JP2023012586A - Method and system for estimating size of earthquake vibrations - Google Patents

Abstract

To provide a method and a system for estimating the size of earthquake vibrations that can accurately estimate the size of the earthquake vibrations entering a building.SOLUTION: A method for estimating the size of earthquake vibrations entering a building comprises: obtaining waveforms that are obtained as earthquake observation records at adjacent locations; calculating an average waveform component of amplitudes of the waveforms and variation waveform components that indicate residual errors between the amplitudes of the waveforms and the average waveform component; calculating a variation level indicator between the waveforms; calculating a horizontal correlation distance and a vertical correlation distance of an S wave velocity in a surface ground, which are non-homogeneous parameters of a three-dimensional model of the surface ground, based on the variation level indicator, a coefficient of variation of non-homogeneity in the S wave velocity in the surface ground, a wavenumber in a non-homogeneous medium, a wavenumber of earthquake waves, a diffusion angle of the earthquake waves, and a distance by which the earthquake waves travel; building the three-dimensional model of the surface ground that reflects the non-homogeneity based on the horizontal correlation distance and the vertical correlation distance; and estimating the size of the earthquake vibrations.SELECTED DRAWING: Figure 2

Description

The present invention relates to an evaluation method and an evaluation system for earthquake motion.

Conventionally, as seen in Patent Literature 1 below, for example, various types of building swaying methods for estimating the magnitude of building shaking based on seismic observation records obtained from seismographs or the like installed on the ground surface during an earthquake. Seismic response evaluation methods have been developed.
In such conventional seismic response evaluation methods for buildings, generally, the building is modeled as a one-mass point system or a multi-mass point system. The response of the building is evaluated by inputting the seismic motion as it is.

Patent Literature 2 discloses a configuration in which the average and standard deviation of physical property values are calculated based on a ground investigation and reflected in the physical property values of a ground model. With such a configuration, the analysis of ground vibration is performed while taking into consideration the influence of variation in physical property values of the ground on the stability analysis of the ground.
In addition, in Patent Document 3, the spectrum of microtremors of the horizontal and vertical components of the ground surface observed by a vibration meter is obtained, the approximate value of the frequency response function is calculated, and the surface layer ground profile set from known data is compared. discloses an arrangement for determining a frequency response function.

Physical properties of actual ground are heterogeneous. Therefore, nonuniform physical properties of the ground cause variations in seismic motion, which may affect the seismic response, especially in high-vibration regions.
It is desired to evaluate seismic motion with higher accuracy.

JP 2013-120139 A JP 2005-336914 A JP-A-7-146373

The problem to be solved by the present invention is to provide a seismic motion evaluation method and a system for evaluating input seismic motion to a building with high accuracy.

によって計算し、前記ばらつき度合い指標ε、表層地盤におけるＳ波速度の不均質性の変動係数μ、不均質媒質の波数ｋ_ｓ、地震波の波数ｋ、地震波の散乱角ψ、地震波の伝播距離ｈを、下記の第２計算式

に適用して、３次元表層地盤モデルの不均質パラメータである、前記表層地盤中のＳ波速度における水平方向の相関距離ａ_１と鉛直方向の相関距離ａ_３とを算出し、前記水平方向の相関距離ａ_１と前記鉛直方向の相関距離ａ_３を基に、不均質性が反映された前記３次元表層地盤モデルを構築し、前記地震動の大きさを評価することを特徴とする。
このような構成によれば、３次元表層地盤モデルの不均質パラメータである、表層地盤中のＳ波速度における水平方向の相関距離ａ_１と鉛直方向の相関距離ａ_３とを基に、不均質性が反映された３次元表層地盤モデルを構築することができる。
ここで、実際の地盤においては、水平方向よりも鉛直方向が、短い間隔で、速度構造が変化する。これに対応して、特に、上記のような構成においては、水平方向の相関距離ａ_１と、鉛直方向の相関距離ａ_３とが、それぞれ別のパラメータとして式中にあらわれて、それぞれが別の値として求まっている。これにより、水平方向と鉛直方向で相関距離を同じ値として、等方的となるような場合に比べて、実際の地盤の特性を忠実かつ正確に、３次元表層地盤モデルに反映させることができる。
このようにして構築された３次元表層地盤モデルは、地盤の不均質性を、正確に評価したものとなる。このような地盤モデルを用いて地震動の大きさを評価することにより、建物への入力地震動を高い精度で評価することが可能となる。 In order to solve the above problems, the present invention employs the following means.
That is, the method for evaluating seismic motion of the present invention is a method for evaluating the magnitude of seismic motion input to a building. The average waveform component <u> of the amplitude and the variation waveform component u _f representing the residual between the amplitude of each waveform and the average waveform component are calculated, and the variation degree index ε between the waveforms is calculated as the following first a formula

