 # SEISCALC

## Coded by Jefferson Williams

##### Seismic Attenuation
Explanation

• The attenuation relationship (Eqn. 5 on p. 302) of Hough and Avni (2009) was primarily derived from estimates of inferred local Intensity due to the 1927 ML 6.25 Jericho Earthquake. It may be less accurate in the far field because news reports were used to infer local Intensity in further afield locations such as Egypt (Amotz Agnon, personal communication, 2020 or 2021).

Calculator

• Conversion from Intensity to PGA using Wald et al (1999)
• Site Effect not considered
Variable Input Units Notes
Magnitude
km. Distance to earthquake producing fault
Variable Output - Hough and Avni (2009) Units Notes
unitless Local Intensity - Inputs are M (unspecified) and Repi
m/s2 Conversion from Intensity to PGA using Wald et al (1999)

##### Magnitude Estimation
Explanations

Wells and Coppersmith (1994)

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.

Ambraseys and Jackson (1998)

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

MS = 5.13 + 1.14 log(L)        (2)

MS = 5.27 + 1.04 log(L)        (3)

MW = 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

Mo = (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 MS or MW ) is of the form

log(Mo)= 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 Mo 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

MW = 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 Mo 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.

Estimating Magnitude from an Isoseismal Map - Ambraseys and Jackson (1998)

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

MS = −1.54+0.65(Ii)+0.0029(Ri)+2.14 log(Ri)+0.32p        (1)

where

• Ri =(ri2+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

Calculators

Moment Magnitude from Strike-Slip Fault Displacement -Wells and Coppersmith (1994)

Source - Wells and Coppersmith (1994)

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

Moment Magnitude from Normal Fault Displacement - Wells and Coppersmith (1994)

Source - Wells and Coppersmith (1994)

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

Moment Magnitude from Reverse Fault Displacement - Wells and Coppersmith (1994)

Source - Wells and Coppersmith (1994)

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

Moment Magnitude from Strike-Slip Fault Rupture Length - Wells and Coppersmith (1994)

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

Moment Magnitude from Normal Fault Surface Rupture Length - Wells and Coppersmith (1994)

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

Moment Magnitude from Reverse Fault Surface Rupture Length - Wells and Coppersmith (1994)

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

Moment or Surface Magnitude from Rupture Length - not Wells and Coppersmith (1994)

Variable Input Units Notes
km. Rupture Length
Variable Output - not considering a Site Effect Units Notes
unitless Moment Magnitude from Ambraseys (1988)
(developed for the Middle East)
unitless Moment Magnitude from Bonilla and Lienkaemper (1984)
unitless Surface Magnitude from Ambraseys and Jackson (1998) Eqn. 2
unitless Surface Magnitude from Ambraseys and Jackson (1998) Eqn. 3
unitless Moment Magnitude from Ambraseys and Jackson (1998) Eqn. 11

Estimating Magnitude from an Isoseismal Map

• Estimate Surface Magnitude (MS) from Avg. radii of isoseismals (ri) at a specific value of Intensity (Ii)
• Source - Ambraseys and Jackson (1998)
Variable Input Units Variable Name
km. Mean isoseismal radius for a given Intensity I
unitless The given Intensity
unitless p=0 for mean values. p=1 for 84 percentile values (Ambraseys, 1992)
Variable Output Units Notes
unitless Surface Magnitude

Moment Magnitude from Fault Length and Width

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

##### VS30 Site Effect
Explanation

VS30 Site Effect Explanation

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 VS30 greater than 655 m/s you will get a positive number, indicating that the site amplifies seismic energy. If you enter a VS30 less than 655 m/s you will get a negative number, indicating that the site attenuates seismic energy rather than amplifying it. Intensity Reduction (Ireduction) is calculated based on Equation 6 from Darvasi and Agnon (2019).

VS30 Explanation

VS30 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 VS30 for a number of sites in Israel. If you get VS30 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 VS30 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.

Calculator - Intensity Reduction due to VS30 Site Effect

Variable Input Units Notes
m/s Enter a value of 655 for no site effect
Equation comes from Darvasi and Agnon (2019)
Variable Output - Site Effect Removal Units Notes
unitless Reduce Intensity Estimate by this amount
to get a pre-amplification value of Intensity

##### Slope Effect
Explanation

Salamon et. al. (2010) noted that seismic amplification could occur on slopes greater than 60 degrees where the slope height is roughly equal to one fifth of a seismic wavelength. Turning this relationship around, the frequency at which this effect will occur is defined as follows :

f = VS/(5*H)

where
f = frequency (Hz.)
VS = Shear Wave Velocity (m/s)
H = slope height in meters

Calculator and Plot

Source - Salamon et. al. (2010)

Slope Effect Calculator
Variable Input Units Notes
m/s Shear Wave Velocity
m Slope Height
Variable Output Units Notes
Hz. Frequency

Plot

##### Ridge Effect (aka Topographic Effect)
Explanation

• Figure 13a from 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. 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.

