The Latest Mathematical Models of Earthquake Ground Motion

Strong motion instrument networks have enabled creation of a large number of databanks ranging from small to regional and world ones. This data is of a great importance for the investigations aimed at prediction of strong earthquake ground motion parameters by application of empirical mathematical models fitted to the databanks. These mathematical models are referred to as ground motion models or attenuation laws. They define the relationships between ground motion parameters and factors that affect the amplitudes of ground motion as are the released energy, the regional characteristics, the local soil characteristics, the type of fault, the radiation pattern, etc.

motion.It includes the focal mechanism, the size of the seismic field represented by an ellipse with a shape dependent on the relative relationship of its semi-axes and with a longitudinal axis in the direction of the projection of the fault plain upon the surface as well as the position of the instrument location (Stamatovska, 1996).Presented for this mathematical model are the idea used in defining the mathematical equation for a single earthquake, the general procedure of definition of the azimuth dependent mathematical model for any selected azimuth and its application in the seismic hazard analyses.The detailed description of the procedure of its development is aimed at its easier understanding and use by other researchers.This also contributes to easier understanding of the procedure by which the author has developed a new mathematical model based on radius vectors.

Mathematical equation
The starting point is a general empirical ground motion model in which ground motion parameter-Y depends on magnitude-M , distance-R and local soil conditions-S .It is given in Equation 1ln ln ln( ) where, Y -peak ground acceleration-PGA , or peak ground velocity-PGV or peak ground displacement-PGD ; parameter of dynamic response of a linear or nonlinear model of a single degree of freedom system-SDOF, as well as Fourier Amplitude Spectrum-FS M -magnitude h R -hypocentral distance in km S -parameter that includes the effect of local soil conditions and has values, for example, 0 for rock, 1 for alluvium, 2 for deep alluvium C -constant by which is defined the shape of the attenuation in the epicentral zone expressed in km ,,, M RS bb b b -regression coefficients lnY σ -standard deviation P -binary variable, which has the value of 0 and 1 for median and median plus one standard deviation, respectively.
The model is based on the following theoretical assumptions: term M bM e involves the relationship between energy and magnitude; coefficient R b has a negative value and accounts for the spherical spreading of the seismic wave energy, while term S bS includes the effect of local soil conditions.
The ground motion model given in Equation 1 is simplified by use of records of occurred strong earthquakes obtained on rock soil type or referent soil with 700 / S Vm s ≥ , by which the parameter defining the effect of the local soil conditions is omitted.With this, the parameters of ground motion under strong earthquake effect are only a function of distance and magnitude.

Mathematical equation for a single earthquake
The solution of the mathematical equation of a single earthquake came from the analyses of the records of an earthquake obtained at two locations, i.e., by instruments situated at equal epicentral distance from the earthquake epicenter.For each of the two locations, the epicentral distance and the focal depth are equal.The difference is in their position in respect to the projection of the fault upon the surface, i.e., the angle between the direction of the fault plane and the direction toward the instrument location.Hence, the differences in the recorded amplitudes at these two locations result from the position of the location in respect to the projection of the fault plane and the characteristics of the region in the direction of that location.If the recorded amplitudes, for example, amplitudes of PGA with equal value are connected by an isoseismal, then it is clear that, although the two considered locations are at equal epicentral distances, due to the different recorded amplitudes, the two locations will not lie on the same isoseismal.This means that the characteristics of the focus and the region in the direction toward the location perform faster or slower attenuation of the energy of the seismic waves by which they define the form of the isoseismals of equal PGA.Since the earthquake depth is the same for both locations, it is clear that the regional characteristics perform correction through the epicentral distances wherefore the form of the seismic field on the surface is not a circle.Therefore, the model of ground motion for each individual earthquake is a function of corrected epicentral distance or epicentral distance divided by a single function, the so called ρ, whose value depends on the form of the isoseismal of equal amplitudes of PGA and the angle between the fault plane and the direction of the location, i.e., the radiation pattern.
During mathematical modelling, particular importance is given to idealization of the form of the seismic field on the surface.For the azimuth dependent mathematical model developed by the author, it is assumed that this form may range from a circle to any shape of an ellipse with a longitudinal axis in the direction of the projection of the fault plane upon the surface (Figure 1).The shape of the ellipse is defined by the ratio of the semi-axes : ab , whereas the position of any two points М and i М lying on it, is defined by radius vectors ρ if and i ρ ii f , whose moduli are equal to ρ and i ρ .