Calculate the degree of dispersion index ε, coefficient of variation μ of inhomogeneity of S-wave velocity in surface ground, wavenumber k _s of inhomogeneous medium, wavenumber k of seismic wave, scattering angle ψ of seismic wave, propagation distance h of seismic wave , the second formula below

is applied to calculate the horizontal correlation distance a ₁ and the vertical correlation distance a ₃ in the S wave velocity in the surface ground, which are inhomogeneous parameters of the three-dimensional surface ground model, Based _on the correlation distance a1 and the vertical correlation distance a3, the _three -dimensional surface ground model reflecting heterogeneity is constructed to evaluate the magnitude of the seismic motion.
According to such a configuration, based on the horizontal correlation distance a ₁ and the vertical correlation distance a ₃ in the S wave velocity in the surface ground, which are inhomogeneous parameters of the three-dimensional surface ground model, inhomogeneity It is possible to construct a three-dimensional surface ground model that reflects the nature of the soil.
Here, in the actual ground, the velocity structure changes at shorter intervals in the vertical direction than in the horizontal direction. Correspondingly, especially in the configuration as described above, the horizontal correlation distance a1 and the vertical correlation distance a3 appear in the equation as separate parameters _, and _each is a separate parameter. sought as a value. As a result, the characteristics of the actual ground can be faithfully and accurately reflected in the 3D surface ground model compared to isotropic cases where the correlation distance is set to the same value in the horizontal and vertical directions. .
The three-dimensional surface ground model constructed in this way is an accurate evaluation of the heterogeneity of the ground. By evaluating the magnitude of seismic motion using such a ground model, it becomes possible to evaluate the input seismic motion to a building with high accuracy.

The seismic motion evaluation system of the present invention is a seismic motion evaluation system that evaluates the magnitude of seismic motion input to a building. Estimate the S-wave velocity of the subsurface ground and the coefficient of variation of the inhomogeneity of the S-wave velocity using the subsurface ground boring survey records , S-wave velocity characteristic estimator, S-wave velocity in the surface ground, which is an inhomogeneous parameter of the three-dimensional surface ground model, based on the variation index and the coefficient of variation of the inhomogeneity of the S-wave velocity An estimating unit for inhomogeneous parameters of surface ground that estimates the horizontal correlation distance and the vertical correlation distance in the velocity, and based on the horizontal correlation distance and the vertical correlation distance, for FEM analysis a three-dimensional surface ground model construction unit for constructing the three-dimensional surface ground model; an input seismic motion evaluation unit for evaluating the magnitude of the seismic motion input to the building by estimating the arriving input seismic motion.
According to such a configuration, based on the degree of variation index between waveforms obtained as seismic observation records and the variation coefficient of inhomogeneity of S-wave velocities estimated using surface ground boring survey records, A three-dimensional surface ground model is constructed by estimating the horizontal and vertical correlation distances of the S-wave velocity in the surface layer, which are inhomogeneous parameters of the three-dimensional surface layer model.
Here, in the actual ground, the velocity structure changes at shorter intervals in the vertical direction than in the horizontal direction. Correspondingly, especially in the configuration as described above, the horizontal correlation distance and the vertical correlation distance appear as separate parameters in the formula, and are obtained as separate values. there is As a result, the characteristics of the actual ground can be faithfully and accurately reflected in the 3D surface ground model compared to isotropic cases where the correlation distance is set to the same value in the horizontal and vertical directions. .
The three-dimensional surface ground model constructed in this way is an accurate evaluation of the heterogeneity of the ground. By evaluating the magnitude of seismic motion using such a ground model, it becomes possible to evaluate the input seismic motion to a building with high accuracy.

According to the present invention, it is possible to provide a seismic motion evaluation method and a system for evaluating input seismic motion to a building with high accuracy.

1 is a diagram showing the configuration of a seismic motion evaluation system according to an embodiment of the present invention; FIG. 4 is a flow chart showing a method for evaluating earthquake motion according to the present embodiment. FIG. 4 is a diagram schematically showing a component of a wave field incident on a heterogeneous medium reaching an observation point from a reference point in the heterogeneous medium, and a scattered field component included in this component. FIG. 10 is a diagram showing the shape of a three-dimensional inhomogeneous ground model used for verification in a verification example; FIG. 10 is a diagram showing distributions of S-wave velocities on the side surface and the ground surface of the heterogeneous ground model in the verification example; It is a figure which shows the time-history waveform of a response wave in a verification example. FIG. 11 is a diagram showing results of calculating a coherent component <u> and a variation waveform component u _f of a response waveform in a verification example; 8 is a diagram showing power spectra of the coherent component <u> and the variation waveform component _uf of the response waveform shown in FIG. 7; FIG. FIG. 10 is a diagram showing a result of sorting out the relationship between frequency f and ln (<ε ² >+1) in a verification example; It is a figure which shows the relationship with the separation distance between observation points in a verification example. FIG. 11 is a diagram showing estimation results of correlation distances in a verification example;

EMBODIMENT OF THE INVENTION Hereinafter, with reference to an accompanying drawing, the form for implementing the evaluation method and evaluation system of the earthquake motion by this invention is demonstrated based on drawing.
FIG. 1 shows the configuration of a seismic motion evaluation system according to an embodiment of the present invention.
As shown in FIG. 1, the seismic motion evaluation system 10 includes a data storage unit 11, a variation index calculator 12, an S-wave velocity characteristic estimator 13, and a surface ground inhomogeneous parameter estimator 14. , a three-dimensional surface ground model constructing unit 15, an input seismic motion evaluation unit 16, and a surface ground analysis result display unit 17. This seismic motion evaluation system 10 evaluates the input seismic motion for a building constructed on the ground.

