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.

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

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

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 .

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

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, .

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

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