What is seismic magnitude?

Seismic magnitude is a measure of the size of an earthquake obtained from the analysis of seismic waves recorded by seismographs, and is related to the amount of energy released during the rupture of the fault that causes the earthquake. Unlike intensity, which describes the effects and damage observed on the Earth's surface, magnitude is an intrinsic property of the earthquake and, therefore, independent of the observation site.

Magnitude is expressed by a numerical value on a logarithmic scale, which allows very large differences in energy to be represented in a simple way. Thus, an increase of one unit of magnitude corresponds approximately to a ten-fold greater amplitude of the seismic waves and to a released energy about 32 times greater.

Magnitude is a fundamental parameter for comparing earthquakes, preparing seismic catalogs and describing seismic activity with rigor.

Why are there different types?

When an earthquake occurs, the energy released propagates in the form of various seismic waves (P, S, Rayleigh, Love, etc.). Seismograms - the time records obtained by seismic instruments - allow us to calculate different types of magnitude based on specific properties of the signal, such as the amplitude of certain waves, the duration of the record or the physical parameters of the rupture process.

Although all magnitudes aim to quantify the size of the same earthquake, their values ​​do not always coincide exactly. In theory, the different magnitudes should converge towards a similar value, but each is based on different observations and has its own limitations. These limitations may depend on factors such as the distance between the earthquake and the seismic station, the depth of the focus, the size of the earthquake, the type of waves analyzed or the characteristics of the instruments used.

For this reason, the same earthquake can have several magnitude values ​​associated with it depending on the calculation method used. In order to have a more robust and comparable measurement between different types of earthquakes, the moment magnitude (Mw) is currently the reference magnitude in most seismic catalogs.

The main types of seismic magnitude are described below. The definitions presented are general in nature and their implementation may require specific adaptations or calibrations depending on the region or seismic network used.

Ml: Local magnitude

The local magnitude, Ml, often called the Richter magnitude, was introduced by Richter (1935) to quantify the size of earthquakes in California. Its original conception was based on a local seismic network consisting of Wood–Anderson seismographs. He defined a scale that assigned Ml = 0 to an earthquake that produced an amplitude of 1 micrometer at a station located 100 km from the epicenter. Although the scale was created for specific conditions and instruments, the concept has become established as a standard measure for local and regional earthquakes. It is especially useful for small and moderate earthquakes, since it is based on information of relatively high frequencies, usually well recorded by nearby stations. The most important limitation is that it saturates for high magnitudes, approximately for Ml > 6.

In the recording of the horizontal components of the seismometer, the synthetic Wood–Anderson instrumental response is applied to simulate how that original instrument would have responded. Once the corrected amplitude (A) in millimeters is obtained, Ml is calculated using the formula:

M_l  =  log₁₀ (A_02P)  -  log₁₀ (A₀)

Where:

• A_02P is the maximum amplitude from zero to peak expressed in mm.
• log₁₀(A₀) is the empirical calibration function that depends on the epicentral distance.

Mb: Internal wave magnitude

The internal wave magnitude, Mb, is a seismic scale that estimates the size of an earthquake from the amplitude of the P, PP or S waves, measured with broadband or long-period instruments in periods between 2 and 20 s. Its original formulation developed by Gutenberg (1945b and c) is expressed as:

Mb = log₁₀⁡(A/T)_max + Q(Δ,h)

Where:

• A is the amplitude of the P, PP or S wave.
• T is its period.
• (A/T)_max is the maximum value of the A/T quotient.
• Q(Δ,h) is a correction term that depends on the epicentral distance (Δ) and the depth of the focus (h).

The magnitude Mb has been widely used in global seismology, with variations, as it allows for rapid estimates of earthquakes recorded at great distances (teleseisms). It also played a relevant role in the monitoring of nuclear tests, due to its sensitivity to P waves.

However, it has an important limitation: saturation in large earthquakes (from Mb > 6), since the high-frequency content does not increase proportionally with the size of the rupture. For this reason, today the moment magnitude (Mw) is the reference scale, although Mb is still used in rapid detection and analysis of small to moderate earthquakes.

Ms: Surface wave magnitude

The surface wave magnitude, Ms, is a historically very relevant magnitude developed by Gutenberg (1945a), useful for characterizing moderate to large earthquakes based on the amplitude of Rayleigh waves. However, it suffers from saturation for magnitudes above approximately 8 and depends on the surface structure. Today it is often replaced by the Mw, which is much more robust and accurate for large events, but there is an extensive database that facilitates historical comparisons.