Regression analysis method
The exploration through analysis of a large number of published ground motion models (Joyner & Boore, 1981;1988;Boore & Joyner 1982;Ambraseys & Bommer, 1992;Ambraseys et al., 1996;Boore et al., 1993;Sabetta & Pugliese, 1987, 1996;Idriss, 1991;Sadigh, 1993;Sadigh at al., 1993;Campbell, 1981) has pointed out the primary importance of the empirical model developed by application of the double regression method.This method (Joyner & Boore, 1981) involves the mode in which earthquakes occur in nature, one at a time, which is encompassed by the first step.Their connection is the objective of the second step.Accordingly, the regression analysis method is carried out in two steps as follows: First step: Definition of ground motion models for each occurred earthquake taken separately, and, Second step: Connection of all occurred earthquakes, i.e., different magnitudes and focal depths.

First step of regression analysis
The first step of the regression analysis involves definition of regression coefficients 0 b and 1 b , and standard deviation ln PGA σ .To carry out the first step, it is necessary to perform parametric analysis in which the value of the parameters affecting function ρ will vary.
These are: the azimuth of the projection of the fault plane upon the surface β and the ratio of the semi-axes of the ellipse of the seismic field : ab .
The procedure itself is reduced to the following: 1.An initial value for the azimuth of the projection of the fault plane on the surface-β (Figure 2a) is selected; 2. The : ab ratio is defined for value of 1. b = , by which the relative ratio of the semi-axes of the seismic field is : 1 a а = (Figure 2a) 3. An initial value of the relative ratio 1 a = .(Figure 2a) is defined; 6.The value of the relative ratio a is changed for an increase of a Δ and the procedure from item 4. (Figure 2b) is repeated; 7. A new value of azimuth β with an increase Δβ is selected and the procedure pursuant to 1 (Figure 2c) is repeated.
A number of solutions is obtained.Out of these, the one for which the standard deviation has the least value is selected.With this, the ground motion model due to an earthquake is defined.In the same way, the ground motion models are defined for all occurred earthquakes originating from a single focus.

Second step of regression analysis
In the second step of the regression analysis, all the occurred earthquakes originating from the same focus are connected and regression coefficients b , R b and M b and the standard deviation ln PGA σ are computed.The data used in the second step of the regression analysis are: earthquake magnitude-M and hypocentral distanceh R as independent variables and PGA as dependent variable (Equation 1).Hypocentral distance is computed according to the following formula: while value e R ρ is computed separately for each occurred earthquake and for all the instrument locations on which the records from that earthquake are obtained.
A key issue in the second step of the regression analysis is the connection of all the earthquakes (Figure 3) and definition of the ground motion model given by Equation 1.

Fig. 3. Connection of earthquakes -second step of regression analysis
The solution is possible only if a ground motion model is defined for a direction toward a location, in which case it is necessary to perform normalization of value e R ρ .The normalization is performed separately for each occurred earthquake with value i ρ defined for the direction toward the selected location by use of the ground motion model computed in the first step of the regression analysis performed for that earthquake (Figure 4).All the normalized values are used in the second step of the regression analysis.
It is possible to compute ground motion models for different directions (azimuths according to locations) in which case it is necessary to perform normalization of e R ρ for each selected direction, separately.
The value of constant C is defined by its variation (for example, from 0 km to 200 km, by a step of 1, or 2, or more km) and execution of the second step of the regression analysis for each of its values.A number of solutions is obtained out of which the one for which the standard deviation in the second step of the regression analysis is minimal, is selected.

