Variation of b-Value of Normal, Reverse and Strike-Slip Faulting Earthquakes with Focal Depth

Abstract


Introduction
The equation log10N = a -b M describes the distribution of seismic magnitudes, where the constant N refers to the total number of earthquakes of magnitude M, a and b constants (Gutenberg and Richter, 1944).The b-value depicts frequency of earthquake size distribution, whereas the a-constant represents the total number of earthquakes.Many seismologists have become interested in the dependency of bvalue on depth.The intermediate and deep events have a higher b-value compared to shallow events (Gutenberg and Richter, 1944).Many researchers have shown that b-value decreases with focal depth of shallow earthquakes (e.g., Al-Heety 2011;Spada et al., 2013;Rigo et al., 2018;Al-Heety and Mohammad, 2021;Sharma et al., 2022).Siangshai et al. (2021) found that b-value of the shallow events that occurred in Shillong Plateau varies at different depths.Based on the faulting style over which the earthquake occurs, shallow depth earthquakes are classified into normal fault earthquakes, reverse fault earthquakes, and strike-slip fault earthquakes (Al-Heety 2020).Variation of the value of b-constant with depth normal, reverse, and strike-slip fault shallow-depth events were investigated by Al-Heety and Mohammad (2021).
The goal of this study is to look at how normal, reverse, and strike-slip faulting earthquakes' focal depths affect value of the b-constant.

Data and Methodology
The Global Centroid Moment Tensor (GCMT) Catalog, which covers the years 2008 to 2017, was used to compile the earthquake database for this study.For the purpose of this work, normal faulting, reverse faulting, and strike-slip faulting earthquakes with magnitudes equal to or higher than 5.1 and focal depths between 0 and 700 km were chosen.The numbers of the selected normal faulting, reverse faulting, and strike-slip faulting earthquakes are 7884, 8260 and 1949, respectively.
Calculating the completeness of the earthquake catalogue is necessary for the accurate estimation of the values of the constants a and b. (Woessner and Wiemer, 2005).Wiemer and Wyss's (2000) maximum curvature approach is used to calculate the earthquake catalog completeness magnitude (Mc).The Mc completeness magnitude for normal, reversal, and strike-slip faulting earthquakes was estimated using maximum curvature approach (Wiemer and Wyss's, 2000).The magnitudes of completeness for the selected events are 5.2, 5.The following maximum likelihood technique (Aki, 1965) is used to calculate the b-value and associated doubt: (1) Where Mmean is the magnitudes average and Mc is the magnitude of completeness.
The following equation has been used to calculate the degree of uncertainty regarding the value of the b-constant: (2) Where N is the earthquakes number.

Results and Discussion
Results of the b-constant value's dependency on the depth of the normal, inverse, and strike-slip faulting earthquakes, whose focal depths range from 0 to 700 km, are listed and shown in Table 1 and Figure 5, respectively.The result of calculating the b-constant value for the normal fault earthquakes (Fig. 5A) showed that the lowest value of the constant was at a depth of 350 km and the highest value at a depth of 400 to 450, and the value descends at depth about of 500 km, then returns and rises again at a depth of 600 km and decreases after that.The results of calculating the value of the b-constant for the reverse faulting earthquakes showed a variation pattern with depth different from the normal fault earthquakes (Figure 5B), where the lowest value was at about 450 km depth and the highest value was at 700 km depth.where the lowest value was at about 450 km depth and the highest value was at 700 km depth.The results of estimating the b-constant value for the strike-slip earthquakes showed a different variation pattern with depth from that of the normal and reverse fault earthquakes, where the lowest value was at the depth of 300 and 600 km and the highest value at the depth of 550 km (Fig. 4C).
The results of estimating the b-constant value for the strike-slip earthquakes showed a different variation pattern with depth from that of the normal and reverse fault earthquakes, where the lowest value was at the depth of 300 and 600 km and the highest value at the depth of 550 km.The results of the regression analysis of the change of b-constant value with depth for normal, reverse and strike-slip faulting earthquakes revealed that there is insignificant poor negative and positive correlation relations between b-value and depth for normal, strike-slip faulting earthquakes and reverse faulting earthquakes, respectively (Fig. 5).
The point at which an alteration in value of the b-constant occurs is called the point of inflection.The results of the b-constant value for normal and reverse faulting earthquakes showed two inflection points, the first inflection point coincides approximately with the first seismic mantle discontinuity at a depth of 410 km.
The second inflection point is near to the second seismic mantle discontinuity at a depth of roughly 520 km.For the strike-slip faulting earthquakes, there are two inflection points of the b-constant value, the first at a depth of 250 km and the second point roughly corresponding to the seismic discontinuity of 520 km.
There are three seismic discontinuities in the mantle at depths 410, 520 and 660 km which may change.It is now generally accepted that variations in the phase of minerals like olivine and other ones cause the majority of velocity and density jumps about 410 and 660 km (Shearer, 2000).More controversial, and its existence has occasionally been questioned, is the 520 kilometer discontinuity (Bock, 1994).However, the vast bulk of seismic data now supports an increased velocity gradient close to 520 km of depth, which most likely corresponds to the phase change from β-olivine to γ-spinel (Rigden et al., 1991).The inflection point's depth of b-constant value of normal and reverse faulting earthquakes may be near the mantle seismic discontinuities at depths 410 and 520 km.The b-value's flection point depth of the strike-slip faulting earthquakes may be also near to the 520 km mantle seismic discontinuity.The other flection point depth of b-value of the strike-slip faulting earthquakes is at 250km.This flection point may be associated with the lithosphere-asthenosphere boundary (LAB).The thickness of the lithosphere (LBD depth) ranges from 40-90km beneath the older seafloor to 130 -200km on average beneath the continental interior and may reach up to 250km (Rychert et al., 2020).

Conclusions
• There is no definite pattern of change in the b-constant value with depth for the three classes of earthquakes.The results of the regression analysis of the change of b-constant with depth show insignificant poor correlation relations.• The depth of inflection point of the b-value for normal and reverse faulting earthquakes is near to the mantle seismic discontinuities 410 and 520 km.• The depth of inflection points of b-constant value strike-slip faulting earthquakes is near to LAB at depth of 250km and the mantle seismic discontinuity of 520km.
2 and 5.1 for normal, reverse faulting and strike-slip faulting events, respectively.The number of earthquakes of magnitudes ≥ Mc used in calculating the b-valuet are 6876, 7588 and 1949 for normal, reverse and strike-slip faulting events, respectively.The temporal and spatial of the selected normal reverse and strike-slip faulting earthquakes are shown in Figs. 1, 2 and 3, respectively.

Fig. 4 .
Fig. 4. Variation of b-constant value of (A) normal faulting earthquakes, (B) reverse faulting earthquakes, and (C) strike-slip faulting earthquakes.The red lines are the seismic discontinuities (410km, 520km and 660km) in the mantle.

Table 1 .
b-constant value for normal, reverse and strike-slip faulting events.