The data storage unit 11 stores externally input data of surface ground boring survey records and seismic observation records. Surface ground boring survey records are obtained by conducting surface ground boring surveys prior to building construction, with the site where the building is to be constructed or a certain area including the building site being the survey target area. Boring surveys are conducted at one or more locations within the survey area. In addition, the seismic observation record is obtained by conducting seismic observation prior to the construction of the building on the site where the building is to be constructed or within the area including this site. A seismic observation record is, for example, a seismic motion waveform obtained from a seismometer installed on the ground surface. The seismic observation record is obtained by installing seismometers etc. at multiple points on the site and conducting seismic observation.

The variation degree index calculator 12 calculates the variation degree index between waveforms obtained as seismic observation records at each of adjacent points among multiple points where seismometers and the like are installed on the premises. calculate.
The S-wave velocity characteristic estimator 13 estimates the S-wave velocity of the surface ground, the average of the S-wave velocities, and the coefficient of variation of heterogeneity using the drilling survey records of the surface ground.
The estimator 14 for the inhomogeneous parameter of the surface ground is an inhomogeneous parameter of the three-dimensional surface ground model, based on the variation degree index and the coefficient of variation of the estimated inhomogeneity of the S-wave velocity. Estimate the horizontal and vertical correlation distances at the S-wave velocity.
The three-dimensional surface ground model construction unit 15 constructs a three-dimensional surface ground model for FEM analysis based on the estimated horizontal correlation distance and vertical correlation distance.
The input seismic motion evaluation unit 16 evaluates the magnitude of the seismic motion input to the building by FEM-analyzing the constructed three-dimensional surface ground. Specifically, the input seismic motion evaluation unit 16 estimates the input seismic motion that enters the lowest end of the three-dimensional surface layer ground model, propagates it in the surface layer ground, and reaches the top end point of the surface layer ground. , to evaluate the magnitude of the seismic motion input to the building.
The analysis result display unit 17 displays the evaluation result of the magnitude of the seismic motion input to the building as the result of the FEM analysis in the input seismic motion evaluation unit 16 .

FIG. 2 is a flow chart showing a method for evaluating seismic motion according to this embodiment.
The earthquake motion evaluation method according to the present embodiment is implemented by performing processing based on a preset computer program in the earthquake motion evaluation system 10 . As shown in FIG. 2, the earthquake motion evaluation method comprises step S11 of acquiring seismic observation records, step S12 of calculating the average and residual of the amplitude of the waveform obtained as the seismic record, and Step S13 of calculating the degree of variation in waveform amplitude, Step S14 of acquiring ground survey data, Step S15 of calculating heterogeneous parameters of the three-dimensional surface ground model, and Step S16 of constructing the three-dimensional surface ground model. , and a step S17 of evaluating the magnitude of the seismic motion input to the building.
In order to evaluate the seismic motion by the seismic motion evaluation system 10, in advance, the boring survey record of the surface layer obtained by conducting the boring survey of the surface layer and the seismic observation record obtained by conducting the seismic observation. Data is stored in the data storage unit 11 .

In step S11 for acquiring the seismic observation record, waveforms u ₁ and u ₂ of the seismic observation record are extracted from any two adjacent points from the data of the seismic observation record stored in the data storage unit 11 and acquired.

地震記録として得られた波形の振幅のばらつき度合いを計算するステップＳ１３では、ばらつき度合い指標の計算部１２において、地震動のばらつき度合い指標εを、次に式（３）として示される第１計算式により計算する。

地盤調査データを取得するステップＳ１４では、建物の施工に先立ち、建物を施工する敷地、または建物の近傍を調査対象エリアとして表層地盤のボーリング調査等を行うことで得た、地盤調査データを、データ記憶部１１から取得する。ステップＳ１４では、Ｓ波速度の特性推定部１３において、表層地盤のボーリング調査記録を使用し、表層地盤における各層のＳ波の（平均）伝播速度ｖ_ｉおよびＳ波速度の不均質性の変動係数μを求める。 In step S12 of calculating the average and residual of the amplitudes of the waveforms obtained as seismic records, the two waveforms u ₁ and u ₂ obtained in step S11 are calculated using the following equations (1) and (2): , the mean waveform component <u> of coherent amplitudes between waveforms of each waveform, and the variation waveform component (residual) u _f representing the residual between the amplitude of each waveform and the mean waveform component <u> calculate.

In step S13 for calculating the degree of variation in the amplitude of the waveform obtained as an earthquake record, in the variation degree index calculator 12, the variation degree index ε of the seismic motion is calculated by the first formula shown as Equation (3). calculate.

In step S14 for acquiring ground survey data, ground survey data obtained by conducting a surface ground boring survey or the like with the site where the building is to be constructed or the vicinity of the building as the survey target area prior to construction of the building, is collected as data. Acquired from the storage unit 11 . In step S14, the S-wave velocity characteristic estimating unit 13 uses the boring survey records of the surface layer, and the (average) propagation velocity v _i of the S-wave in each layer in the surface layer and the coefficient of variation of the inhomogeneity of the S-wave velocity Find μ.

