SEISCALC

Wells and Coppersmith (1994) examined a global dataset of earthquakes and developed a series of empirically based equations to estimate moment magnitude of an earthquake from local observations of displacement or fault rupture length and the type of fault (e.g., strike-slip, normal, reverse). Such equations are known as scaling relationships. Other researchers have also developed scaling relationships. It is anticipated that future updates to this Encyclopedia will include a multiplicity of scaling relations along with suggested criteria to help decide which equation is most appropriate for a given situation.

For Surface Magnitude (M_S) and Moment Magnitude (M_W) estimates based on Rupture Length (L) in km., Ambraseys and Jackson (1998) presented the following equations:

M_S = 5.13 + 1.14 log(L)        (2)

M_S = 5.27 + 1.04 log(L)        (3)

M_W = 4.9 + 1.33 log(L)         (11)^note

while noting that Equation (3) is almost identical to that derived by Wells and Coppersmith (1994).

Ambraseys and Jackson (1998:397) noted the following

it is important, particularly for palaeoseismological investigations, to have some indication of whether the rupture length and offset estimated from historical sources are likely to be seriously under- or overestimated, given the magnitude of the event. This is a principal use of magnitude—length relationships. For an assessment of individual events or particular regions, it may be more informative to make such estimates from a combination of first principles and more closely constrained empirical relationships, along the following lines:

1. for earthquakes that rupture the entire thickness (d) of the seismogenic upper crust, the downdip width of the fault is d/sinϴ, where ϴ is the fault dip, and the moment is then
   
   M_o = (μcd/sin ϴ)L²       (8)
   
   where
   
   
   • µ is the rigidity modulus
   • c is the ratio of average displacement (u) to fault length (L), which is observed to be close to 5 x 10^-5 for intracontinental earthquakes (Scholz 1982; Scholz et al. 1986)


2. both observationally and theoretically it is known that for such earthquakes the relationship between moment and magnitude (M, whether M_S or M_W ) is of the form
   
   log(M_o)= A + BM       (9)
   
   where A and B are constants, with B ≈ 1.5 (e.g. Kanamori and Anderson 1975; Ekstrom and Dziewonski 1988))


3. combining these expressions gives a relationship between moment and fault length of the form
   
   M = (1/B) log(µcd/sin ϴ) — (A/B) + (2/B) (log L)       (10)
   
   For illustration, if we take
   
   
   • µ=3 x 10¹⁰ N m^-2
   • c = 5 x 10^-5
   • A = 9.0 (for M_o in units of N m, see Ekstrom and Dziewonski 1988)
   • B=1.5
   
   then for a seismogenic layer of thickness d=15 km and a vertical strike-slip fault (ϴ = 90°), the relationship is
   
   M_W = 4.9+1.33L        (11)
   
   with L in kilometres, which is similar to the empirical relationships given above and in Wells & Coppersmith (1994) and is a reasonable fit to the earthquakes of M ≥ 6.0 in Fig. 3

The advantage of this approach over some global empirical relationship is that it is more explicit where the assumptions are: A is known to vary regionally (Ekstrom & Dziewonski 1988) and so is d. Moreover, for earthquakes in which the fault length is small compared with the seismogenic thickness, the relationships between moment and magnitude and between moment and fault length are both known to be different from those given above, such that B≈1.0 (Ekstrom and Dziewonski 1988) and M_o is proportional to L³. Thus a single relationship over the whole magnitude range of Fig. 3 (and over the magnitude ranges discussed by Wells & Coppersmith 1994) is not likely to be valid anyway. The explicit approach illustrated here is therefore more likely to be useful for detailed palaeoseismological investigation of specific events.

Ambraseys and Jackson (1998) produced an equation to estimate Surface Magnitude (M_S) from the average radii of isoseismals (r_i) at a specific value of Intensity (I_i).

M_S = −1.54+0.65(I_i)+0.0029(R_i)+2.14 log(R_i)+0.32p        (1)

where

• R_i =(r_i²+9.7²)^0.5
• r, in kilometres, is the mean isoseismal radius of intensity I
• p is zero for mean values and one for 84 percentile values (Ambraseys 1992).

