Analysis of Gutenberg-Richter b-value and mmax. Part III: Non-positive Gutenberg-Richter b-value

Análisis del parámetro b y mmax del Modelo de Gutenberg-Richter. Parte III: valor no positivo del parámetro b de Gutenberg-Richter

Universidad Católica de Salta, Argentina.

Abstract

English

When we analyzed the Gutenberg-Richter distribution function in our earlier works, we assumed that the   -value is positive.  Using generalized estimators, we found that in some cases the -value can be also negative. This paper gives a theoretical background for the negative -value. We also expand (the expansion of) the KS functions on the interval

Keywords: Gutenberg-Richter distribution function, Gutenberg-Richter b-value, Kijko-Sellevoll functions.

Español

Cuando en trabajos anteriores analizamos la función de distribución de Gutenberg-Richter, asumimos un valor positive para el parámetro . Con el uso de distintos estimadores, encontramos que este parámetro puede tomar también valores negativos. En este artículo se establece un marco teórico para el caso de valor negativo de  y demostraremos la expansión de la función Kijko-Sellevoll (KS) al intervalo  

Palabras clave: Función de distribución de Gutenberg-Richter, parámetro b de Gutenberg-Richter, funciones de Kijko-Sellevoll

Probabilidad – Ingeniería Sísmica / artículo científico

Citar: Haarala, M. (2021). Analysis of Gutenberg-Richter b-value and mmax. Part III: Non-positive Gutenberg-Richter b-value. Cuadernos de Ingeniería (13). Recuperado de: http://revistas.ucasal.edu.ar

1. Introduction

In earlier works (Haarala and Orosco, 2016a, 2016b, 2018 we have studied the double truncated exponential probability density function (PDF), or called also as the Gutenberg-Richter probability density function (GR),

                  

(where ) assuming that the  is always positive. Even if we generate the data with positive -value, the generalized estimators can give negative -values. We gave an example in a previous work (Haarala and Orosco, 2016b) where we could not get the values with the generalized Page estimator with our program. We set the estimates   to zero (as we can see the Page estimates at the points for  in the Figure 1) without knowing that they are negative values.

Figure 1: Example of Generalized Aki-Utsu (GAU) and Page (GP) estimators (Haarala and Orosco, 2016b)

A reason for this “failure” was our assumption that the -value is always positive. Another reason was the discontinuity   of the PDF  at . When we proved more general and simple results for the Kijko-Sellevoll (KS) functions, we found their real convergence interval  even though we gave the proof only to the positive interval  (Haarala and Orosco, 2018).  In this article we focused to the negative part of the Kijko-Sellevoll (KS) functions, which will yield the solutions for the interval .

2. Generalization of Gutenberg-Richter distribution function

Let’s consider the distribution function  which has the cumulative distribution function (CDF)

                  

where  and . The difference  is positive always, so the factor  can be negative only if  (i.e. ). We can see that CDF  has a discontinuity at , where both the numerator and the denominator are zero.

If  is negative, it still holds that  for all  in the  PDF  because of  and . The CDF  holds also, since  for all  because of both the nominator and the denominator are negative at the same time. It is not difficult to see from   that

          

and  is a non-decreasing right continuous function.

If , the limit of the PDF of GR distribution function can be gotten as

          

when . This is a Uniform Distribution function. It’s CDF is well known, but we can get it also by

          

          

Now we can complete the definition of the General Gutenberg-Richter (GGR) distribution function. The PDF is defined as

                  

with CDF

                  

where  and . We will show later that    is always bounded for practical applications. That is to say, we could assume directly .

Figures 2a and 2b illustrate this process with parameters , ,  using different values for parameter . In the figures 2c and 2d, the parameters are , ,  with different values of parameter . We can see from these figures how the probability decreases in small values and concentrate to  when . Even in the case of the Uniform distribution function, the events in the interval  become so rare that it is more probable to get a lot of huge values than small values when . This fact made us suspect that  is bounded, when  is negative.

The negative   has an opposite behavior than the positive one. While the positive  concentrates the events close to the minimum limit, the negative  concentrates the events close to the maximum limit. The figure 3, which was generated using , , and the -values  and  (both figures have  events), illustrate this situation.

Figure 2: Some PDFs and CDFs for non-positive b-values

Figure 3: Distribution of events with different b-values

3. Kijko-Sellevoll functions

Let  be a set of random variables from the catalogue. We assume that these random variables are independently and identically distributed (iid) with CDF of  given by . Moreover, let  to be a sample of magnitudes having a CDF

                  

for all .