ここで、ｋ_ｓは不均質媒質の波数、ｋは地震波の波数、ψは地震波の散乱角である。
また、上式（４）におけるｈは、地震波の伝播距離、すなわち、隣接地点間の地震動のばらつきに影響を与える地震深さである。本実施形態においては、地震波の伝播距離ｈを、２点間の水平方向の離間距離ξにより、下記の式（５）のように表現している。

表層地盤の不均質パラメータの推定部１４は、例えば、上式（４）の左辺と右辺の差を目的関数として、これが０に近づくような最適化問題を解くことにより、水平方向の相関距離ａ_１と、鉛直方向の相関距離ａ_３とを算出するように構成することができる。 In step S15 for calculating the inhomogeneous parameters of the three-dimensional surface ground model, the estimating unit 14 of the inhomogeneous parameters of the surface ground performs the following calculations based on the degree of variation index ε and the variation coefficient μ of the inhomogeneity of the S-wave velocity. The inhomogeneous parameters of the three-dimensional surface ground model are calculated based on the second calculation formula shown as formula (4) in . Equation (4) shows the relationship between the ground inhomogeneity parameter and the degree of variation index ε of seismic motions between adjacent points. In step S15, based on the second calculation formula of formula (4), the horizontal correlation distance a ₁ and the vertical correlation distance in the S-wave velocity in the surface ground, which are inhomogeneous parameters of the three-dimensional surface ground model Calculate and estimate _a3 .

Here, _ks is the wave number of the inhomogeneous medium, k is the wave number of the seismic wave, and ψ is the scattering angle of the seismic wave.
Also, h in the above equation (4) is the seismic wave propagation distance, that is, the seismic depth that affects variations in seismic motion between adjacent points. In this embodiment, the propagation distance h of seismic waves is expressed by the horizontal separation distance ξ between two points as shown in Equation (5) below.

The estimating unit 14 for the heterogeneous parameter of the surface layer, for example, uses the difference between the left side and the right side of the above equation (4) as the objective function, solves the optimization problem such that this approaches 0, and obtains the horizontal correlation distance a ₁ and a vertical correlation distance _a3 .

In step S16 for constructing a three-dimensional surface ground model, a _three -dimensional surface ground model reflecting inhomogeneity is constructed based on the horizontal correlation distance a1 and the vertical correlation distance _a3 calculated in step S15. to build.
More specifically, when building a _three -dimensional surface ground model, randomness is given to each of the physical property values of the _ground elements so as to satisfy the horizontal correlation distance a1 and the vertical correlation distance a3. By doing so _, the values of the horizontal correlation distance a1 and the vertical correlation distance a3 are reflected in the _three -dimensional surface ground model.
In step S17 for evaluating the magnitude of seismic motion input to the building, FEM analysis is performed on the three-dimensional surface ground model in which heterogeneity is reflected, constructed in step S16. Evaluate the magnitude of seismic motion input to the building. In the FEM analysis at this time, the seismic motion is incident on the lowest end of the three-dimensional surface layer model, propagated in the surface layer ground, and the input seismic motion reaching the uppermost point of the surface layer ground is estimated. The estimation result of the FEM analysis in the input earthquake motion evaluation unit 16 is displayed in the analysis result display unit 17 as the evaluation result of the magnitude of the earthquake motion input to the building.

これにより、ε^２は、上式（３）を用いて、次の式（７）のように表現できる。

対象とする地盤内における波動場のエネルギーが一定の場合（内部減衰がない場合）、（７）式のＩ_ｔ／Ｉ_ｃは、次の式（８）のように、距離の指数関数で表現できる。

ここで、αは散乱係数である。 The second calculation formula, shown as formula (4) above, is formulated as described below.
First, the amplitude I _t of the entire waveform can be expressed by the following equation (6) using the amplitude I _c of the coherent waveform and the amplitude I _f of the waveform accompanying scattering. .

Thus, ε2 can be expressed as in the following equation (7) using the above equation ( ³ ).

When the energy of the wave field in the target ground is constant (when there is no internal attenuation), I _t /I _c in Equation (7) is expressed as an exponential function of distance as shown in Equation (8) below. can.

where α is the scattering coefficient.

ここで、ｒ´は不均質媒質Ｖ内の基準点から不均質媒質Ｖ内の点Ｑまでの位置ベクトル、ｉは虚数単位である。
式（９）中のＦ（ｒ´）は、不均質媒質Ｖにおける散乱ポテンシャルで、次の式（１０）のように表現できる。

ここで、ｎ１（ｒ´）は伝播速度の変動分である。
ｎ１（ｒ´）≪１の場合、式（１０）は、次の式（１１）のように近似できる。

ｒがｒ´に比べて十分に大きい場合、式（９）中の｜ｒ－ｒ´｜は、次の式（１２）のように近似できる。

ここで、ｏは不均質媒質Ｖ内の基準点Ｏから観測点Ｐまでの単位ベクトルである。 Next, the scattering coefficient α in Equation (8) is obtained. Based on scattering theory, the scattering coefficient α of seismic waves incident on a heterogeneous medium is expressed by the correlation length and variation coefficient of the heterogeneous medium.
As shown in FIG. 3, the component of the wave field incident on the inhomogeneous medium V in the direction of the unit vector s that reaches the observation point P located on the vector r from the reference point O in the inhomogeneous medium V is U(r ) and the component of the scattered field included in U(r) is U ^s (r), U ^s (r) can be expressed by the following equation (9).

Here, r' is the position vector from the reference point in the inhomogeneous medium V to the point Q in the inhomogeneous medium V, and i is the imaginary unit.
F(r') in Equation (9) is the scattering potential in the inhomogeneous medium V and can be expressed as in Equation (10) below.

