Fundamentals of surface deformation and application to construction monitoring

Abstract

The rigorous approach to plane deformation developed by the author is extended to the case of curved surfaces and is applied to the monitoring of surface-like constructions by repeated surveys. From coordinates of discrete points at two survey epochs, interpolation of the displacements produces displacement functions expressed in terms of surface curvilinear coordinates, which can be used for the computation of coordinate-invariant deformation parameters which are meaningful from the strength-of-materials point of view. In addition, it is shown how to incorporate information from additional in situ measurements by strainmeters, extensiometers, tiltmeters, etc. in the interpolation process. The computed invariant parameters at any surface point are the dilatation, the maximum shear strain, and the maximum bending expressed in terms of the maximum change of radius curvature among all surface directions. In addition to the theoretical tools, a complete algorithm is presented for direct practical implementation.

Appendices

Appendix 1: Determination of the angle θ of maximum radii of curvature variation for corresponding normal sections

The maximum of the difference in the radii of curvature

$$ \Delta R(\theta ) = R\prime - R = \frac{1}{{k\prime (\theta )}} - \frac{1}{{k(\theta )}} = \max $$

(65)

is obtained at the angle θ for which $\frac{{\partial \Delta R}}{{\partial \theta }} = 0$, or explicitly

$$ \frac{{\partial \Delta R}}{{\partial \theta }} = - \frac{1}{{k{\prime^2}}}\frac{{\partial k\prime }}{{\partial \theta }} + \frac{1}{{{k^2}}}\frac{{\partial k}}{{\partial \theta }} = 0. $$

(66)

From the differentiation of Euler’s equation, $k = {k_1}{\cos^2}\theta + {k_2}{\sin^2}\theta$ follows that $\frac{{\partial k}}{{\partial \theta }} = ({k_2} - {k_1})\sin 2\theta$ and $\frac{{\partial k\prime }}{{\partial \theta \prime }} = (k_2^{\prime } - k_1^{\prime })\sin 2\theta \prime$. Since $\frac{{\partial k\prime }}{{\partial \theta }} = \frac{{\partial k\prime }}{{\partial \theta \prime }}\frac{{d\theta \prime }}{{d\theta }} = (k_2^{\prime } - k_1^{\prime })\sin 2\theta \prime \frac{{d\theta \prime }}{{d\theta }}$ and $\frac{{d\theta \prime }}{{d\theta }} = \frac{{{\beta_1}\sin \theta - {\beta_2}\cos \theta }}{{\sin \theta \prime }}$, Eq. 66 becomes

$$ \begin{array}{*{20}{c}} {\frac{{{k_2} - {k_1}}}{{{k^2}}}\sin 2\theta - \frac{{k_2^{\prime } - k_1^{\prime }}}{{k{\prime}^{2}}}\sin 2\theta \prime \frac{{d\theta \prime }}{{d\theta }}} \\{ = \frac{{({k_2} - {k_1})\sin 2\theta }}{{{{\left[ {({k_1} - {k_2}){{\cos }^2}\theta + {k_2}} \right]}^2}}} - \frac{{(k_2^{\prime } - k_1^{\prime })\sin 2\theta \prime }}{{{{\left[ {(k_1^{\prime } - k_2^{\prime }){{\cos }^2}\theta \prime + k_2^{\prime }} \right]}^2}}}\frac{{{\beta_1}\sin \theta - {\beta_2}\cos \theta }}{{\sin \theta \prime }}} \\{ = \frac{{({k_2} - {k_1})\sin 2\theta }}{{{{\left[ {({k_1} - {k_2}){{\cos }^2}\theta + {k_2}} \right]}^2}}} - \frac{{2(k_2^{\prime } - k_1^{\prime })\cos \theta \prime ({\beta_1}\sin \theta - {\beta_2}\cos \theta )}}{{{{\left[ {(k_1^{\prime } - k_2^{\prime }){{\cos }^2}\theta \prime + k_2^{\prime }} \right]}^2}}} = 0.} \\\end{array} $$

(67)

Replacing $\cos \theta \prime = {\beta_1}\cos \theta + {\beta_2}\sin \theta$, our fundamental equation for the determination of $\theta$ becomes

$$ \frac{{({k_2} - {k_1})\sin 2\theta }}{{{{\left[ {({k_1} - {k_2}){{\cos }^2}\theta + {k_2}} \right]}^2}}} - \frac{{2(k_2^{\prime } - k_1^{\prime })\left( {{\beta_1}\cos \theta + {\beta_2}\sin \theta } \right)\left( {{\beta_1}\sin \theta - {\beta_2}\cos \theta } \right)}}{{{{\left[ {(k_1^{\prime } - k_2^{\prime }){{({\beta_1}\cos \theta + {\beta_2}\sin \theta )}^2} + k_2^{\prime }} \right]}^2}}} = 0 $$

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Appendix 2: Interpolation of plane functions f(X,Y) using the collocation method

We shall present shortly the method of interpolation known in geodesy as collocation which is based on the stochastic interpretation of the unknown plane function f(X,Y) as a random field so that its values at points other than those where the values f(X _i ,Y _i ), i = 1,…,n, have been observed are random variables predicted using the principle of minimum mean square error of prediction among all inhomogeneous linear function of the available data.