It is not necessary to assume that the magnitudes are ordered. Actually, we are using here the maximum function, . The formula  can be expressed as

            

Similar way for the minimum function, , (4) results as

        

with a CDF

                  

for all .

We have showed in our earlier work (Haarala and Orosco, 2016a, 2018), that the expected value of the maximum  in case of positive  is

                    

where  is a Kijko-Sellevoll function 1 (KS-1)

                  

and  is a Kijko-Sellevoll function 2 (KS-2)

                  

These relationships are valid for all  and for all  (Haarala and Orosco, 2018). The relation between KS-1 and KS-2 functions is

                  

Note that we have integer valued  in the CDFs  and , when we have a set of events. The real valued  is a useful feature in the applications, where the estimate of the number of events is a real value. For example, if we estimate 7.5 events by year, rounding this value into 7 or 8 we are producing a numerical bias for the results. It is to remember that the value 7.5 does not mean that there are really 7.5 events by year. The 7.5 is an average number of events by year, when we are considering a long interval of time. Our formulae make it possible to directly calculate those results without rounding.

We will give our proofs using variable  instead of  giving general results for the formulae. In the Appendix A it can be seen that    can be also negative even though the proofs are given only for positive real values, .

4. The series for the expected values

Kijko-Sellevoll functions

First of all, we will show that the KS functions  and  are valid also on the interval . Actually, our earlier proof (Haarala and Orosco, 2018) holds on this interval, if . Because  is not defined generally when   (it is defined only for ), we must consider  for all . We have

        

                  

The  is a geometric series which gives  when   . (Actually, the convergence interval is , but we consider only the negative part since the proof of  is different when the positive part is considered.) This geometric series diverges at . Thus, . Owing to this it holds that  in the interval  and we have the limit , when , we could define . This limit could be seen like an expected value. Because of  and  for all ; it is like the case of a coin, which has expected value  when . This definition is related with the fact that the alternating series

          

converges when .

The conclusion is that equality  holds for all  and it gives an integration formula

                  

Applying this integration formula for the expected value, we have

          

                  

because  and

          

when . The KS functions are alternating series in this interval.

The result  is the same than  with  in the non-negative interval. It means that we can use the relations

          

for all  and .

Extension for the first Kijko-Sellevoll function

It is much more complicated to solve the case when . In this case, it is  (or in others words ). In like manner as before, we get the geometric series as

          

which is true for all . Thus,

                  

There are two observations when . Firstly, we have in the case

          

where we have set . Secondly, the integration in the case  gives

                  

Hence,

                  

where  is a switch function giving , if  is false, and , if  is true.

When we worked with KS functions in negative side, we had  (or in other words ). This means that the  must be close enough to  when we integrate over the interval . If the difference between  is bigger, we have . Integrating over the interval , we get

                  

Due to , when , we can get the limit

                  

This is the same as . Hence, we say that the series  in  holds for all , where we replace the discontinuity term by  in the case  of our calculus.

Using integration formula , the integration over  gives

          

The final result of the integral can be written as

          

So, we call the function

                  

as an Extension for the Kijko-Sellevoll function 1 (EKS-1). This function is valid, when .

This function  does not look like a KS-1 function, but it is a reflection of it (Appendix A). Also, we could show that EKS-1 function yields

                  

when . This expression was found anterior work (Haarala and Orosco, 2016a) by showing

          

The proof of relation (19) is given in Appendix A.

Even though we have the discontinuity term in the series, there is no discontinuity as we showed above. In the numerical calculus, it is to replace the discontinuous term with the logarithmic term . Because we use the acceleration method to calculate the series, we do not need to   mind this correction if  is bigger than the number of terms in the accelerated sum (for example,  in double precision systems).

There is an alternative way to solve the problem of the expected value in the case of the negative -value; we will show that in the Appendix B.

Extension for the second Kijko-Sellevoll function

The Extension for the Kijko-Sellevoll function 2 (EKS-2) could be found by

          

To find the EKS-2, the  and  have the series

                  

and

                  

respectively. Using , we get from

          

Finally, the expected value yields to

          

where we need to replace the discontinuity term of the series by

          

when . The Extension for the Kijko-Sellevoll function 2 (EKS-2) is now

          

for .