Here, n1(r') is the variation of the propagation velocity.
If n1(r')<<1, Equation (10) can be approximated as Equation (11) below.

When r is sufficiently larger than r', |rr'| in equation (9) can be approximated by the following equation (12).

Here, o is a unit vector from the reference point O to the observation point P in the inhomogeneous medium V.

式（１３）を式（９）に代入すると、観測点Ｐにおいて観測される散乱場Ｕ^ｓ（ｒ）は、次の式（１４）のように表現できる。

ここで式（１４）の右辺のｆ（ｏ，ｓ）は、不均質媒質Ｖに単位ベクトルｓの向きに入射した平面波に対する単位ベクトルｏの向きの散乱波の振幅を意味する。
式（１４）のＦ（ｒ´）に式（１１）を代入すると、式（１４）のｆ（ｏ，ｓ）は、次の式（１５）のように表現できる。

式（１５）のＵ（ｒ´）は、不均質媒質Ｖの伝播速度の変動が小さく、散乱が弱い場合、Ｂｏｒｎ近似を適用し、次の式（１６）のように表現できる。

The following equation (13) is obtained from the equation (12).

Substituting equation (13) into equation (9), the scattered field U ^s (r) observed at observation point P can be expressed as in equation (14) below.

Here, f(o, s) on the right side of Equation (14) means the amplitude of the scattered wave in the direction of the unit vector o with respect to the plane wave incident on the inhomogeneous medium V in the direction of the unit vector s.
When formula (11) is substituted for F(r') in formula (14), f(o, s) in formula (14) can be expressed as in formula (15) below.

U(r') in Equation (15) can be expressed as in Equation (16) below by applying the Born approximation when the inhomogeneous medium V has a small variation in propagation velocity and weak scattering.

式（１７）において、ｋ_ｓ＝ｋ（ｓ－ｏ）とした。またベクトルｋ_ｓの大きさは、図３に示すように、単位ベクトルｓと単位ベクトルｏのなす角をθとすると、次の式（１８）のようになる。

ここで、不均質媒質Ｖにおいて、単位立体角から散乱される波動場の強さを、微分散乱断面積ｄσ（ｏ、ｓ）とすると、ｄσ（ｏ、ｓ）は散乱振幅ｆ（ｏ、ｓ）との間に、次の式（１９）のような関係がある。

ここで、ｆ^＊（ｏ、ｓ）は、ｆ（ｏ、ｓ）の共役複素数を示す。また、右辺の＜＞は平均を示す。 Substituting equation (16) into equation (15) yields equation (17) below.

In equation (17), k _s =k(s−o). As shown in FIG. 3, the magnitude of the vector k _s is given by the following equation (18), where θ is the angle formed by the unit vector s and the unit vector o.

Here, in the inhomogeneous medium V, if the strength of the wave field scattered from a unit solid angle is the differential scattering cross section dσ(o, s), then dσ(o, s) is the scattering amplitude f(o, s ), there is a relationship such as the following equation (19).

where f ^* (o,s) denotes the complex conjugate of f(o,s). Moreover, <> on the right side indicates the average.

また、不均質媒質Ｖ内における基準点Ｏから位置ベクトルｒ_１´の地点およびｒ_２´の地点における伝播速度の変動分の相関関数をＢｎ（ｒ_ｄ）とすると、Ｂｎ（ｒ_ｄ）は、次の式（２１）のようになる。

ここで、ｒ_ｄ＝ｒ_１´－ｒ_２´である。
ベクトルｒ_ｄの大きさが、伝播速度の変動の相関距離よりも大きい範囲では、相関関数Ｂｎ（ｒ_ｄ）の値はほぼ無視できる。
（ｒ_１´＋ｒ_２´）／２＝ｒ_ｃとし、式（２１）を式（２０）に代入すると、微分散乱断面積ｄσ（ｏ、ｓ）は、次の式（２２）のように表現できる。

ここで、ｄＶ_ｃ、ｄＶ_ｄはそれぞれｒ_ｃ、ｒ_ｄによる微小体積である。 Substituting equation (17) for f(o, s) in equation (19) yields the following equation (20).

Also, let Bn(r _d ) be the correlation function of the variation of the propagation velocity at the points of the position vectors r ₁ ' and r ₂ ' from the reference point O in the inhomogeneous medium V, then Bn(r _d ) is It becomes like the following formula (21).

where r _d =r ₁ '-r ₂ '.
The value of the correlation function Bn(r _d ) is almost negligible in the range where the magnitude of the vector _rd is larger than the correlation distance of the variation of the propagation velocity.
(r ₁ '+r ₂ ')/2=r _c , and substituting the formula (21) into the formula (20), the differential scattering cross section dσ(o, s) is expressed as the following formula (22) can.

where dV _c and dV _d are the microvolumes due to r _c and r _d respectively.

式（２３）を式（２２）に代入すると、ｄσ（ｏ、ｓ）は、次の式（２４）のように表現できる。

不均質媒質Ｖからの散乱断面積σ（ｏ、ｓ）は、次の式（２５）のように、微分散乱断面積ｄσ（ｏ、ｓ）を全立体角で積分することにより得ることができる。

Assuming that φ _n (ks ) is the spectral density of the variation in propagation velocity in the inhomogeneous medium V, φ _n ( _ks ) can be expressed by the Fourier transform of the correlation function Bn( _{r d} ). )become that way.

