthquake rates and probabilities for Hawaii are discussed in
(ref. to be supplied).

How do we combine the above source-uncertainty information into a single estimate of
rate and probability of earthquakes at any given site? We do this by multiplying the
rate of occurrence as a function of *M* and *R* by the epistemic weight
*w*_{i} associated with that source model. We then add contributions
for all relevant source models:

(3)

to get our rate estimate of earthquake sources at distance *R* and magnitude
*M*. Equation (3) is the approach, that is, sum across epistemic models of
variability to get a single rate of earthquakes. The result might be thought of
as a mean or average model, but it should be understood that each of the estimates
is also in theory a viable model of the mean rate of earthquakes of a given
magnitude at a given distance from the site. Thus the rate estimates given in
the written reports at this web-site, and in USGS seismic hazard maps and
related products, are weighted averages of several distinct and somewhat
incompatible models' earthquake mean rate estimates.

An example might help. Suppose there are two models of mean recurrence interval
for source *S*, 500 years and 1500 years, and that these have each been
assigned a weight 0.5, because we are equally confident of their correctness.
Then, for *S*, we calculate hazard using an effective rate of 0.5(1/500+1/1500)
= 0.001333 or a mean recurrence interval of 750 years. This averaging of opinions
has the advantage of simplicity but the disadvantage of obscuring features of the
input models, one of which may capture a reasonably good approximation of the mean
rate (or size, or location) in nature. We just don’t know which one is best,
so we average them.

Precomputed maps are available. These include maps of expected numbers of earthquakes within a given distance of each site on the map and maps of probability of one or more earthquakes. The expected numbers maps are usually for a 50-year time span and the probabilities are computed for either a 30-year or 50-year interval.

The interactive web site allows the user to compute probability of earthquakes within
a fixed radial distance and user-specified time span for a user-specified site (which
you enter with Zip Code *or* latitude and longitude) using the current USGS PSHA
models for the U.S. and PRVI region. This Web site uses the same method for computing
rates and probabilities as are outlined in Equations 1, 2, and 3 above.

Three additional details may be important for the distance calculations:

First, all distances are epicentral or distance to surface projection of the fault. That is, if you imagine a cylinder with radius R centered at your site, any point source that is within this cylinder or any finite source that has some part of the fault or rupture within this cylinder is counted in the rate of earthquakes calculation. The source will generally be crustal, but deep intraplate earthquakes are also counted in many regions such as Puget Sound and southern Alaska.

Second, the gridded finite sources are potentially counted only if the center of the notional fault is within 15 km of R, e.g., if R=50 km, we consider grid cells up to 65 km away for potential inclusion of finite random sources. In fact a M7 source might have length greater than 30 km, and thus some larger random sources might be missed using this cutoff criterion. If you are concerned about R=50 km, try R=60 km to insure that you count these outlying large but unknown sources. In the PSHA for the USA, Rmax is never less than 200 km, and any sources whose center points are more than 200 km away from the site are simply omitted. These somewhat more distant sources may become more significant, i.e., relatively more frequent compared to closer sources, when Rmax is reduced, which it is in this report.

Finally, source rates for background seismicity are generally sampled at 0.1 degrees in latitude and longitude. That is, a source cell dx and dy is 0.1°. Sampling can be more dense in Puerto Rico and Hawaii. A given cell's rate is added to the total rate if the cell's northwest corner is inside the cylinder with radius Rmax, but not otherwise (except for the finite-source effect discussed in previous paragraph). The intersection of a cylinder with a grid cell near its circumference yields a fractional volume. Calculating this fractional volume might provide a somewhat better rate than the simple binary "in or out" count that is performed here. Because of distance smoothing and the tendency for these edge-effect biases to average out over the 360°angle, we are not attempting more refined calculations for edge-intersecting source cells here. Another reason that more careful handling of edge effects is omitted is that the rate calculations are being performed with the computer code algorithms of the PSHA model, which does not perform these extra edge calculations. However, the PSHA model uses a large Rmax, and sources at the circumference tend to be of little importance to seismic hazard because of attenuation at those distances. This approximation may result substantial rate bias (as much as a factor of four) when Rmax is on the order of dx or dy, i.e., about 10 km for most U.S. locations. (For Puerto Rico and Hawaii, the source-cell increment is 0.05° in latitude and longitude.)

Some comments about magnitude may help. First, we report earthquake size in moment
magnitude (**M**) consistently here. In the WUS, we consider earthquakes with
**M**≥5 to be large enough to include in PSHA calculations, and in the CEUS
we consider earthquakes with **M**≥4.5 to be large enough. Thus, the lower
limit magnitude on the maps that follow vary by region. In fact there is an overlap
region roughly between 115 degrees and 100 degrees W where both lower limits may occur
in the source files. This region is considered WUS in the maps that follow, but one
might keep in mind that lower magnitude sources may be included in the expected numbers
calculations for sites in this region. The eastern Rocky Mountain range front roughly
defines the transition zone where western sources end and eastern sources begin.

Rates of earthquakes that are estimated in PSHA calculations are in part based on
historical seismicity rates and rate estimates. We present epicenters of historical
earthquakes on the expected number of earthquake maps to show the correlation. At
a few locations we list historical data and compare observed and expected numbers.
This kind of work can be useful in efforts to "validate" the PSHA model.
In the WUS, smaller historical earthquakes, those with magnitude less than ≈6.5,
are best correlated with random or gridded seismicity, and larger earthquakes are best
correlated with mapped Quaternary fault earthquakes. There are several caveats to
this rule, for example, as we go back in time, when regions were sparsely settled,
location and size estimates may be fairly rough, and correlation with known faults
is somewhat conjectural. Furthermore, the PSHA model does not always limit itself
to Quaternary fault sources and smoothed historical seismicity, but also includes,
in some regions, earthquake rates inferred from GPS -determined rates of regional
contraction or extension of the Earth's crust. Currently these regions include the
Puget Sound source zone of Washington and several zones near the California-Nevada
boundary (Frankel *et al*, 2002).

Many parts of the U.S. have known (catalogued) earthquakes with maximum size lower
than the PSHA model *Mmax*. Thus at most locations in the U.S., we cannot use
local historical data to validate the PSHA-model-derived probabilities of larger
earthquakes. There is ongoing debate about the validity of different *Mmax*
models for many parts of the U.S.A.