Ambraseys and Jackson (1998:395-396) noted that with few exceptions, macroseismic data for the historical period are scanty and the magnitudes that can be calculated from eq. (1) are rather uncertain. They suggested in such cases to use the magnitude estimate to group earthquakes into three broad categories

• V, very large events with M. values probably exceeding 8.0
• L, large shocks of magnitude between 7.0 and 8.0
• M, medium events with M. ranging between 6.0 and 7.0

Pavlides and Caputo (2004) examined a dataset of earthquakes from the broader Aegean Region (see Fig. 2 above) and developed a series of empirically based equations to estimate surface magnitude (M_S) of an earthquake from local observations of surface rupture length (SRL), maximum vertical displacement (MVD), and/or average displacement (AD). This resulted in the following regression equations:

M_S = 0.90*log₁₀(SRL) + 5.48          Equation 1

M_S = 0.59*log₁₀(MVD) + 6.75         Equation 2

These equations showed good correlation coefficients (0.84 and 0.82 respectively). Pavlides and Caputo (2004:1) noted that co-seismic fault rupture lengths and especially maximum displacements in the Aegean Region have systematically lower values than the same parameters worldwide, but are similar to those of the Eastern Mediterranean–Middle East region. Pavlides and Caputo (2004) also developed equations to estimate the max and min surface magnitude (M_S) from local observations of surface rupture length (SRL), maximum vertical displacement (MVD), and/or average displacement (AD). These are shown below:

M_Smax = 1.00*log₁₀(SRL) + 5.60          Equation 5

M_Smax = 0.48*log₁₀(MVD) + 7.04         Equation 6

M_Smin = 1.42*log₁₀(SRL) + 4.36          Equation 7

M_Smin = 0.87*log₁₀(MVD) + 6.53         Equation 8

Pavlides and Caputo (2004) also presented equations to solve for surface rupture length (SRL) and maximum vertical displacement (MVD) from surface magnitude (M_S):

log₁₀(SRL) = 0.78*M_S - 3.93          Equation 3

log₁₀(MVD) = 1.14*M_S - 7.82         Equation 4

Figures 5 and 6 (below) show differences between the equations of Wells and Coppersmith (1994), Ambraseys and Jackson (1998), Pavlides and Caputo (2004), and Papazachos and Papazachou (1989).

Source - Wells and Coppersmith (1994)

Variable	Input	Units	Notes
Average Displacement		cm.	Strike-Slip displacement
Maximum Displacement		cm.	Strike-Slip displacement
Variable	Output - not considering a Site Effect	Units	Notes
MW		unitless	Moment Magnitude for Avg. Displacement
MW		unitless	Moment Magnitude for Max. Displacement

Calculate   

Source - Wells and Coppersmith (1994)

Variable	Input	Units	Notes
Average Displacement		cm.
Maximum Displacement		cm.
Variable	Output - not considering a Site Effect	Units	Notes
MW		unitless	Moment Magnitude for Avg. Displacement
MW		unitless	Moment Magnitude for Max. Displacement

Calculate   

Source - Wells and Coppersmith (1994)

Variable	Input	Units	Notes
Average Displacement		cm.	Seismic slip on the ramps
Maximum Displacement		cm.	Seismic slip on the ramps
Variable	Output - not considering a Site Effect	Units	Notes
MW		unitless	Moment Magnitude for Avg. Displacement
MW		unitless	Moment Magnitude for Max. Displacement

Calculate   

• Source - Wells and Coppersmith (1994)

Variable	Input	Units	Notes
Rupture Length		km.	Rupture Length
Variable	Output - not considering a Site Effect	Units	Notes
MW		unitless	Moment Magnitude

Calculate   

• Source - Wells and Coppersmith (1994)

Variable	Input	Units	Notes
Rupture Length		km.	Rupture Length
Variable	Output - not considering a Site Effect	Units	Notes
MW		unitless	Moment Magnitude