Substituting equation (23) into equation (22), dσ(o, s) can be expressed as in equation (24) below.

The scattering cross section σ(o, s) from the inhomogeneous medium V can be obtained by integrating the differential scattering cross section dσ(o, s) over all solid angles, as in the following equation (25). .

ここで、φは平面波の入射方向（ｓ）に対する偏角である。
なお、上記の式（６）～式（２６）は、「光学の原理 第7版 ＩＩＩ」（Ｂｏｒｎ， Ｍ．，Ｗｏｌｆ，Ｅ．著、草川徹訳）および「Ｗａｖｅ ｐｒｏｐａｇａｔｉｏｎ ａｎｄ ｓｃａｔｔｅｒｉｎｇ ｉｎ ｒａｎｄｏｍ ｍｅｄｉａ」（Ｉｓｈｉｍａｒｕ，Ａ著）を参考に導出している。 By substituting equation (24) into equation (25) and converting dΩ into polar coordinates, the following equation (26) can be obtained.

Here, φ is the deflection angle with respect to the incident direction (s) of the plane wave.
It should be noted that the above formulas (6) to (26) are the same as those described in "Principles of Optics, 7th Edition III" (Born, M., Wolf, E., translated by Toru Kusakawa) and "Wave propagation and scattering in random media". (Ishimaru, A).

式（２７）のγを用いて、一次散乱の範囲における散乱断面積σ（ｏ、ｓ）と散乱係数α（ｏ、ｓ）の関係を、次の式（２８）のようにする。

式（２８）のスペクトル密度φ_ｎ（ｋ_ｓ）について、非等方的なガウス型として、次の式（２９）を定義する。

ここで、ａ_１は、不均質媒質における、水平方向に延在する一の軸線方向であるｘ軸方向の相関距離、ａ_２は、水平面内においてｘ軸方向に直交する方向であるｙ軸方向の相関距離、ａ_３は、水平面と直交する鉛直方向であるｚ軸方向の相関距離である。また、ｋ_ｓ１（＝ｋ_ｓｓｉｎθｃｏｓφ）は、不均質媒質におけるｘ軸方向の波数、ｋ_ｓ２（＝ｋ_ｓｓｉｎθｓｉｎφ）は、不均質性を定義するｙ軸方向の波数、ｋ_ｓ３（＝ｋ_ｓ（１－ｃｏｓθ））は、不均質性を定義するｚ軸方向の波数である。 Assuming that the scattering angle threshold that affects the magnitude of the variation waveform component u _f is ψ, the ratio γ of the surface area within the range of the threshold ψ to the surface area of the entire sphere is given by the following equation (27).

Using γ in Equation (27), the relationship between the scattering cross section σ(o, s) and the scattering coefficient α(o, s) in the primary scattering range is given by Equation (28) below.

For the spectral density φ _n ( _ks ) of Equation (28), the following Equation (29) is defined as an anisotropic Gaussian.

Here, a1 is the correlation distance in the x-axis direction, which is _one axial direction extending in the horizontal direction in the inhomogeneous medium, and _a2 is the y-axis direction, which is the direction orthogonal to the x-axis direction in the horizontal plane. is the correlation distance in the z _- axis direction, which is the vertical direction perpendicular to the horizontal plane. In addition, k _s1 (=k _s sin θ cos φ) is the wave number in the x-axis direction in the inhomogeneous medium, _ks 2 (= k _s sin θ sin φ) is the wave number in the y-axis direction that defines the inhomogeneity, _ks 3 (= k _s (1−cos θ)) is the wave number along the z-axis that defines the inhomogeneity.

ここで式（１８）より、ｋ_ｓをθで微分すると、次の式（３１）が得られる。

式（１８）および式（３１）を、式（３０）のｓｉｎθおよびｃｏｓθに代入すると、α（ｏ、ｓ）は、次の式（３２）のように、ｋ_ｓで積分した形式で表現できる。

式（８）のＩ_ｔ／Ｉ_ｃの関係を式（７）に代入すると、＜ε^２＞とαの関係は、次の式（３３）のようになる。

式（３３）のαに式（３２）を代入し、両辺に対数をとると、式（４）のように、ガウス型による地盤の不均質性および地盤深さと地震動のばらつきとの関係式が得られる。 The horizontal inhomogeneity is assumed to be isotropic, ie a ₁ =a ₂ holds.
Substituting equation (29) into equation (28), the coefficient α(o, s) is given by equation (30) below.

By differentiating ks with respect to _θ from equation (18), the following equation (31) is obtained.

Substituting equations (18) and (31) for sin θ and cos θ in equation (30), α(o, _s ) can be expressed in the form integrated by ks as in equation (32) below. .

Substituting the relationship of I _t /I _c in Equation (8) into Equation (7), the relationship between <ε ² > and α is given by Equation (33) below.

Substituting equation (32) for α in equation (33) and taking the logarithm on both sides yields equation (4), the relational expression between ground heterogeneity and ground depth and seismic ground motion variation according to Gaussian type. can get.