Advantages
The advantages of the azimuth dependent ground motion model are: -Definition of separate ground motion models for different directions - The mathematical form of the azimuth dependent ground motion model (Equation 1) is applicable in seismic hazard methodology; -Application in definition of ground motion models for spectral characteristics of ground motion expressed by response spectra and the Fourier Amplitude Spectrum.
In this case, the results from the first step of the regression analysis (Stamatovska, 2008) ( β , a , 0 b , 1 b and ln PGA σ from the first step) defined for PGA are used, and it is only in the second step that the PGA value is replaced by the value of the spectral characteristic of the earthquake, as for example, the spectrum of the linear model of SDOF (absolute acceleration-SA , relative velocity-SV , relative displacement SD ), the Fourier Amplitude Spectrum-FS and the spectrum of the nonlinear model of SDOF (acceleration spectrum, displacement spectrum, ductility factor and alike); -In case of a new earthquake, only the ground motion model for the new earthquake is defined in the first step.All the previous results from the first step obtained for the preceding earthquakes are used (preceding earthquakes + the new earthquake) and the second step of the regression analysis is carried out; -Improvement of the azimuth dependent ground motion model is possible through idealization of the seismic field upon the surface via including irregular forms defined by radius vectors.

Application in probabilistic seismic hazard analyses -PSHA
The application of the azimuth dependent ground motion model in PSHA is based on the following two steps: -Definition of azimuth dependent ground motion models for different azimuth directions; -Definition of sub-sources in a seismic source.
To define the ground motion model for any azimuth direction of a seismic source, it is necessary to pre-define ground motion models for each occurred earthquake from that source by application of the first step of the regression analysis of the azimuth-dependent empirical mathematical model (Stamatovska, 1996(Stamatovska, , 2002(Stamatovska, , 2006(Stamatovska, , 2008;;Stamatovska & Petrovski, 1996, 1997) presented by Equation 1.
Important parameters from the first step of the regression analysis for each occurred earthquake are: the azimuth of the projection of the fault upon the surface-β and the value of the relative ratio а .By using these parameters, the value of function i ρ can be computed for each selected direction i defined by azimuthi β .In doing so, angle-i α , as an angle between the azimuth of the projection of the fault plane upon the surface-β and the selected azimuthi β is defined by using Equation 6.
With the value of function i ρ normalization for the selected azimuth is performed.Each corrected epicentral distance е R ρ in which ρ is the value computed for the azimuth of the instrument location, is multiplied by i ρ .
This procedure is iterated separately for each occurred earthquake originating from the investigated seismic focus (for example, if four strong earthquakes took place, it is iterated 4 times).All the normalized values are used in the second step of the regression analysis and the regression coefficients b , M b and R b as well as the standard deviation lnY σ are computed.With this, the ground motion model for that azimuth is defined.By selection of a new azimuth (new location) and iteration of the entire procedure described in this part, ground motion models for different azimuth directions are obtained.This step is schematically presented in Figure 5.
The computed ground motion models can directly be applied in analyses of seismic hazard for all the software packages in which the ground motion model is assigned or reduced to the mathematical form presented in Equation 1 in the case of a point seismic source.In all other cases of seismic sources, it is necessary to model sub-sources.

Definition of sub-sources in seismic source
In the methods for computation of seismic hazard (Cornell, 1968), the seismic source is modelled as point, line or area source.Each point of the seismic source, defined by Fig. 5. Application of the results obtained in the first step of the regression analysis for definition of the model of ground motion at a selected location coordinates (x, y) where x is east longitude, while y -north latitude, is a potential epicenter of a future earthquake from that focus.The possibility that the model of the seismic source be represented by a point (in the case of a point seismic source), or a number of points (in the case of a linear or an area model of seismic source) facilitates the procedure to be applied if a software package is developed for the purpose of avoiding a large number of computations.Then, the area of the seismic source is modelled by sub-sources with very small areas Sx y ΔΔ Δ = , to be harmonized with the computed ground motion models for different azimuths (Figure 6).
The above means that the azimuths of the end points of the small seismic sub-source computed in respect to a single point in region-i for which the seismic hazard is computed should tend to a single azimuth value.This is possible in all cases where the seismic hazard is computed for a point in the region that is sufficiently distant to reach an azimuth (Figure 6, point 1).However, particular attention should be paid to a point of the region that is very close to the seismic source (Figure 6, point 2) when the azimuth of the end points of the small seismic sub-source do not tend to an azimuth but there is a considerable difference among them.It is further necessary to reduce the area of the seismic sub-source

Mathematical model based on radius vectors
The mathematical model based on radius vectors represents an advanced azimuth dependent mathematical model.It is developed as an azimuth dependent model of a random shape of a seismic field defined by radius vectors in different azimuth directions.