Calculate   

• Source - Wells and Coppersmith (1994)

Variable	Input	Units	Notes
Rupture Length		km.	Fault Break
Variable	Output - not considering a Site Effect	Units	Notes
MW		unitless	Moment Magnitude

Calculate   

Variable	Input	Units	Notes
Rupture Length		km.	Rupture Length
Variable	Output - not considering a Site Effect	Units	Notes
MW		unitless	Moment Magnitude from Ambraseys (1988) (developed for the Middle East)
MW		unitless	Moment Magnitude from Bonilla and Lienkaemper (1984)
MS		unitless	Surface Magnitude from Ambraseys and Jackson (1998) Eqn. 2
MS		unitless	Surface Magnitude from Ambraseys and Jackson (1998) Eqn. 3
MW		unitless	Moment Magnitude from Ambraseys and Jackson (1998) Eqn. 11
MS		unitless	Avg. Surface Magnitude from Pavlides and Caputo (2004) Eqn. 1 (developed for the Aegean)
MSmin		unitless	Min. Surface Magnitude from Pavlides and Caputo (2004) Eqn. 7 (developed for the Aegean)
MSmax		unitless	Max. Surface Magnitude from Pavlides and Caputo (2004) Eqn. 5 (developed for the Aegean)

Calculate   

Variable	Input	Units	Notes
Max. Vertical Displacement		m	Max. Vertical Displacement
Variable	Output - not considering a Site Effect	Units	Notes
MS		unitless	Avg. Surface Magnitude from Pavlides and Caputo (2004) Eqn. 2 (developed for the Aegean)
MSmin		unitless	Min. Surface Magnitude from Pavlides and Caputo (2004) Eqn. 2 (developed for the Aegean)
MSmax		unitless	Max. Surface Magnitude from Pavlides and Caputo (2004) Eqn. 2 (developed for the Aegean)

Calculate   

• Estimate Surface Magnitude (M_S) from Avg. radii of isoseismals (r_i) at a specific value of Intensity (I_i)
• Source - Ambraseys and Jackson (1998)

Variable	Input	Units	Variable Name
ri		km.	Mean isoseismal radius for a given Intensity I
Ii		unitless	The given Intensity
p		unitless	p=0 for mean values. p=1 for 84 percentile values (Ambraseys, 1992)
Variable	Output	Units	Notes
MS		unitless	Surface Magnitude

Calculate   

Variable	Input	Units	Notes
Fault Length		km.
Fault Width		km.
Variable	Output	Units	Notes
Mw		unitless	Moment Magnitude computed using Wesnousky (2008)
Mw		unitless	Moment Magnitude computed using Hanks and Bakun (2008)
Fault Area		km.2

Calculate   

Plot M vs. SRL   

Plot M vs. MVD   

The value given for Intensity with site effect removed is how much you should subtract from your Intensity estimate to obtain a pre-amplification value for Intensity. For example if the output is 0.5 and you estimated an Intensity of 8, your pre-amplification Intensity is now 7.5. An Intensity estimate with the site effect removed is helpful in producing an Intensity Map that will do a better job of "triangulating" the epicentral area. If you enter a V_S30 greater than 655 m/s you will get a positive number, indicating that the site amplifies seismic energy. If you enter a V_S30 less than 655 m/s you will get a negative number, indicating that the site attenuates seismic energy rather than amplifying it. Intensity Reduction (I_reduction) is calculated based on Equation 6 from Darvasi and Agnon (2019).

V_S30 is the average seismic shear-wave velocity from the surface to a depth of 30 meters at earthquake frequencies (below ~5 Hz.). Darvasi and Agnon (2019) estimated V_S30 for a number of sites in Israel. If you get V_S30 from a well log, you will need to correct for intrinsic dispersion. There is a seperate geometric dispersion correction usually applied when processing the waveforms however geometric dispersion corrections are typically applied to a borehole Flexural mode generated from a Dipole source and for Dipole sources propagating in the first 30 meters of soft sediments, modal composition is typically dominated by the Stoneley wave. Shear from Stoneley estimates are approximate at best. This is a subject not well understood and widely ignored by the Geotechnical community and/or Civil Engineers but understood by a few specialists in borehole acoustics. Other considerations will apply if you get V_S30 value from a cross well survey or a shallow seismic survey where the primary consideration is converting shear slowness from survey frequency to Earthquake frequency. There are also ways to estimate shear slowness from SPT & CPT tests.