In general, a function $f(P) = f(X,Y)$ with discrete data ${f_i} = f({P_i}) = f({X_i},{Y_i})$ can be interpolated by modeling it as a linear combination $f(P) = {a_1}{\varphi_1}(P) + {a_2}{\varphi_2}(P) + ... + {a_m}{\varphi_m}(P)$ of known basis functions ${\varphi_k}(P)$, $k = 1,...,m$. For all the available data ${f_i} \equiv f({P_i})$, $i = 1,...,n$, the corresponding set of n equations ${f_i} = {a_1}{\varphi_1}({P_i}) + {a_2}{\varphi_2}({P_i}) + ... + {a_m}{\varphi_m}({P_i})$ in the m unknowns a _k takes the matrix form f = Fa, with ${F_{{ik}}} = {\varphi_k}({P_i})$. When n > m, there is no exact solution, but instead we may set ${\mathbf{f}} = {\mathbf{Fa}} + {\mathbf{v}}$ and obtain a least-squares smoothing interpolation through $\widehat{{\mathbf{a}}} = {({{\mathbf{F}}^T}{\mathbf{PF}})^{{ - 1}}}{{\mathbf{F}}^T}{\mathbf{Pf}}$ for a positive-definite weight matrix P. For n = m, an exact interpolation is obtained through $\widehat{{\mathbf{a}}} = {{\mathbf{F}}^{{ - 1}}}{\mathbf{f}}$. However, the most flexible case is that of exact interpolation with n < m and an infinite number of solutions, in which case a unique one $\widehat{{\mathbf{a}}} = {{\mathbf{W}}^{{ - 1}}}{{\mathbf{F}}^T}{({\mathbf{F}}{{\mathbf{W}}^{{ - 1}}}{{\mathbf{F}}^T})^{{ - 1}}}{\mathbf{f}}$ satisfying a ^T Wa = min can be obtained, for some preferably diagonal weight matrix W. Setting ${({{\mathbf{f}}_P})_k} = {\varphi_k}(P)$, the interpolated function takes the form

$$ f(P) = \sum\nolimits_{{k = 1}}^m {{{\widehat{{\mathbf{a}}}}_k}{\varphi_k}(P)} = {\mathbf{f}}_P^T\widehat{{\mathbf{a}}} = {\mathbf{f}}_P^T{{\mathbf{W}}^{{ - 1}}}{{\mathbf{F}}^T}{({\mathbf{F}}{{\mathbf{W}}^{{ - 1}}}{{\mathbf{F}}^T})^{{ - 1}}}{\mathbf{f}}. $$

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If the two-point function $k(P,Q) = \sum\nolimits_{{k = 1}}^m {W_{{kk}}^{{ - 1}}{\varphi_k}(P){\varphi_k}(Q)}$ is introduced, the solution can be also written in the compact form $f(P) = {\mathbf{k}}_P^T{{\mathbf{K}}^{{ - 1}}}{\mathbf{f}}$, where ${K_{{ij}}} = k({P_i},{P_j})$ and ${({{\mathbf{k}}_P})_i} = k(P,{P_i})$. Thus, the solution can be obtained even without explicitly defining the weights W _kk or the base functions $\{ {\varphi_k}(P)\}$ by introducing rather directly the function k(P,Q). A probabilistic interpretation is possible if we set ${W_{{kk}}} = 1/\sigma_k^2$ and interpret $\sigma_k^2$ as the variances of the zero mean uncorrelated coefficients a _k . When a _k are considered random variables, $f(P) = \sum\nolimits_{{k = 1}}^m {} {a_k}{\varphi_k}(P)$ becomes a zero mean stochastic process (random function) and $k(P,Q) = E\{ f(P)f(Q)\}$ is simply the covariance function C(P,Q) of f(P). The interpolating equations $f(P) = {\mathbf{k}}_P^T{{\mathbf{K}}^{{ - 1}}}{\mathbf{f}}$ become in this case the (minimum mean square error) prediction equations

