東北日本の地殻歪(I)

―解析方法―

抄録

A least square adjustment is developed in order to deduce earth's horizontal strain from displacement vector. Let (_xi, _yi) (i=1, 2, …, P) be the coordinates of p triangulation stations. These coordinates are determined with respect to the centroid of the geodetic figure consisting from p stations. Let d (_xi, _yi) (i=1, 2, …, P) be the displacement vectors at the triangulation stations. These vectors are given as the deviations from translation vector of the centroid that is estimated by any suitable method from original displace ment vectors. We assume homogeneous strain u inside the geodetic figure as
u^t=[εx, εy, εxy, ω], (1)
where εx and εy, are normal strains, γxy, shear strain in the sense of engineering shear strain, and w rotation. Displacement vectors d are
d^t=[dx₁, dy₁, …, dx_p, dy_p] . (2)
All the components are assumed to be independent.
The relation between u and d is given from elasticity theory
d=Bu, (3)
where B is design matrix and has a form
B=(4)
Least square solution of (3) is
u=(B^tP_dB)^-1B^tP_dd . (5)
P_d is weight matrix of d. Variance-covariance matrix Σ(u) of u is
Σ(u)=σ₀²Ωuu (6)
Cofactor matrix Q_uu is, using notation N=(B^tP_dB)
Q_uu = N^-1 (7)
and a posteriori reference variance σ₀² is
σ₀²=V^tP_dV|c-u=V^tP_dV|γ. (8)
V is residual of displacement vectors calculated using the determined strain u. c is number of observation equations, u number of unknowns.
Let maximum and minimum principal strains be _ε1 and _ε2, and azimuth of maximum principal strain _ε1 be α, then(9)
Vector of principal strain E is defined as
E[ε1, ε2, α] (10)
Variance-covariance matrix of E is Σ(E), and this is propagated from 3×3 elements inΣ(u), Using Jacobian J_eu, Σ(E)=JeuΣ(u)euJeu^t(11)(12)Dilatation Δ and maximum shear strain γ_max is obtained as Δ=ε₁+ε₂ λmax=ε₁-ε₂(13)Variance of Δ and γγγ_max, covariance between Δ and γ_max are σ_Δ², σΔ ², σ²γmax, and σΔγmax(14)where, σ_ε1² and σ_ε2² are variances of _ε1 and _ε2, and _ε1ε2 is covariance between _ε1 and _ε2.