• Source - Rodkin and Korzhenkov (2018)
• Conversion from PGV to Intensity is made using Equation 2 of Wald et al (1999) (only valid for I between V and IX).

Variable	Input	Units	Notes
α		degrees	Critical Tilt Angle
H		m	Wall Thickness
Variable	Output - not considering a Site Effect	Units	Notes
PGV		m/s	Peak Ground Velocity
I		unitless	Intensity

• Source - Rodkin and Korzhenkov (2018)
• Conversion from PGV to Intensity is made using Equation 2 of Wald et al (1999) (only valid for I between V and IX).

Variable	Input	Units	Notes
k		unitless	Coefficient of friction
Displacement		cm.	Displacement of masonry
Variable	Output - not considering a Site Effect	Units	Notes
PGV		m/s	Peak Ground Velocity
I		unitless	Intensity

Static Analysis

A Factor of Safety (FS) for the slope is estimated first to determine if the slope was stable under aseismic conditions. Two methods are considered here - the Fellenius method and the modified Bishop method. Both methods divide the soil model into vertical slices and determine Factor of Safety as a ratio from a sum of moment balances performed on each vertical slice. This accounts for changes in topography and lateral changes in unit thicknesses and elevation of the water table. The generalized equation is shown below.

FS = Σ M_r / Σ M_d

where

M_r = Resisting Moment
M_d = Driving Moment

Resisting Moment is the ability of the soil mass to avoid moving. The driving moment is the forces such as gravity and water saturated layers pushing the soil mass down slope. FS less than 1 indicates that the slope is unstable. FS greater than 1 indicates that the slope is stable. Factor of Safety tends to be sensitive to the height of the water table. Lower values of FS are associated with shallower water tables. If you know the time of year an earthquake struck, this could help you estimate the height of the water table. Well data can also be helpful. Ideally, mechanical tests on soil layers can inform your 2 model of the various layers. Wechsler et al (2009) used the commercial software Slope/W™ to perform the Static Analysis at Umm Qanatir.

Dynamic Analysis

The Newmark Displacement method can be used to evaluate slope stability during an earthquake. First a critical acceleration is estimated. Critical acceleration is the minimum earthquake induced acceleration that initiates slope movement. Critical acceleration is computed as follows:

a_c = (FS-1)sinβ

where

a_c  = critical acceleration measured in g's (1 g = 9.81 m/s²)
FS = Factor of Safety (computed previously for a range of conditions)
β    = Thrust angle (degrees) If strong motion records from the area are lacking, the Newmark displacement D_N can be estimated using an empirically derived formula from data from Southern California. The empirical relationship comes from Jibson et al (2000: p. 8 eqn. 3) which estimates the critical displacement D_N at which the slope begins to fail.

lnD_N = 1.521*ln(I_α) - 1.993ln(a_c) - 1.546

where

D_N = Newmark Displacement (cm.)
I_α  = Arias Intensity (m/s)
a_c  = Critical acceleration (g's)

Arias Intensity (I_α) can be related to Magnitude and Fault Distance using an equation derived for the Dead Sea by Katz and Crouvi (2007):

log₁₀I_α = 1.2M_W - 2.2log₁₀R - 4.9

where

I_α  = Arias Intensity
M_W = Moment Magnitude
R  = Fault Distance (km.)

By assuming a D_N of 5 or 10 cm., one can solve for I_α and then M_W as a function of R.

• Figure 13a from Massa et al (2010)
  Figure 13. (a) Residuals between predicted acceleration response spectra (5% damping) for reference station (NRN2) and for station NRN7, for periods up to 1 s. The results are reported for the west—east component and are presented in terms of logarithmic differences. PGHA indicates peak ground horizontal component (in this case, the east—west component).
  
