Introduction to Earthquakes

Abstract

Seismology comes from the Greek term “Seismos” for earthquake and “Logos” for science which relates to the study of generation.

Appendices

Appendix 1: Relationship Between Earthquake Magnitude and Energy from Seismogram Records

In this appendix the origin of the relationship between earthquake magnitude and energy from seismogram records is provided. The records of the seismographs for a given earthquake can be observed in twofolds: time reading and amplitude reading. The time reading gives an idea of the exact location, its focal depth, and the origin time of an earthquake , whereas the amplitude reading gives the information on the total energy released during an earthquake . There are two different ways in which the energy released in an earthquake can be quantified either from seismograms or from field observations in the epicentral area in combination with theoretical studies.

To bring more clarity on the different types of energy that is released during an earthquake , Yoshima (1963) has chosen a spherical source in a perfectly elastic solid as his model and has divided the energy released from the spherical source into three components, W _R = E ₀ + W _t + W _D, where E ₀ is the seismic wave energy , W _t and W _D represent the deformation energy . W _R is the outward flow of energy (or the strain energy ) across a sphere of radius ‘r’ from a spherical source which can be obtained from Eq. (1.26) as given below:

$$ W(r) = - 4\pi r^{2} \int\limits_{{t_{1} }}^{{t_{2} }} {\sigma_{r} \frac{\partial u}{\partial t}{\text{d}}t} $$

(1.26)

As $\sigma_{\text{r}}$ is the radial stress per unit surface, the term 4πr ² (surface area of a sphere) comes into picture, and the distance corresponds to du = {du/dt) dt. The negative sign makes the energy positive when the direction for $\sigma_{\text{r}}$ is chosen to be inward normal to the sphere. Similarly, the seismic wave energy , E ₀, can be expressed by Eq. (1.27) as given below:

$$ E_{0} = 4\pi r^{2} \rho V_{\text{p}} \int {\left( {\frac{\partial u}{\partial t}} \right)^{2} {\text{d}}t} $$

(1.27)

where V _P is the compression wave velocity. This is the expression of the seismic wave energy which can be used for the determination of wave energy from seismic records.

Consider a spherically symmetrical source of energy situated at a point F which emits P or S wave as shown in Fig. 1.38. The total seismic energy emitted from this source (only body waves) is E ₀ = E _P + E _s which is calculated from Eq. (1.28) as given below:

$$ E_{0} = 8\pi^{3} \rho \left( {h^{2} + 4r_{0} \left( {r_{0} - h} \right)\sin^{2} \left( {\frac{\Delta }{2}} \right)} \right)e^{kR} \frac{(1 + q)}{{Q^{2} }}\int\limits_{0}^{t} {V_{\text{p}} \left( {\frac{A}{T}} \right)^{2} {\text{d}}t} $$

(1.28)

where

ρ :

is the density of the medium,

Q :

is the fraction of the recorded to the incident energy ,

h :

is the depth of the source,

q :

E _s/E _p (fraction of S wave to P wave energy ),

E ₀ :

E _p(1 + q) or E ₀ = E _s(1 + 1/q),

$e^{kR}$ :

is the absorption factor (in which absorption of waves is taken into account),

k :

is a value which varies with the type of wave, and

t :

is the duration of the wave group (=nT) where n is the number of cycles (or number of wave trains)

Fig. 1.38

A section through the earth which explains the calculation of energy of body waves

To make the calculations simple, the sum of all the wave trains are replaced by a single measurement of amplitude and time period, such that the energy computed remains the same. One of the most preferred choices is to measure the maximum amplitude (A _m) with its corresponding time period (T _m) within any given wave group [17]. The integrals over the wave trains (or the corresponding summations) can then be written as:

$$ \int\limits_{0}^{t} {\left( {\frac{A}{T}} \right)^{2} {\text{d}}t} = \left( {\frac{{A_{{m}} }}{{T_{{m}} }}} \right)^{2} t_{0} $$

(1.29)

It should be noted that (A _m) is the maximum amplitude in each wave group; i.e., for body waves, this means that (A _m) is not the amplitude of the first swing but can be measured up to about 10 s after the onset of the wave, just to correspond to the maximum in the group.

There may be an adequate risk to include some other phases of the body waves (as, e.g., there may be interference from pP for shocks shallower than about 40 km, or from PcP at distances beyond about 75–80° as given in Fig. 1.14) when the maximum amplitude measurement corresponds to 10-s window, but such complications are generally of no great consequence as the errors usually do not exceed those which anyway are inevitable in energy determinations. Another approach is that in practice, the period at which the magnitude of body waves is usually determined is 1 s [18]. Most of the calculations of the energy –magnitude relationships depend on the energy of a wave group emanating from a point source as shown below [9],

$$ E_{0} = 2\pi^{3} h^{2} V_{\text{p}} \rho \left( {\frac{{A_{{m}} }}{{T_{{m}} }}} \right)^{2} t_{0} $$

(1.30)

where t ₀ is the duration of the wave group. This formulation applies at the epicenter (Δ = 0) where h is the hypocentral distance. A factor of 3/2 is applied to the energy equation for calculating the total energy in terms of transverse waves. As at short distance, the amplitude of transverse wave dominates; hence it is more appropriate to represent Eq. (1.31) in terms of wave energy of S-group as shown below,

$$ E_{0} = 3\pi^{3} h^{2} V_{\text{s}} \rho \left( {\frac{{A_{{m}} }}{{T_{{m}} }}} \right)^{2} t_{0} $$