Uniform distribution

The Uniform Distribution function results are well known, we will show here how we can also get them from the GGR CDF. Let start with the KS-2 function

                  

Since

                  

the equation  gives

          

If , then we have the KS-2 estimator for the Expected value as

                    

which is known as an unbiased estimator for the maximum of the Uniform Distribution function in the form

          

The KS-1 estimator at  can be directly got by

          

Now we have shown all possible cases to calculate the expected value. We give new definitions to the KS functions:

          

and . The name KS max associates better the KS function or its extension to the maximum, because KS-1 and EKS-1 are measures of the distance from the maximum to the expected value. Similarly, because KS-2 and EKS-2 are measures of the distance from the minimum to the expected value, the name KS min associates the KS function or its extension to the minimum. We can see the examples of the  and  in Figure 4.

Figure 4: Example of the KS functions

5. Series for the variance

The third Kijko-Sellevoll function

As we saw in the case of the expected values above, it is only a technical detail to prove that the KS functions work also in the negative side. If we assume that , we need no changes to the earlier proofs. We can see that in this case the KS-3 is valid in the interval , but we need to assume  in more general case.

Let’s start with

                  

where the geometric series gives (when )

          

Thus,

          

when  and . Following our earlier work (Haarala and Orosco, 2016), the second moment can be integrated by parts as

                  

Hence,

                  

because

          

We must point out that we integrate  instead of  because the exponent function  does not exist in the real axes for all .

The obtained result  is the same as that obtained for  positive  (Haarala and Orosco, 2016b), so the KS-3 holds in the interval  just like other KS functions. Because of the series in  have an absolute convergence in the open interval , so we can rearrange those series. Thus, we can calculate the variance as

                  

We need to check yet the point at . Because we know that (Haarala and Orosco, 2016b)

          

for all , where  is a Harmonic Number of order 2. This shows that the  has an absolute convergence at , so it converges at the same point. So, the variance (and the KS-3) holds for all .

It is worth noting that a General Harmonic Number of order 2 can be defined as

          

It holds  for all .

Extension for the third Kijko-Sellevoll function

To find extension for the KS-3 in the case , we get

          

We applied the same procedure which resulted in  except that here the integral has divided into two parts. The second integral can be got directly (alike the function KS-3)

          

With the first integral, we start with the geometric series

                    

Integrating all terms which has , we find

          

where the term  must be calculated as

          

Thus,

          

Similar way as before, the variance yields

          

We call the function  as an Extension for the Kijko-Sellevoll function 3 (EKS-3).

As it can be seen, we did not solve the problem at the discontinuity point  (or  in the case of EKS-3) as we made before. Even the formula of EKS-3 is valid in any neighborhood of , it will be unstable to calculate numerically when .

Variance for the Uniform distribution

We have proved above, that KS-3 is valid when , so they are valid in the neighborhood of zero. We can apply the  in to the , so

        

This gives  in the case .

6. Some analysis of the GR distributed data

Let assume that . Then the expected value gives

        

This trivial result shows that the expected value is constant for all .

Symmetrically distributed data (Uniform distribution case)

Let  to be fixed. We have now

          

thus,

                  

Because , it indicates

        

Thus,  because of the expected value function is increasing.

The formulae  shows that if we can find estimators for the expected values  and , we can calculate the estimates for the  and  quite simple way. We will give an example in the section 8 how to use the formulae .

It is not to be forgotten that all expected values lie between then minimum and maximum as

                   

for all . The limits are  and . If  or for all  or  and  of some , then the limits are unbounded.

If both expected values  and  are bounded, the right-hand side in both equations are bounded and the data has bounded limits. We can see also from  that if one expected value is bounded and another is unbounded, the limits are unbounded. But if it happens, there exists only one bounded expected value because other way we can find two bounded expected values showing bounded limits.

We have seen that two bounded expected values at distinct points guaranties bounded limits for the data in case of .

We will see next that the maximum estimates of  and  are bounded in case of the uniform distribution. Suppose that we have  events . Without losing generality,  is assumed to be a maximum estimator for  and  to be a mean estimator for . In this case, we have  and . So, we can find from

        

The maximum or minimum could reach when . Thus,  and

                 

We can see from here that

                 

If all events are equal, , we can see from  that  indicating that the upper and lower limits are equal. In other words, if then  is a constant (this case the distribution function is a delta function).

Asymmetrically distributed data

The case  is different, because the expected values are bounded also in the unbounded case of maximum as we will see later.