  Massa et al (2010)

Output with site effect removed assumes that PGA is higher than it would be if there was no site effect. In this situation, Intensity (I) with site effect removed is calculated pre-amplification (i.e. it will be lower). This is because an Intensity estimate with the site effect removed is helpful in producing an Intensity Map that will do a better job of "triangulating" the epicentral area.

Site Effect is based on Equation 2 and Figure 13 a of Massa et al (2010). In their study, they estimated a frequency dependent additional PGA (St in Eqn. 2) which is added by a topographic site effect. The additional topographic site effect PGA varied from ~0.1 g to 0.5 g for dominant frequencies of approximately 1 - 5 Hz.. Higher PGA's were shown to be present for higher frequencies which are more likely to occur when the earthquake producing fault is closer to the site. They also noted that a greater topographic effect was observed when the seismic energy arrived orthogonal (perpendicular in their words) to the ridge. Both of these considerations suggest that a topographic ridge effect should be considered when other evidence suggests that a nearby fault broke during the earthquake. The additional Site Effect PGA is linearly scaled from 0 - 0.5 g for site effects where amplitude increases from 1x to 10x. It's not the greatest transform to remove site effect from the Intensity estimate but may be useful for crude estimates.

Variable	Input	Units	Notes
FS		unitless
β		degrees
DN		cm.	Wechsler et al (2009) recommends a value of 5 or 10
R		km.	Distance to nearest earthquake producing fault
Variable	Output (No Site Effect)	Units	Notes
ac		g	minimum acceleration to induce slide
I		unitless	Conversion from ac to I using Wald et al (1999)
M		unitless	Attenuation relationship of Hough and Avni (2009) used to calculate Magnitude from I and R
Iα		m/s	Calculated from eqn. 2 of Wechsler et al (2009)
Iα		m/s	Calculated from eqn. 3.17 of (Kramer, 1996:87)
Mw		unitless	calculated from eqn. 3 of Wechsler et al (2009) which comes from Katz and Crouvi (2007)
Variable	Input	Units	Notes
Seismic Amplification		unitless	Site Effect due to Topographic or Ridge Effect (set to 1 to assume no site effect)
Variable	Output (Site Effect)	Units	Notes
I		unitless	Intensity with Topographic Effect removed
M		unitless	Magnitude with Topographic Effect removed - using Hough and Avni (2009)
MW		unitless	Moment Magnitude with Topographic Effect removed MW from eqn. 3 of Wechsler et al (2009) which comes from Katz and Crouvi (2007)

Calculate   

CAVEAT

Plot M_W vs. R for D_N = 5    Plot M_W vs. R for D_N = 10   

Salamon and Di Manna (2019) created a bounding envelope for landslide tsunamis based on a curated data set.

• Bounding Envelopes for landslide tsunamis from Salamon and Di Manna (2019)
  Fig. 4.
  
  Empirical ‘Magnitude-Distance’ bounding envelopes for tsunamis generated by seismogenic submarine landslides. Filled circles denote the distance from the seismogenic fault to the landslide scar (R_f), and empty circles denote the distance between the earthquake epicenter and the landslide scar (R_e). The bounding envelope of the maximal distance between the seismogenic fault and the landslide scar is delineated by line A and the bounding envelope between the epicenter and the landslide scar is denoted by the dashed line B.
  
  Salamon and Di Manna (2019)

Plot R_f    Plot R_e   

Variable	Input	Units	Notes
PGA		g	Peak Horizontal Ground Acceleration
Variable	Output - Site Effect not considered	Units	Notes
I		unitless	Conversion from PGA to Intensity using Wald et al (1999)

Calculate   

• Source - Scordilis (2006) as shown in Darawcheh et al (2021)

Variable	Input	Units	Notes
MS		unitless	Surface Magnitude
Variable	Output	Units	Notes
MW		unitless	Moment Magnitude

Calculate