[CEUS-earthquake-hazards] Comment on China quake and hazards map

[Julio J. Hernández]

Dear Zhenming,
I don't know if people on the bulletin board are tired of these exchanges. But I will try to answer.
 
Regarding "Your explanation .... is a deterministic one and inconsistent with what is known.":
I chose a deterministic earthquake recurrence as an illustration for combining spatial variability with 
temporal recurrence. In this case it results a Bernoulli model of repeated trials. Each earthquake is a trial.
Each PGA >= 0.4 g is a success and each PGA < 0.4 g is a failure. In each trial Prob(success) = 0.5.
On average we have one success each two trials; therefore, on average every 500 years.
[If you flip a perfect coin every minute, you have a return period (of getting one head) = 2 minutes].
Then we can apply Bernoulli equations. For example: Prob(>= 4 successes in 8 trials) = 63.7%.
But, I didn't say that something always occurs. I used the word "expect" in the sense of "expect on  
average" (according to statistical custom). Obviously, Prob(m failures in n trials) does not equal zero. 
This is a known probabilistic model and it is not inconsistent. 
  
On the other hand, if we consider random occurrences of M7 earthquakes modeled as a Poisson 
distribution with mean return period (MRP) = 250 y., the selection of PGA with any probability of 
exceedance is also a Poisson distribution. If that probability is 0.5 we get a model with MRP = 500 y.
Here, for example, Prob( >= 1 earthq. in 250 y.) ~ 63.2%.  Prob( >= 1 earthq. in 500 y.) ~ 86.5%.
And Prob( >= 1 times "PGA >= 0.4 g". in 500 y.) ~ 63.2%..
Notice that the last one is not 0.5*86.5%, because that is a problem of conditional probabilities over 
all possible number of earthquakes. For a formal demonstration, see for example, Sec. 3.2.1 of the 
book "Probability, Statistics, and Decision for Civil Engineers", Benjamin & Cornell (1970), McGraw-Hill.
This is a consistent probabilistic model of general application (floods, vehicles, etc.).

Regarding  "There are very limited observations for characteristic earthquakes ...":
I used a characteristic earthquake only as an example. 
In an actual situation we have many possible earthquakes in many seismogenic sources; each magnitude 
that drives a defined PGA value in a site with certain probability of exceedance, is Poisson-distributed. 
By combining, we get a Poisson distribution of ">= PGA" with a computed rate of recurrence.
You know the details.  This is the same consistent probabilistic model. 

Regarding "... demonstrates that how PSHA can mislead to a zero PGA (no seismic design consideration) 
if a 250-year return period is considered.":
Effectively, in the example zero-PGA has MRP = 250 years. This means that we expect PGA >= 0 
(¡any value of PGA!) every 250 years. That is absolutely true, in fact PGA > 0 for every earthquake.
This is not inconsistent. Simply, any PGA > 0 has MPR > 250 years.
[If you flip a coin every minute, any event has MPR > 1 minute].
In a Poisson process it happens a similar thing. Any event can happen any time, but its MRP > 250 y. 

Best regards, Julio J.
 
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----- Original Message ----- 
From: Wang, Zhenming 
To: Central and Eastern U.S. Earthquake Hazards Listserve 
Sent: Thursday, May 22, 2008 3:16 PM
Subject: [CEUS-earthquake-hazards] Comment on China quake and hazards map


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----- Original Message ----- 
From: Wang, Zhenming 
To: Central and Eastern U.S. Earthquake Hazards Listserve 
Sent: Wednesday, May 21, 2008 7:47 PM
Subject: Re: [CEUS-earthquake-hazards] Comment on China quake and hazards map

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