Theoretical background
The ground motion model defined on the basis of radius vectors has the same mathematical form as the azimuth dependent model, or, ln ln( ) The Latest Mathematical Models of Earthquake Ground Motion 123 where: Y is the ground motion parameter (peak acceleration, velocity, displacement, horizontal vector, spectral amplitude, etc.), i ρ is the modulus of the radius vector in respect to any instrument location, whereas L ρ is the modulus of the radius vector in respect to the location/or the direction for which the ground motion model is defined.The effect of the local soil conditions is not included in this mathematical model due to usage of records obtained on one type of local soil conditions (for example, rock with 700 / s Vm s ≥ ).

Method
The method for definition of this model consists of two parts.The first part involves preparation of data to be used in the regression analysis.In this part, the shape of the recorded seismic field defined by radius vectors (Fig. 7) is established.Each radius vector begins at the earthquake epicentre and runs in the direction from the epicentre to the instrument location.Its modulus is equal to the absolute value of peak acceleration /or velocity/ or displacement/ of ground or vector defined for horizontal direction under the earthquake effect.Applying the normalized seismic field for a selected azimuth/ or direction toward a selected location, the value of the relative relationship of L i ρ ρ moduli (Fig. 8) is defined.This relationship is a dimensionless number and enables obtaining the regional characteristics in different directions.It is used to correct the epicentral distances.This is carried out separately for each earthquake that has occurred from a single seismic focus.
In the second part, the multi linear regression analysis method is used.The data for the regression analysis are: PGA -dependent variable, M and h R -independent variables.Each regression analysis results in regression coefficients b , M b , R b and standard deviation -lnY σ . The number of regression analyses depends on the number of variations of constant C (for example, 27 analyses with variable C ranging from 0 to 130 km, with a step of 5 km).From the multitude solutions, the one for which the standard deviation is minimal is selected.
The second part is equal to the second step of the regression analysis applied in the azimuth dependent model.In this way, the simplest mathematical model for prediction of characteristics of future earthquakes from a single seismic focus is obtained.According to the author, this model is the closest to the physical model since it includes a realistically occurred seismic field recorded by strong motion instruments.
The described procedure is based on the idea that the amplitudes of ground motion obtained for different epicentral distances and different azimuths result from the effect of the amount of the energy released by the earthquake, the focal mechanism and the regional characteristics at different azimuths from the earthquake hypocenter.

Method verification
The method verification has been performed on the basis of the created data bank of available three-component records of strong earthquakes that occurred on March 4, 1977(epicenter 45.8N and 26.8E, M=7.2, h=109 km), August 30, 1986(epicenter 45.52N and 26.49E, M=7.0, h=131 km), May 30, 1990 (epicenter 45.872N and 26.885E, M=6.7, h=99.1 km) and May 31, 1990 (epicenter 45.852N, 26.882E, M=6.1, h=89.1 km).The data bank includes data from records of occurred deep earthquakes at the Vranchea focus (Romania) obtained by the instruments of the Romanian, Bulgarian and Former Yugoslav strong motion networks.
The isoseismals of the recorded PGA seismic field (in 2 / cm s for 700 /

S
Vm s ≥ ) referring to the earthquakes that occurred at the Vranchea focus are given in figures 9, 10 and 11.Two separate investigations have been performed.In the first one, the ground motion parameter are the peak ground accelerations from the two horizontal components, while in the second investigation, the ground motion parameter is the higher value of the two horizontal components of the peak ground acceleration.Mathematical models of ground motion have been defined for seven azimuths toward the following instrument locations: BUC (Bucharest), CFR (Carcaliu), CVD (Chernavoda), IASI (Iasi), VLM (Valeni de Munte) and VRI (Vrincioaia).For all these, the regression coefficients and standard deviations are given (Tables 1 and 2).The results shown in Table 1 refer to two horizontal components, whereas those in Table 2 refer 1 and 2, the PGA values have been computed with a non-exceedance of 50% and 84%, or as median and median + 1 standard deviation (Table 3).
The obtained results point to good fitting of the data from the mathematical model based on radius vectors, particularly in the case of use of the higher component from the two horizontal components.This is confirmed by the small values of the computed standard deviations ( ln 0.4 Y σ ≤ ) as well as the values of the median and median+1 standard deviation for the predicted PGA (PGA-L in Table 3).
The obtained PGA values depend on the instrument type, its transmission characteristics, maintenance, knowledge of the characteristics of the local profile of the instrument location, the procedures for processing of records, etc.The effect of the mathematical operations is reduced to minimum since only one multi linear regression analysis is performed.

