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• from Zohar and Marco (2012:19)

We present a new approach for re-estimating an epicenter of historical earthquake using the spatial distribution of intensity data. We use macroseismic data related to the 1927 Jericho earthquake since this is the first strong earthquake recorded by modern seismographs and is also well documented by historical evidence and reports. The epicenter is located in two sequential steps:

1. Correction of previously-evaluated seismic intensities in accordance with the local site - attributes:construction quality,topographic slope, groundwater level,and surface geology
2. Spatial correlation of these intensities with a logarithmic variant of the epicentral distance.

The resulted location (approximated to 35.5°/31.8°) is consistent with the seismogram-based location calculated by Avni et al. (2002) and also of BenMenahemetal.(1976) with a spatial error of 50 km.T he proposed method suggests an additional approach to the formers based mainly upon spatial analysis of intensity data.

• from Zohar and Marco (2012:21-22)

Several site-attributes are considered:

1. Construction quality – Equal ground acceleration will cause different damage to manmade structures depending on their material and construction (Bolt, 1978). In general, the quality of construction at the beginning of the 20th century is considered to be low and this is well described by Michaeli (1928) and Willis (1928), both experts that advised the government of necessary repairs after the earthquake. However, one could distinguish between two levels of construction quality. Towards the end of the 19th century, massive growth in Jewish population (Ben-Arieh, 1981) resulted establishments of new settlements (e.g., Tel Aviv, Petah Tiqwa) or expansion of existing such as Jerusalem and Haifa. These were built with contemporaneous new techniques and materials such as iron casting, cement and concrete (Fuchs, 1998; Krivoruchko, 1996) while Arabic settlements were less developed and were based primary upon pre 20th century traditional construction techniques, considered to be less resistant to shaking because of extensive use of raw materials such as adobe and wood. Accordingly, we classify localities into two groups of construction quality in which the Jewish settlements are considered stable (Table 1) and thus, ascribe exaggeration of intensities to the Arabic settlements.


2. Topographic slope – the relation between damage and the underlying topographic slope is found to be significant (e.g., Athanasopoulos et al., 1988; Avni, 1999; Evernden et al., 1973; Wust-Bloch, 2002; Zaslavsky et al., 2000). Moreover, the larger the ratio between the height of the mountain peak above its surroundings to its width, the higher is the amplification (Salamon et al., 2010). Slope was calculated using a digital elevation model (DEM) of Hall and Cleave (1998) attributed with resolution of 25 m per pixel along with horizontal and vertical errors ranging between 10–50 m and less than 10 m for 95%o f the data respectively. Because the resolution is too low to use this DEM for calculating topographic slopes of individual structures, the slope of the area of each locality is calculated. Having a max slope value of 31°, we decided to use the median (15°) as a threshold of steep slope in which localities attributed with slope greater than this threshold are presumed to be overestimated (Table 1).


3. Groundwater level depth – P and S waves behave differently as they pass through solid and liquid volumes. P waves travel trough both volumes, while the S waves travel only trough solid and tends to be absorbed in liquid (Bolt, 1978; Evernden et al., 1973).Researchers(e.g.,Yang and Sato, 2000)found that depth of underground groundwater level, mainly in the manner of saturation and thickness, affects the movement of the ground in soft soils such as alluvial areas. Full analysis of this issue is rather complex and beyond the scope of our paper. Yet, we examine the groundwater level depth as a possible influencing factor. This is held using data of drills and boreholes collected in 1933/4 during a comprehensive survey throughout mandatory Palestine (Blake and Goldschmidt, 1947). Comparison of precipitation recorded in 1927 and 1933/4 enables to trace the fluctuation of groundwater level depth of the two sequences and estimate their 1927 contemporaneous depth. Nevertheless, accuracy of the data is limited and fraught with uncertainties whereas many of the localities are attributed with interpolated-based values. Furthermore, the 1933–4 groundwater level could have been influenced by the earthquake itself. Consequently, only intensity in localities with very shallow groundwater depth i.e., between 0 and 10 m (Table 1), were considered to be amplified.


4. Surface geology – amplification occurs when there is a considerable decrease in shear wave velocity (Vs) toward the surface layers. Localities built on unconsolidated soils are likely to suffer more damage than those built on rocks (Yeats et al., 1997). Accordingly, we categorized localities into three surface geology classes (Table 1):
   
   
   1. unconsolidated soil
   2. soft rock (clay and marlstone)
   3. solid rock
   
   which tend to attenuate damage and thus, lead to underestimation of intensity.