The earthquake motion evaluation method described above is a method of evaluating the magnitude of the earthquake motion input to a building. and the variation waveform component u _f representing the residual between the amplitude of each waveform and the average waveform component <u>, and the variation degree index ε between the waveforms is calculated by equation (3). is calculated by the first calculation formula shown as, the degree of dispersion index ε, the coefficient of variation μ of inhomogeneity of S-wave velocity in surface ground, the wave number k _s of inhomogeneous medium V, the wave number k of seismic waves, the scattering angle of seismic waves ψ , the seismic wave propagation distance h is applied to the second calculation formula shown as Equation (4) to obtain the horizontal correlation distance a ₁ and the vertical correlation distance a3 are calculated, and based on the horizontal correlation distance a1 and the vertical correlation distance _a3 , _{a three} -dimensional surface ground model that reflects the heterogeneity is constructed, and the seismic motion Evaluate the size of
According to such a configuration, based on the horizontal correlation distance a ₁ and the vertical correlation distance a ₃ in the S wave velocity in the surface ground, which are inhomogeneous parameters of the three-dimensional surface ground model, inhomogeneity It is possible to construct a three-dimensional surface ground model that reflects the nature of the soil.
Here, in the actual ground, the velocity structure changes at shorter intervals in the vertical direction than in the horizontal direction. Correspondingly, especially in the configuration as described above, the horizontal correlation distance a1 and the vertical correlation distance a3 appear in the equation as separate parameters _, and _each is a separate parameter. sought as a value. As a result, the characteristics of the actual ground can be faithfully and accurately reflected in the 3D surface ground model compared to isotropic cases where the correlation distance is set to the same value in the horizontal and vertical directions. .
The three-dimensional surface ground model constructed in this way is an accurate evaluation of the heterogeneity of the ground. By evaluating the magnitude of seismic motion using such a ground model, it becomes possible to evaluate the input seismic motion to a building with high accuracy.

The earthquake motion evaluation system 10 as described above evaluates the magnitude of the earthquake motion input to the building. On the other hand, using the variation degree index calculation unit 12 that calculates the variation degree index ε between these waveforms and the boring survey record of the surface layer, the S wave velocity of the surface layer and the heterogeneity of the S wave velocity Inhomogeneous parameters of the three-dimensional surface ground model based on the S-wave velocity characteristic estimation unit 13 that estimates the variation coefficient μ, the degree of variation index ε, and the variation coefficient μ of the inhomogeneity of the S-wave velocity, A surface ground inhomogeneous parameter estimator 14 for estimating the horizontal correlation distance a ₁ and the vertical correlation distance a ₃ in the S-wave velocity in the surface ground, and the horizontal correlation distance a ₁ and the vertical correlation distance a 3 Based on the correlation distance a ₃ , a three-dimensional surface ground model construction unit 15 constructs a three-dimensional surface ground model for FEM analysis, and a seismic motion is incident on the lowest end of the three-dimensional surface ground model, and an input seismic motion evaluation unit 16 that evaluates the magnitude of the seismic motion input to the building by estimating the input seismic motion that propagates and reaches the uppermost end point of the surface layer ground.
According to such a configuration, based on the degree of variation index ε between waveforms obtained as seismic observation records and the coefficient of variation μ of heterogeneity of S-wave velocity estimated using the boring survey records of the surface layer Next, the horizontal correlation distance a1 and the vertical correlation distance a3 in the S _- wave velocity in the surface ground, which are inhomogeneous parameters of the 3D surface ground model, are estimated to construct a _3D surface ground model. .
Here, in the actual ground, the velocity structure changes at shorter intervals in the vertical direction than in the horizontal direction. Correspondingly, especially in the configuration as described above, the horizontal correlation distance a1 and the vertical correlation distance a3 appear in the equation as separate parameters _, and _each is a separate parameter. sought as a value. As a result, the characteristics of the actual ground can be faithfully and accurately reflected in the 3D surface ground model compared to isotropic cases where the correlation distance is set to the same value in the horizontal and vertical directions. .
The three-dimensional surface ground model constructed in this way is an accurate evaluation of the heterogeneity of the ground. By evaluating the magnitude of seismic motion using such a ground model, it becomes possible to evaluate the input seismic motion to a building with high accuracy.

(Verification example)
Here, in order to verify the applicability of the above formula (4), we calculated the variation of the seismic motion for the response wave between adjacent points calculated by the seismic motion simulation using the 3D FEM inhomogeneous ground model, and expressed the formula ( 4) was used to estimate the inhomogeneous parameters of the ground model.
Figure 4 shows the shape of the three-dimensional inhomogeneous ground model used for verification. The size of the model was 300m in two horizontal directions and 200m in the depth direction. Also, the mesh size of the FEM model was set to 1 m×1 m.
The average S-wave velocity structure of the ground was 300 m/s, and the S-wave velocity was varied so that the variation pattern followed a Gaussian autocorrelation function. The correlation distance a1 in the horizontal direction of the _ground was 30 m, and the correlation distance a3 in the vertical direction was ₃ m. The coefficient of variation μ was set to 0.15. Also, by changing the initial random numbers, five types of ground models were created in all cases studied.
FIG. 5 shows an example of the distribution of S-wave velocities on the side and surface of the inhomogeneous ground model.
For the input seismic motion, for the purpose of giving an impulse-like vibration, a displacement waveform of a triangular function with an excitation time of 0.02 seconds having an amplitude only in the y-direction was vertically incident from the bottom surface of the ground model. In addition, as indicated by the thick line L in FIG. 4, response waveform extraction points were set at intervals of 1.0 m on a line of 100 m in length in both the x and y directions at the center of the ground model ground surface, and the time history waveform of the displacement response was obtained. Extracted. The response wave extraction time was about 2.5 seconds from the start of the input from the bottom of the model.

