Earthquakes

This fundamental theory, first proposed by H. F. Reid in 1906 following the great San Francisco earthquake, elucidates the mechanics of earthquake generation through the accumulation and sudden release of elastic strain energy stored in rocks near a fault.

1. Initial condition: Two adjacent rock blocks, labeled A and B, lie on opposite sides of a fault plane. Due to tectonic forces, shear stress acts parallel to the fault, attempting to displace these blocks relative to each other. However, frictional forces along the fault surface prevent immediate slippage, locking the fault in place.
2. Elastic deformation and stress accumulation: Continued tectonic loading causes elastic deformation of the rocks, storing strain energy. The displacement of the blocks is hindered, but stress ($\tau$) builds up, increasing the potential energy in the system.
3. Rupture and slip: When the accumulated shear stress surpasses the frictional resistance, a sudden slip occurs along the fault plane, releasing the stored elastic energy. The point within the crust where this rupture initiates is the focus (or hypocenter), and the point directly above on the Earth's surface is the epicenter.
4. Elastic rebound: Following rupture, the rock blocks snap back approximately to their original undeformed shapes, a process called elastic rebound, which generates seismic waves propagating through the Earth.

Seismic waves generated by the earthquake rupture propagate through the Earth and are classified into two broad categories:

a. Body Waves (Travel through the Earth's interior)

Primary waves (P-waves): Longitudinal compressional waves causing particle displacement in the direction of wave propagation. They can propagate through solids, liquids, and gases due to their compressional nature. Travel at the highest velocity among seismic waves. The velocity $V_p$ of P-waves in an isotropic, homogeneous medium is given by:

where:

• $K$ = Bulk modulus (incompressibility of the material)
• $\mu$ = Shear modulus (rigidity)
• $\rho$ = Density of the medium

Secondary waves (S-waves): Transverse shear waves causing particle motion perpendicular to wave propagation direction. Only propagate through solids since shear modulus $\mu$ is zero in fluids. Slower than P-waves. The velocity $V_s$ is given by:

b. Surface Waves (Travel along the Earth's surface)

1. Rayleigh waves: Characterized by elliptical, retrograde rolling particle motion in the vertical plane, causing both vertical and horizontal ground displacement similar to ocean waves.
2. Love waves: Cause horizontal, transverse ground shaking perpendicular to the direction of propagation, often producing strong lateral movement.

Surface waves generally travel slower than body waves but cause more structural damage due to their larger amplitudes and longer durations.

1. Seismic wave velocities are sensitive to the physical properties of Earth's materials, including density, elastic moduli, pressure, temperature, and mineral composition.
2. Seismologists use variations in travel times, reflection, and refraction of seismic waves at different internal layers to infer the Earth's stratified structure, such as the crust, mantle, and core.

Earthquakes are analyzed using seismometers, instruments that measure the amplitude and frequency of seismic waves.

How a Seismometer Works

1. Principle: The seismometer operates on the principle of differential motion between a free mass and a supporting frame fixed to the ground.
2. Structure:
   • A heavy mass (also called a pendulum or inertial mass) is loosely attached to a frame fixed to the Earth.
   • The mass carries a pen or stylus.
   • The frame supports a revolving, clock-driven drum wrapped with special paper.
3. Operation: When seismic waves reach the location, the ground and frame vibrate. The heavy mass, due to its inertia, tends to remain stationary relative to the moving frame. This difference in motion between the mass and the frame is recorded by the pen on the revolving drum as a seismogram.

Types of Seismometers

1. Vertical motion recorder: A mass attached to a vertical spring records vertical ground movements. (Refer to Figure 5.37 (a))
2. Horizontal motion recorder: A mass attached to a hinge detects horizontal ground movements. (Refer to Figure 5.37 (b))

Seismograms and Their Uses

The seismogram shows the ground motion as a function of time, with the recorded amplitude representing the strength of shaking. Analysis of seismograms allows scientists to:

1. Determine the magnitude (strength) of the earthquake.
2. Identify the direction of the seismic waves.
3. Locate the epicenter and focus (hypocenter) of the earthquake.
4. Understand the properties of the medium (rock layers) through which the waves have traveled.

When an earthquake occurs, P-waves (Primary waves) and S-waves (Secondary waves) are generated. Since P-waves travel faster than S-waves, they arrive first at a seismograph station.

Step-by-step Method

a. Measure the Time Difference (S–P Time)

Use a seismogram from a station (e.g., Station A) to find the arrival times of P- and S-waves.

Example:

• P-wave arrives at 6.5 minutes
• S-wave arrives at 11.5 minutes
• S–P time = 11.5 – 6.5 = 5 minutes

b. Determine the Epicentral Distance

• Use a Seismic Travel-Time Graph (like Figure 5.40 in your text) to convert the S–P time difference into distance.
• For a 5-minute difference, the distance is ≈2300 km.
• This means the epicentre lies somewhere on a circle of 2300 km radius centered at Station A.

c. Repeat for Two More Stations

Do the same for two other stations (e.g., Station B and C):

• Station B: S–P time → 7000 km
• Station C: S–P time → 4800 km

d. Triangulation Method

• Draw three circles:
  • Centered at each station
  • With radii equal to the epicentral distances
• The point where all three circles intersect is the epicentre of the earthquake.

The magnitude of an earthquake is a quantitative measure of the energy released at the source of the earthquake. Several magnitude scales are used, but one of the most widely known is the Richter scale, proposed by Professor Charles F. Richter in 1935.

Richter Local Magnitude ($M_L$)

The local magnitude $M_L$ of an earthquake is calculated using the formula:

where:

• $A(\Delta)$: Maximum amplitude of the seismic wave at a distance $\Delta$
• $A_0(\Delta)$: Amplitude of a reference or standard earthquake at the same distance

This formulation ensures that the magnitude remains independent of the station's distance from the epicenter.

Standard Wood-Anderson Seismometer Correction

For standardization, a Wood-Anderson torsion seismometer is used. At a standard epicentral distance of 100 km, the local magnitude is corrected using:

where:

• $A_r$: Observed amplitude at a distance $r$
• $A_{100}$: Reference amplitude at 100 km

For other distances, a correction factor is introduced due to attenuation and spreading of seismic waves. This leads to empirical formulas like:

where $c_1$, $c_2$, and $c_3$ are constants determined from calibration data, and $D$ is the epicentral distance in kilometers.

Lillie's Empirical Formula

A commonly used empirical formula derived by Lillie is:

This equation effectively adjusts the measured amplitude for distance-based attenuation and standardizes the magnitude estimation.

Earthquake Magnitude and Effects Table

Source: British Geological Survey

Modern seismology also utilizes other magnitude scales, such as:

1. Moment Magnitude Scale ($M_w$): More accurate for large earthquakes. Based on seismic moment $M_0$, where:

   $$M_0 = \mu A D$$

   and:

   • $\mu$: Shear modulus of rocks
   • $A$: Area of the fault
   • $D$: Average slip on the fault

2. The moment magnitude is given by:

   $$M_w = \frac{2}{3} \log_{10}(M_0) - 6.0$$