5. Population – Ambraseys (1971) found a correlation between density of populations to post-earthquake reports in which reports of densely populated area tend to be exaggerated. Feldman and Shapira (1994) found a tendency for higher estimation of seismic intensity in large cities such as Jerusalem and Tel Aviv. Consequently, we categorize localities into two classes defined by a population threshold of 5000 residents (Table 1)

To determine which of the site-attributes is significant, a multi- linear regression is carried out:

Y = α₁*X₁ + α₂*X₂ + ... α_n*X_n + β          (1)

Whereas Y is the mode intensity of a given locality, X_1-n represents a given site-attribute and α, β are the regression's slope and intercept respectively.

The significance of each site-attribute's correlation with intensity is listed in Table 1. Besides population, the correlation of all other four site-attributes with the intensity is found to be significant, meaning that we can reject the null hypothesis of no impact over the intensity. Accordingly, the population site-attribute was excluded from of the correction procedure.

Correction rate was set to the standard error of the mode intensity of (Avni, 1999). Being almost identical to one MSK unit (STD=1.2419), the correction rate was refined accordingly to a value of 1 (Table 1)and is based upon linearity assumption that is, the difference between intensity 3 and 4 is identical as between 8 and 9. Out of 133 inspected localities, 111 were corrected.

• from Zohar and Marco (2012:22)

Most of the seismic activity in the area is related to the Dead Sea Transform (DST) (Garfunkel et al., 1981). Thus, our inspected geographic area covers the central of DST region extending between 33.96˚ to 36.06˚ and 30.94˚ to 33.26˚ in the east–west and south–north axis respectively (Fig. 1). Initially, a set of assumed epicenters is generated and arranged in a 5*5 km grid-cell structure in which each polygon cell represents an assumed epicenter, totaling 2501 inspected cells (Fig. 2). Then we iterate these cells and calculate epicentral distances to the intensity localities (Fig. 3). However, the decay of intensity with distance is non linear and depends also on the azimuthal variations of the radiated energy (Howell and Schulz, 1975), anisotropic wave propagations (Stromeyer and Grunthal, 2009) and local site-effects (Bolt, 1978). Therefore, logarithmic proportionality (e.g., Bakun, 2006; Bakun and Scotti, 2006; Howell and Schulz, 1975; Stromeyer and Grunthal, 2009) is commonly used to describe this relationship. Following this approach, we adopt the attenuation relation developed by Stromeyer and Grunthal (2009) after Sponheuer (1960):

I = I^* - a*log(SQRT[(R²+h²)/h²]) -b*[SQRT(R²+h²)-h]          (2)

Where I is the given intensity, I^* is an estimate of the I₀ (Pasolini et al., 2008), a and b are constants representing the geometric spreading and absorption of energy respectively, R is the epicentral distance and h is the rupture depth. Assuming that I^*, a and b are constants, I depends on the following logarithmic proportionality:

I = log(SQRT[(R²+h²)/h²]) - [SQRT(R²+h²)-h]                   (3)

Taking into consideration that most earthquakes that are related to the DST are generated in depth of up to 15 km (Shapira, 1979), h was set to 15 km and accordingly, a logarithmic factor for each of the epicentral distances calculated previously was generated.

Finally, for each cell we have correlated the intensities and their adjacent logarithmic factor of distance and use the ‘Pearson’ estimate as a quality indicator.

• from Zohar and Marco (2012:23)

Upon completeness of all 2501 cell correlations, we scaled and examined their Pearson indicators. These values range between 0.2061 and 0.6478 whereas the highest values are related to cells located at the north of the Dead Sea. Fig. 4 demonstrates a graduated colored grid in which each of its cells represents an adjacent Pearson indicator of an assumed epicenter. The highest correlation values, covering an area of approximately 50 km² , are emphasized indicating the statistically most appropriate location of the epicenter. The center of mass of this area is located within the DST region and is about 35 km north of the epicenter calculated by Avni et al. (2002) and ~25 km south of the one calculated by Ben Menahem et al. (1976). Considering that the expected subsurface rupture length of such magnitude is approximately 20 km (Wells and Coppersmith, 1994), we conclude that this model is consistent with previous suggested locations with possible error of up to 50 km.

To demonstrate the importance of intensity correction we implement a similar model only with intensity data that was not corrected i.e., the raw mode intensity of Avni (1999). The result of this model is presented in Fig. 5 in which the suggested location is some 25 km east of the DST region. Furthermore, this indication is nearly 45 km northeast to Avni's calculation and 35 south-east of Ben-Menahem. That is, correction of the intensity yields a location that is more consistent with previous calculations than using uncorrected data.

Sensitivity of the model is also to be tested. This is done using exaggerated correction rate i.e., correcting the intensity with 2 MSK units instead of 1. The results are shown in Fig. 6 and demonstrate a location which is distant by far from the DST and the two previous calculations.