(1.31)

This form of equation is arrived in a similar fashion by putting (Δ = 0, q = 2 and Q = 2) in Eq. (1.28), and the final energy expression in the form of transverse wave is,

$$ E_{0} = 3\pi^{3} h^{2} V_{\text{s}} \rho \left( {\frac{{A_{{m}} }}{{T_{{m}} }}} \right)^{2} t_{0} e^{kh} $$

(1.32)

where the integration of the wave group is carried out in terms of transverse waves by replacing (1 + q) by (1 + 1/q) in Eq. (1.28). This equation agrees with the equation given by Gutenberg and Richter [9] with the exception that it did not incorporate the absorption factor, e ^kh.

As the energy varies over a large range, taking the logarithm on both sides of Eq. 1.28, the expression reduces to:

$$ \begin{aligned} \log \left( {E_{0} } \right) & = \log \left( {8\pi^{3} \rho V_{\text{p}} } \right) + \log \left( {h^{2} + 4r_{0} \left( {r_{0} - h} \right)\sin^{2} \left( {\frac{\Delta }{2}} \right)} \right)e^{kR} ) \\ & \quad + \log (1 + q) + \log \left( {t_{0} } \right) + 2\log \left( {\frac{{A_{{{\text{m}}1}} }}{{T_{{{\text{m}}1}} }}} \right) \\ \end{aligned} $$

(1.33)

which is equivalent to the expression $\log \left( {E_{0} } \right) = a_{1} + b_{1} m_{\text{b}}$, and is taken as the definition of the body wave magnitude which is generalized as

$$ m_{\text{b}} = \log \left( {\frac{{A_{{{\text{m}}1}} }}{{T_{{{\text{m}}1}} }}} \right) + F_{1} \left( {\Delta ,h} \right) + C_{{{\text{s}}1}} + C_{{{\text{r}}1}} $$

(1.34)

where F ₁ is a correction term for distance (Δ) and depth (h) of the source, C _s1 is the station correction, taking the conditions at the respective stations into account, and C _r1 is the regional correction, taking the focal mechanism (radiation pattern) and path properties into account. The function F ₁ (Δ, h) has been determined by Gutenberg [19, 20] and Gutenberg and Richter [21] by a combination of theoretical and empirical findings, for the following waves: Ρ vertical, Ρ horizontal, PP vertical, PP horizontal, S horizontal. The constants a ₁ and b ₁ are then determined empirically from m and logE ₀. The combination of $F_{1} (\Delta ,h) + C_{{{\text{s}}1}} + C_{{{\text{r}}1}}$ is $Q(\Delta ,h)$ in which the density of the medium, depth of focus of earthquake , radius of the earth, and epicentral distances are all taken into consideration as shown in Fig. 1.36, and the plot of $Q(\Delta ,h)$ is shown in Fig. 1.21.

In order to grasp the total energy of surface waves (Rayleigh and Love waves respectively), we have to carry out a threefold integration in space and time: along the circle through Β around F (with circumference = 2πr ₀ sin (Δ), along the duration of the wave train, and from the surface down to (theoretically) infinite depth. Hence, for surface waves the corresponding expression for seismic wave energy is given below in Eq. (1.35).

$$ \begin{aligned} \log \left( {E_{0} } \right) & = \log \left( {4\pi^{3} \rho V_{\text{L}} r_{0} H} \right) + \log \left( {\sin (\Delta \Delta^{n} e^{k\Delta } } \right)e^{kR} ) \\ & \quad + \log (1 + q) + \log \left( {t_{0} } \right) + 2\log \left( {\frac{{A_{{{\text{m}}2}} }}{{T_{{{\text{m}}2}} }}} \right) \\ \end{aligned} $$

(1.35)

where H = 1.1λ (λ is the wavelength of the wave), V _L is the velocity of the surface wave, $\Delta ^{n}$ is the dispersion factor, and q = E _R/E _L (fraction of Rayleigh wave to Love wave energy ). This is equivalent to the expression $\log \left( {E_{0} } \right) = a_{2} + b_{2} M_{\text{s}}$ and is taken as the definition of the body wave magnitude which is generalized as:

$$ M_{\text{s}} = \log \frac{{A_{2} }}{{T_{2} }} + F_{2} (\Delta ,h) + C_{{{\text{r}}2}} + C_{{{\text{s}}2}} . $$

(1.36)

In Eq. (1.36), usually ‘h’ is omitted and surface waves are used for magnitude determinations only for shallow shocks. Another approach is that in practice, the period at which the magnitude of body waves is usually determined is around 20 s (Kramer 1996). The constants a ₂ and b ₂ are found out in a similar fashion as for body waves. The earthquake magnitude that is determined depends on seismic wave measured. There are different magnitude scales for P waves , for Rayleigh waves, and for different periods of motion.

Appendix 2: Subroutines

1.2.1 A.2.1    A User Subroutine for Computing the Fourier and Power Spectra of a Given Earthquake

1.2.2 A.2.2    A User Subroutine for Computing the Arias Intensity of a Given Earthquake

1.2.3 A.2.3    A User Subroutine for Computing the Response Spectrum of a Given Earthquake