$$ f(P) = {\mathbf{c}}_P^T{{\mathbf{C}}^{{ - 1}}}{\mathbf{f}} $$

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with ${C_{{ij}}} = C({P_i},{P_j})$ and ${({{\mathbf{c}}_P})_i} = C(P,{P_i})$ determined from the single choice of a positive-definite covariance function $C(P,Q) = C({X_P},{Y_P};{X_Q},{Y_Q})$. Since the prediction point P appears only in the vector c _P , it is possible to obtain the required first- and second-order partial derivatives of $f(P) = f(X,Y)$ by differentiating directly the elements ${({{\mathbf{c}}_P})_i} = C(P,{P_i}) = C(X,Y;{X_i},{Y_i})$. For example, $\frac{{{\partial^2}f}}{{\partial X\partial Y}}(X,Y) = {{\mathbf{c}}^T}{{\mathbf{C}}^{{ - 1}}}{\mathbf{f}}$ with ${c_i} = \frac{{{\partial^2}}}{{\partial X\partial Y}}{({{\mathbf{c}}_P})_i} = \frac{{{\partial^2}}}{{\partial X\partial Y}}C(X,Y;{X_i},{Y_i})$ and similar relations hold for the other partial derivatives. Replacing f(X,Y) with Z(X,Y), U(X,Y), V(X,Y), W(X,Y), we may interpolate–predict the required derivatives, e.g., ${U_X} = \frac{{\partial U}}{{\partial X}}$, ${U_{{XY}}} = \frac{{{\partial^2}U}}{{\partial X\partial Y}}$, etc. If instead of ${f_i} = f({P_i})$ we have observations ${y_i} = f({P_i}) + {v_i}$ contaminated with zero mean noise v _i with covariance matrix Σ, Eq. 70 is simply replaced by $f(P) = {\mathbf{c}}_P^T{({\mathbf{C}} + {\mathbf{\Sigma }})^{{ - 1}}}{\mathbf{y}}$. In order to achieve independence from the reference system used, we need to assume that the random field f(P) is not only homogeneous but also isotropic, in which case the covariance function depends only on the distance r between the points P and Q, i.e., $C(P,Q) = C(r)$. An example of such a covariance function is $C(r) = {\sigma^2}{e^{{k{r^2}}}}$, where σ ² is the variance of the random field, approximated, e.g., by ${s^2} = \frac{1}{n}\sum\nolimits_i {f_i^2}$, where it is assumed that a constant mean m, approximated by $\bar{f} = \frac{1}{n}\sum\nolimits_i {{f_i}}$, has already been removed from the data (${f_i} \to {f_i} - \bar{f}$). The parameter $k = - \ln 2/{R^2}$ relates to the correlation length R defined by $C(R) = \frac{1}{2}C(0)$, which must be of the same order of magnitude as the mean distance between neighboring points. The parameters σ ² and k are selected so that C(r) best fits an empirical covariance function estimated from the data.

Before interpolating (in particular when coordinates X _i and ${\mathbf{X}}_i^{\prime }$ do not refer to the same reference system), it is necessary to perform a “trend removal” by a least-squares fitting where the coordinates ${\mathbf{X}}_i^{\prime }$ are rotated (R) and translated (d) into a new set ${\mathbf{\tilde{X}}}_i^{\prime } = {\mathbf{RX}}_i^{\prime } + {\mathbf{d}}$, satisfying $\sum\nolimits_i {|{{\mathbf{X}}_i} - {\mathbf{\tilde{X}}}_i^{\prime }{|^2} = \min }$. While the deformation parameters themselves are invariant under changes of the coordinate systems at both epochs under comparison, the same is not true for the interpolation–prediction process. The suggested trend removal satisfies the theoretical requirement that the interpolated displacement fields are zero mean random fields and at the same time secures that the final results are not affected by the adopted reference systems at the two epochs.

The strongest point of the above interpolation prediction is the fact that it can accommodate as input data not only values f _i  = f(P _i ) of the interpolated field at particular points but also any type of observable depending on the unknown field, which after linearization can be expressed as a continuous linear functional L _i (f). In this case, the elements of the involved matrices become

$$ {C_{{ij}}} = {L_{{i,P}}}{L_{{i,Q}}}C(P,Q),\quad {({{\mathbf{c}}_P})_i} = {L_{{i,Q}}}C(P,Q), $$

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where the additional subscripts P and Q clarify with respect to which of the two variables of C(P,Q) the functionals L _i and L _j are acting, according to a scheme that is called “law of covariance propagation.”