Let’s assume that . We can use the KS-2 function

          

where . When , we can get

                    

where  is a General Harmonic number (Abramowitz and Stegun, 1972; Haarala Orosco, 2016a).

The factor  is unbounded if ,  or both of them are unbounded. If we assume that  is bounded. The equation  shows that the  is bounded if and only if the expected value is bounded.

We can write the equation  of unbounded case, as

          

By reason of , when , then . Thus,  for any fixed , no matter if the expected values  are from bounded or unbounded data. If  is unbounded, then  for all .

The same can also be shown when . Again, the factor  is unbounded if ,  or both of them are unbounded.  In that case we will use the EKS-1 function because . To find the limit in the unbounded case, we have

          

When , then ,  and we get the expected value

                  

and the maximum

          

Similar way than before,  is bounded if and only if the expected value is bounded. Moreover, all the expected values from bounded or unbounded data are bounded because for fixed . If  is unbounded, then  for all .

The analysis above sounds quite theoretical. But we showed that if we have two distinct bounded estimators, , then the  is bounded and at least one of the limits  or  is bounded. This means that at most the maximum  can be unbounded in the earthquake catalogue, where , and the minimum is bounded.

As we saw above, the expected values are bounded even the data is bounded or unbounded. This makes so difficult to estimate bounded . The recurrence formula gives one idea to show, if data is unbounded. It can find from the Appendix A.

The equation  can be written in the classical form

          

Owing to  always, it implies the  for all . This means, if data comes from unbounded system, that the b- value is always positive. It cannot get negative values.

Similar way, the equation  gives

          

Because of   , then  for all .

We have shown above that the b -value do not change the sign if the data is unbounded. It means that getting positive and negative -values within generalized estimators, is possible only in the case of bounded data and enough big  as we could see in figure 1.

Similar way as in the Uniform distribution case, we can create the new estimators. Let’s consider the case . If , then

        

gives

        

This also shows that and  are bounded and we can calculate them if there exist two   different and bounded expected values.

In the case of , , we have

        

Thus,

        

In this case, and  are bounded and possible to evaluate with two different and bounded expected values.

7. Expected value for the minimum

We will change the variable setting , when  and . It implies that we flip the axes in such a way that the minimum will be the new maximum and the maximum will be the new minimum. Then the integral yields

                  

The KS-1 gives the minimum for . Having the expected value for the minimum, we have

          

Taking into account that if , the KS-1 must be replaced by EKS-1.   We see that the KS-1 function does not measure only the distance from the  to the expected value for the maximum, it is also a measure from the  to the expected value for the minimum with negative .

We can see something more with these equations. Using  , the expected value for the maximum can be considered as

                  

In other words, the expected value curve for the maximum in the case of the positive  is equal than the expected value curve for the minimum in the case of negative .

We saw that

        

These equations give straightforwardly

                 

This symmetry is easy to understand from Figure 3.

8. Examples

Figure 5 shows the behavior of the expected value curve with different b -values. All the curves start from the minimum the lower the value of b, the faster it reaches the maximum value.

Figure 6 shows how the minimum and maximum follows the expected value curve in case of .  The green and red lines present the expected values curves for minimum and maximum, respectively. The expected value curve for minimum is calculated by . The blue lines are acquired from the catalogues of the random samples. Each catalogue size  has generated a sample of 100 events, where the mean of maximums and mean of minimums are calculated from.

By comparison between Figures 5 and 6 we realize that the curves  and are mirror images at the point .

Figure 7 shows some examples of the distribution for the maximum and minimum estimators and at , where we set

        

The first is a maximum estimator and the second is a mean estimator. The left column of figures presents the distribution of the minimum estimators for cases of the catalogue size 2, 3, 5 and 10. Similar way, the right column of figures presents the distribution of the maximum estimators for cases of the catalogue size 2, 3, 5 and 10. The mount of catalogues in the sample are . The minimum, maximum, mean and the median for the sample of the minimum estimator are

        

The same statistic for the maximum estimators gives

        

We can see that the mean value gives the unbiased estimate for the maximum and minimum. In case on the minimum estimator, the median gives also unbiased estimate for the minimum because the distribution is symmetric. Moreover, the distributions are bounded with the limits .

9. Conclusion

We have given a general definition for the Gutenberg-Richter distribution function and the new series in the case of negative -value. Moreover, we showed that if we have two bounded estimates for the expected values, then  is bounded and at least one of limits, or , is bounded. We showed some results which gives the relation between positive and negative . This work gives more perspective to understand the behavior of the Gutenberg-Richter distributed data.