Calculator - Estimate Magnitude and Intensity with and without a Site (Ridge) Effect

• Sources
• Conversion from PGA to Intensity using Wald et al (1999)
• Magnitude is calculated from Intensity (I) and Fault Distance (R) based on Hough and Avni (2009) who did not specify the type of Magnitude scale they were using.
Variable Input Units Notes
g Peak Horizontal Ground Acceleration
km. Distance to earthquake producing fault
unitless Site Effect due to Topographic or Ridge Effect
(set to 1 to assume no site effect)
Variable Output - Site Effect not considered Units Notes
unitless Conversion from PGA to Intensity using Wald et al (1999)
unitless Attenuation relationship of Hough and Avni (2009)
used to calculate Magnitude
Variable Output - Site Effect removed Units Notes
unitless Conversion from PGA to Intensity using Wald et al (1999)
unitless Attenuation relationship of Hough and Avni (2009)
used to calculate Magnitude

##### Archaeoseismic Calculators
Explanation

Rodkin and Korzhenkov (2018) presented two ways to estimate Peak Ground Velocity (PGV) - the Tilt Method (my name) and the PGV estimation method (PGVEM - their name).

###### Tilt Method
• Fig. 11 from Rodkin and Korzhenkov (2018) -
• Figure 4 from Korzhenkov and Mazor (2014) -
This method requires as input the Critical Tilt angle (α) of a wall in order for it to collapse. If the tilt is large enough, the projection of the wall's center of gravity will be located outside its base (see Fig. 11) and the wall will fall over. In order to estimate the Critical Tilt angle (α), we need to come up with some wall dimensions and input them into the calculator below. Calculated critical tilt angle is for a rigid wall. If the wall is composed of blocks which are mortared together, as the seismic forces cause the wall to tilt the top of the wall may start to bend and fail. If the top of the wall is destructed, the lower part of the wall will have a different effective geometry and require a larger tilt to fall over. Archaeoseismic observations can help produce a realistic tilt angle. For example, Rodkin and Korzhenkov (2018) noted that tilt angles in the lower parts of walls in Rehovot ba-Negev reached 15° to 20° (e.g. see Figure 4) while critical title angle based on geometry of the site and the calculator below leads to a values of 11°-12°. In this case, the larger observed angles are preferred.
Structure Height (m) Thickness (m) α - Critical Tilt Angle
Church
House

###### PGV Estimation Method
• Fig. 13 from Korzhenkov and Mazor (2014) -
• Figure 16 from Korzhenkov and Mazor (2014) -
The PGV Estimation Method also requires two inputs - the coefficient of friction (k) of the sliding masonry block and the observed displacement of the block. At Rehovot ba-Negev, Rodkin and Korzhenkov (2018) estimated that k varied from 0.8 - 1.0 and displacement went as high as 10 - 15 cm. (the larger values are more important in their method). An example of the larger observed shift of ~15 cm. at Rehovot ba-Negev can be seen in Figure of 13 of Korzhenkov and Mazor (2014). Another example can be seen in Figure 16 from the same article. Although the PGV Estimation Method is their preferred method, there were apparently a limited number of displacement measurements in Rehovot ba-Negev. Thus, they included the Tilt method to help constrain reasonable PGV values. Inputting their suggested range of k values and displacement values leads to PGV values between 1.3 and 1.7 m/s - higher than what one obtains with the Tilt Method.

Calculators

Tilt Method Calculator

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

PGV Estimation Method Calculator

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

##### Landslide Calculators
Explanation

2D Model used by Wechsler et al (2009) for Umm Qanatir

###### 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 Mr / Σ Md

where

Mr = Resisting Moment
Md = 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:

ac = (FS-1)sinβ

where

ac  = 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)
Thrust Angle

• Newmark (1965)'s slope stability model showing thrust angle (β)
For a planar slide, thrust angle (β) is the direction the COG of the sliding block moves when displacement initially occurs. Ideally this is the dip of the bedding plane. In regional studies this is typically approximated by the slope angle (Katz and Crouvi, 2007 citing Miles and Keefer, 2001). For rotational movement on a circular surface, Newmark (1965) showed that the thrust angle is the angle between the vertical and a line segment connecting the center of gravity of the landslide mass and the center of the slip circle ( Jibson, 1996:310).

If strong motion records from the area are lacking, the Newmark displacement DN 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 DN at which the slope begins to fail.

lnDN = 1.521*ln(Iα) - 1.993ln(ac) - 1.546

where

DN = Newmark Displacement (cm.)
Iα  = Arias Intensity (m/s)
ac  = Critical acceleration (g's)

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

log10Iα = 1.2MW - 2.2log10R - 4.9

where

Iα  = Arias Intensity
MW = Moment Magnitude
R  = Fault Distance (km.)

By assuming a DN of 5 or 10 cm., one can solve for Iα and then MW as a function of R.

Ridge Site Effect Removal Methodology

• Figure 13a from 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.

Calculators

2D Model used by Wechsler et al (2009)

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

CAVEAT

##### Tsunami Calculators
Explanation

Salamon and Di Manna Plot

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

Calculators and Plotters

Salamon and Di Manna Plot

• Bounding Envelopes for landslide tsunamis from Salamon and Di Manna (2019)

##### Sand Boils
Explanation

Obermeier (1996) supplied a chart which can be used to estimate PGA (Peak Ground Acceleration) of earthquake induced Sand Boils.

Sand Boil Chart to estimate PGA Fig. 9. - Proposed boundary curves relating thickness of nonliquefiable surface layer to thickness of the liquefiable zone as a function of peak earthquake accelerations required to induce venting or ground rupturing at the surface. From Ishihara (1985). Obermeier (1996)

Convert PGA to Intensity
Variable Input Units Notes
g Peak Horizontal Ground Acceleration
Variable Output
(No Site Effect)
Units Notes
unitless Conversion from PGA to Intensity using Wald et al (1999)

##### Various Converters
Calculators

Convert PGA to Intensity

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

Convert Surface Magnitude to Moment Magnitude

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