Advantages and disadvantages
The advantages and disadvantages of the ground motion model based on radius vectors are: - The advantage of the mathematical model based on radius vectors is that it uses a recorded seismic field.In this case, the uncertainties that are incorporated in the computation of the mathematical model of the earthquake ground motion result from the accuracy of the records. - The disadvantage of this model is the case of use of a small number of records of occurred earthquakes and their non-uniform distribution in respect to the different azimuths.In such a case of a small number of records, the irregular closed polygon of the seismic field upon the surface will represent a polygonal figure with longer sides.This is not a deficiency of the method itself but a deficiency related to the available number of records and position of instruments.As such, it will be overcome by gradual increase of the number of instruments and records.

Conclusions and recommendations
The conclusions and recommendations referring to the presented ground motion models are the following: -The azimuth dependent ground motion model defined by application of the double regression analysis contains all the specificities of the occurred individual earthquakes originating from a single seismic source; -In an indirect way, by application of a parametric analysis, it includes in itself the characteristics of the seismic focus and the position of the location in respect to the projection of the fault plane upon the surface, or radiation pattern; - The results obtained in the first step of the regression analysis can be controlled by the results computed by use of seismological data-seismograms.An example for this is the azimuth of the projection of the fault plane on the surface -β ; -It is possible to develop a method for computation of azimuth dependent ground motion model by use of results from seismological investigations, or taking the direction of the projection of the fault plane on the surface from the seismological investigations.This will extensively simplify the computation of the azimuth dependent ground motion model since the first step of the regression analysis will involve only parametric analysis of the relative ratio of the semi-axes of the ellipse of the seismic field : 1 a а = ; -Two models are applicable in seismic hazard analyses; - The ground motion model based on radius vectors will yield even better results if the position of the instrument within an observation network is permanent, if it is regularly maintained and calibrated, if there are as many as possible instruments within the network and if the triggering thresholds are such that records of a number of occurred earthquakes are obtained from as many as possible instruments.So, the more exactly the recorded seismic field is defined, the more reduced will be the values of the standard deviations in the mathematical model of ground motion based on radius vectors.
The author believes that, in future, advantage will be given to the model based on radius vectors particularly due to the increasing number of recording instruments and number of records of occurred strong earthquakes generated from single seismic foci.

Acknowledgement
The author wishes to extend her gratitude to the Ministry of Education and Science of R. Macedonia and to UKIM-IZIIS for permanent moral and financial support of her investigations.

4.
The values of function ρ for all instrument locations and the values of the corrected epicentral distances e R ρ are computed; 5. Linear regression is carried out for dependent random variable PGA and independent random variable a e R ρ .Then, the regression coefficients 0 b and 1 b and the standard deviation ln PGA σ from the first step is computed.

Fig. 2 .
Fig. 2. Procedure referring to the first step of the regression analysis Fig. 4. Normalization over selected azimuth

Fig. 6 .
Fig. 6.Effect of modelling of seismic source and epicentral distance upon the extent of deviation from an azimuth

Fig. 7 .
Fig. 7. Recorded seismic field of PGA at rock to the larger component of the two horizontal components.The March 4, 1977 earthquake is included only for an azimuth toward the INC (INCERC-Bucharest) location.

Table 3 .
Comparison between recorded and predicted values of PGAApplying the regression coefficients and standard deviations from Tables

Table A1 .
(continues on next page) Data used for definition of mathematical model based on radius vectors for the VLM azimuth