References

Abramowitz, M., and I. A. Stegun (1972). “Handbook of mathematical functions”, 10 th  ed., Dover Publ., New York.

Haarala, M. and L. Orosco (2016a). Analysis of Gutenberg-Richter b -value and m max . Part I: Exact solution of Kijko-Sellevoll estimator m max , Cuadernos de Ingeniería . Nueva Serie , (9), 51-78 .   http://revistas.ucasal.edu.ar/index.php/CI/article/view/145

Haarala, M.   and L.   Orosco (2016b). Analysis of Gutenberg-Richter b -value and m max . Part II: Estimators for b -value and exact variance, Cuadernos de Ingeniería . Nueva Serie, (9), 79-106 . http://revistas.ucasal.edu.ar/index.php/CI/article/view/146

 

Haarala, M.   and L.   Orosco (2019). Generalized proofs of the Kijko-Sellevoll functions, Cuadernos de Ingeniería . Nueva Serie,  (11), 55-66.   http://revistas.ucasal.edu.ar/index.php/CI/article/view/262

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Appendix A

Reflection

In this section, we assume that  Let start with formula

                  

where . It yields

          

Using the cosine series (Abramowitz and Stegun, 1972)

                  

we get

                  

If we set

          

we find

          

Now, when , then it is  and we find

                  

This shows that we could use the KS function also for the negative exponent. The results  and  let us to write  as

                  

Similarly, we can find the reflection formulae for the KS functions. For the KS-1 is

                  

where  is called as a Generalized Cosine function (GC). The GC has the limits

          

because of  and (Abramowitz and Stegun, 1972)

          

Hence, we can write

                  

This is a reflection formula between EKS-1 and KS-1 functions. Moreover, because the EKS-1 and KS-1 are continuous functions, the subtraction

          

is bounded at the discontinuous points.

In a similar way as above, we can get the reflection formula for the KS-2 from  as

                  

Because

          

where  is a Euler constant and  is a General Harmonic number, the relation  gives a reflection formula of the Psi function (Abramowitz and Stegun, 1972)

          

These reflection formulae - are not so nice in numerical calculus, even though they are possible to use. They become unstable in the neighborhood of the discontinuous point and we found that the formula  is quite powerful in the calculus having only one discontinuity point at , which we do not need to consider in the case of .

Recurrence

The recurrence formula for the KS-1 function can be attained as

          

or we can write it as

                  

For the KS-2 function, the recurrence formula can be found as

          

This gives the recurrence formula for the Psi function (and General Harmonic number)

          

The recurrence formula  gives an interesting result for the maximum. Let’s assume that . The formula  can be written now as

          

or another way as

                  

In the same way we have

                  

We see from  and , that

          

From this, we can find the maximum

                  

If all expected values ,  and  are bounded and the denominator is non-zero, the right-hand side is bounded showing that the  is bounded. This gives a condition

            

for the bounded data. Thus the upper bound of the data depends on the shape of the expected value curve , .

Proof for the equation

Let´s assume that . Then   can be written as

                  

We can see that

          

Firstly, we have

        

Secondly, we get

          

Thirdly, we can rewrite the partial sum as

          

Now,  yields

          

which was to proof.

Appendix B

We will show here an alternative way to solve the integral of the expected value for negative . First of all, it is to change the variable as . It means that we flip the negative numbers to positive part and vice versa. The integral gives

            

where  and . It means that , because . Multiplying the denominator and numerator by , we have

          

Because , it is

          

when . Of course, this gives  at , but we do not consider it because the integral is the same, if we integrate over  or . We can apply now the Binomial Series (Abramowitz and Stegun, 1972) as

                  

where

          

and . This is true, since

            

but also because of

          

The terms of the alternating series  are all positive term series of KS-1 functions, because of we have . If we consider that , we can write the final result as

                  

It is interesting to see that we could carry the calculus from negative side to positive side. Anyway, this series is not so desirable because the magnitude of the binomial factor  increases quickly, and then decreases quickly, especially when  is big. Anyway, this relation  can be used only for small  because of the behavior of the binomial factor, but also because of the time to solve each KS function.

This kind of behavior of the factors is a problem. The binomial factor produces an overflow in the double precision system when  becomes big, for example,

            

meanwhile,

          

 

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