Reference seismological agencies, such as the International Seismological Center (ISC) or the United States Geological Service (USGS), use the formula:

Ms = log₁₀⁡(A/T)_max + 1.66·log₁₀ (D) + 3.30

Where:

• A is the amplitude of the Rayleigh waves.
• T is the period (s).
• (A/T)_max is the maximum value of the A/T ratio.
• D is the epicentral distance (°).

Md: Duration magnitude

The duration magnitude (Md) is a seismic magnitude that is calculated from the observable duration of the seismic signal recorded at a station, differentiating itself from magnitudes based on the maximum amplitude of a wave type. It is very useful for small earthquakes, with very low amplitudes and difficult to separate from noise. The larger the earthquake, the longer the duration of the signal, especially the late part. This provides an indirect but stable measure of the size of the earthquake.

Although the specific formulas may vary by agency, the general process for calculating Md is:

Md = a·log⁡₁₀ (T) + b

Where:

• T is the total measured duration of the signal.
• a and b are empirical constants defined by the seismic network.

Mw: Moment magnitude

The moment magnitude (Mw) is a seismic scale that aims to more accurately represent the energy released by earthquakes. Unlike other magnitudes based on seismic wave amplitudes (such as Ml, Mb, or Ms), Mw is based on the seismic moment (M₀), a physical quantity directly related to the actual strength of the fault rupture process, making it highly reliable and consistent.

One of the problems with traditional magnitudes, such as Ms, Mb or Ml, is that they suffer from saturation, that is, from a certain size of the earthquake, the amplitude of the waves no longer grows proportionally, which prevents distinguishing between very large earthquakes. On the other hand, the moment magnitude (Mw) does not saturate. It provides consistent values ​​from small earthquakes to the largest ever recorded and better reflects the energy released, especially in earthquakes of Mw > 7, where classical magnitudes fail.

These advantages have made Mw the reference scale in modern seismology and is the standard magnitude used by international catalogs.

The general mathematical relationship is:

Mw = 2/3·log₁₀ (M₀) - 10.7

The seismic moment (M₀) is defined as:

M₀= µ·A·D

Where:

• μ is the rock stiffness (shear modulus), which indicates how stiff the rock is. A stiffer rock requires more force to deform.
• A is the area of ​​the rupture zone.
• D is the average displacement between the fault blocks.

How do we calculate seismic magnitude at the ICGC?

The ICGC currently calculates two types of seismic magnitude, local magnitude and moment magnitude. This section describes how the ICGC performs these calculations based on the general definition of each of these magnitudes.

Ml: Local Magnitude

The methodology for calculating the local magnitude used by the ICGC is an adaptation of the general definition to the particularities of the Seismic Network of Catalonia and our territory.

The synthetic Wood-Anderson instrumental filter is applied to the signal of the horizontal components of the seismometer, in order to simulate the original response of this instrument, considering the characteristics provided by its manufacturer: natural period Ts = 0.8s, damping factor Ds = 0.8 and static amplification V_max = 2800. Once the corrected amplitude (A) in millimeters has been obtained, the Ml for each of the components is calculated using the formula:

M_l = log₁₀ (A_P2P / 2) - log₁₀ (A₀)

Where:

• A_P2P is the maximum peak-to-peak amplitude in mm.
• log(A₀) is the empirical calibration function that depends on the epicentral distance, adjusted for distance steps. The one defined by Richter (1958) for California is currently used.

The final value of the local magnitude is determined by calculating the median of all the Ml values ​​calculated for each of the components.

Mw: Moment magnitude

The moment magnitude is calculated at the ICGC using the method developed and described by Delouis (2014).

Using this method, the moment magnitude (Mw) and the double-couple focal mechanism (direction, slope and slip inclination) are determined, which provides a first-order estimate of the rupture length and directivity. Finally, a more refined calculation is computed to determine the depth of focus.

References

• Delouis, B. (2014). FMNEAR: determination of focal mechanism and first estimate of rupture directivity using near source records and a linear distribution of point sources. Bulletin of the Seismological Society of America, 104 (3): 1479-1500, doi:10.1785/0120130151.
• Gutenberg, B. (1945a). Amplitudes of surface waves and magnitudes of shallow earthquakes. Bulletin of the Seismological Society of America, 35: 3-12.
• Gutenberg, B. (1945b). Amplitudes of P, PP, and S and magnitude of shallow earthquakes. Bulletin of the Seismological Society of America, 35: 57-69.
• Gutenberg, B. (1945c). Magnitude determination of deep-focus earthquakes. Bulletin of the Seismological Society of America, 35: 117-130.
• Richter, C. F. (1935). An Instrumental Earthquake Magnitude Scale. Bulletin of the Seismological Society of America, 25 (1): 1-32.
• Richter, C. F. (1958). Elementary Seismology. W.H. Freeman, San Francisco, 768 p.