• from Zohar and Marco (2012:23-25)

• The core of the process utilizes spatial distribution of intensities related to the 1927 Jericho earthquake. This is the first significant earthquake in the region for which we have both seismograph measurements and macroseimic data. Destructive earthquakes in the region had not occurred ever since and thus, the model could not be tested on additional local events.


• Though subjective and incomplete, the correction of intensities in accordance to local site-attributes is of great importance and in several cases, calibrates extreme intensities to their actual size. We concentrate in four site-attributes we find representatives of significance influence: construction quality, topographic slope, groundwater level depth and surface geology. Obviously, additional site-attributes could be added and thus improve the model accuracy.


• The MSK intensity scale we use is suited for European construction (mainly wood) whereas typical construction in the Levant area includes also soft material such as mud and bricks. Furthermore, there is a difference between the two regions in which surface geology and ground soils respond differently to ground shaking (Ramazi and Haghani, 2007). Nevertheless, we use the European scale for two reasons:
  
  
  1. there is no other scale suited specifically for the Levant and the MSK scale is the most similar
  2. errors that occur in estimating the intensity values will probably reflect most localities equally, and will not affect dramatically on the epicenter location.


• The result of the proposed method coincides with the seismogram based calculation of Avni (1999) and also of Ben Menahem et al. (1976) with spatial error of 50 km. However, it is not aimed at distinguishing between these former calculations; rather, it provides an additional method for estimating the earthquake epicenter.

can you summarize zohar and marco's 2012 article in journal of applied geophysics ?

Sure! The article by Motti Zohar and Shmuel Marco titled Re-estimating the epicenter of the 1927 Jericho earthquake using spatial distribution of intensity data was published in the Journal of Applied Geophysics in 2012.

The study presents a "new approach" for re-estimating the epicenter of historical earthquakes using "macroseismic data" (information about earthquake intensity and effects). Specifically, they focused on the "1927 Jericho earthquake", which was the first strong earthquake recorded by modern seismographs and is well-documented by historical evidence.

The authors used two main steps to locate the epicenter:

1. "Correction of previously-evaluated seismic intensities" based on local site attributes such as construction quality, topographic slope, groundwater level, and surface geology.

2. "Spatial correlation" of these intensities with a logarithmic variant of the epicentral distance.

Their findings suggest that the epicenter is located at approximately "35.5°N, 31.8°E", which is consistent with previous seismogram-based locations.

Does this help with what you were looking for?

can you write an algorithm in python to do what was done in the article ?

Sure! Here’s a simplified version of an algorithm in Python that can be used to estimate the epicenter of an earthquake based on intensity data:

Python

import numpy as np
import pandas as pd
from scipy.optimize import minimize

# Example data (intensity, latitude, longitude)
data = [
{'intensity': 6, 'lat': 31.7667, 'lon': 35.2333}, # Jerusalem
{'intensity': 8, 'lat': 31.8833, 'lon': 35.2000}, # Jericho
{'intensity': 5, 'lat': 32.0796, 'lon': 34.7806}, # Tel Aviv
# Add more data points as needed
]

def distance(lat1, lon1, lat2, lon2):
# Calculate the Haversine distance between two points on the Earth
R = 6371 # Earth radius in km
dlat = np.radians(lat2 - lat1)
dlon = np.radians(lon2 - lon1)
a = np.sin(dlat / 2) ** 2 + np.cos(np.radians(lat1)) * np.cos(np.radians(lat2)) *
np.sin(dlon / 2) ** 2
c = 2 * np.arctan2(np.sqrt(a), np.sqrt(1 - a))
distance = R * c
return distance

def intensity_error(params, data):
lat_epicenter, lon_epicenter = params
error = 0
for point in data:
dist = distance(lat_epicenter, lon_epicenter, point['lat'], point['lon'])

predicted_intensity = point['intensity'] - 2 * np.log(dist)
error += (predicted_intensity - point['intensity']) ** 2
return error

# Initial guess for the epicenter location
initial_guess = [31.8, 35.5]

# Optimization to find the epicenter
result = minimize(intensity_error, initial_guess, args=(data,), method='L-BFGS-B',
bounds=[(31, 32), (35, 36)])
epicenter = result.x

print(f'Estimated epicenter location: Latitude {epicenter[0]}, Longitude {epicenter[1]}')
```

This example uses a basic method to calculate the epicenter by minimizing the difference between predicted and observed intensities. Of course, for a more accurate model, you would incorporate the specific corrections and spatial correlations mentioned in the article by Zohar and Marco.

Let me know if you'd like further details or help with anything else!