As an example of the time history waveform of the response wave, FIG. 6 shows the result of drawing the time history waveform of the response wave at intervals of 5 m from the response wave extraction points located on the line of x=100 to 200 m and y=150 m on the surface layer.
In FIG. 6, it can be confirmed that the waveforms of the initial motion are different between the points. Further, among the time history waveforms shown in FIG. The results of calculating u _f are shown in FIG. The initial amplitude of the variation waveform component is larger in the waveform with the separation distance of 50 m than in the waveform with the separation distance of 5 m.
FIG. 8 shows power spectra of the coherent component <u> and the variation waveform component _uf of the response waveform shown in FIG. When the separation distance is 5 m, the coherent waveform component <u> is larger than the variation waveform component u _f over the entire frequency band in the power spectrum amplitude. They are almost at the same level on the high frequency side. That is, it can be confirmed that the power spectrum of the variation waveform component u _f becomes relatively large from the low frequency side as the separation distance increases.

Among all the response waves extracted from the ground surface of five types of inhomogeneous ground models with different initial random numbers, the power spectrum of the average amplitude and the FIG. 9 shows the result of calculating the average <ε ² > of the power spectrum ratios of the variation waveform components and sorting out the relationship between the frequency f and ln (<ε ² >+1). The value of ln (<ε ² >+1) increases as the separation distance increases. There is a tendency for ln (<ε ² >+1) to increase as the frequency f increases. Also, it can be confirmed that ln (<ε ² >+1) becomes smaller as the separation distance between the observation points becomes smaller.
The results of FIG. 9 are applied to Equation (4) to estimate the correlation distances (a ₁ , a ₃ ) of the heterogeneous ground model. The search range was set in the range of 1 m to 50 m for a ₁ and in the range of 0.01, 0.1, 1.0, 10, and 100 times as large as a ₃ for a ₁ . Also, in Equation (4), h is set based on the relationship with the separation distance ξ between the two observation points where the seismometers are installed, as shown in FIG. ψ was set to 20°.
FIG. 11 shows the estimation result of the correlation distance. Both a ₁ and a ₃ can roughly estimate the correlation distance (a ₁ =30 m, a ₃ =3 m) of the heterogeneous ground model set in the seismic motion simulation. Also, the result of calculating ln (<ε ² >+1) on the left side of Equation (4) based on the estimated results of a ₁ and a ₃ is shown by a dashed line in FIG. The evaluation of ln (<ε ² >+1) by Equation (4) (broken line in Fig. 9) can roughly express the result of ln (<ε ² >+1) by the response waveform of the seismic motion simulation (solid line in Fig. 9). there is

10 Seismic motion evaluation system 15 3D surface ground model construction unit 12 Variation index calculation unit 16 Input seismic motion evaluation unit 13 S-wave velocity characteristic estimation unit V heterogeneous medium 14 Surface ground heterogeneous parameter estimation unit

Claims (2)

A method for evaluating the magnitude of seismic motion input to a building, comprising:
For the waveforms obtained as seismic observation records at each of the adjacent points, the average waveform component <u> of the amplitude between these waveforms, and the dispersion representing the residual between the amplitude of each waveform and the said average waveform component The waveform component u _f is calculated, and the variation degree index ε between the waveforms is calculated by the following first calculation formula

calculated by
The degree of dispersion index ε, coefficient of variation μ of inhomogeneity of S-wave velocity in surface ground, wave number k _s of inhomogeneous medium, wave number k of seismic waves, scattering angle ψ of seismic waves, and propagation distance h of seismic waves are calculated as follows: 2 Formula

is applied to calculate the horizontal correlation distance a ₁ and the vertical correlation distance a ₃ in the S wave velocity in the surface ground, which are inhomogeneous parameters of the three-dimensional surface ground model,
Based _on the horizontal correlation distance a1 and the vertical correlation distance a3, the _three -dimensional surface ground model reflecting heterogeneity is constructed, and the magnitude of the seismic motion is evaluated. evaluation method of seismic motion. A seismic motion evaluation system for evaluating the magnitude of seismic motion input to a building,
a variation degree index calculation unit that calculates a variation degree index between waveforms obtained as seismic observation records at each of adjacent points;
an S-wave velocity property estimator that estimates the S-wave velocity of the subsoil and the coefficient of variation of the inhomogeneity of the S-wave velocity using subsoil boring survey records;
Based on the degree of variation index and the coefficient of variation of the inhomogeneity of the S-wave velocity, the horizontal correlation distance and the vertical direction in the S-wave velocity in the surface layer, which are inhomogeneity parameters of the three-dimensional surface layer model an estimating unit for inhomogeneous surface ground parameters that estimates the correlation distance of
a three-dimensional surface ground model construction unit that constructs the three-dimensional surface ground model for FEM analysis based on the horizontal correlation distance and the vertical correlation distance;
By injecting seismic motion into the lowest end of the three-dimensional surface layer ground model, propagating it in the surface layer ground, and estimating the input seismic motion reaching the top end point of the surface layer ground, the magnitude of the seismic motion input to the building an input seismic motion evaluation unit that evaluates the
A seismic motion evaluation system comprising:

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