Alignment and breakthrough errors in tunneling

Misalignment and breakthrough errors have been a threat for tunnelers for 2500 years, and they still represent a problem that has been studied in detail on the occasion of certain rather recent major tunnels.From these analyses precise formulae permitting to compute the uncertainties in tunnel alignment and the breakthrough error have been proposed. Since these formulae are complex, require geodetic details and do not permit direct estimation of the order of magnitude of these errors, the latter have remained an obscure point for non-specialists.In this paper we explain why such errors occur and, based on the theory of error propagation and numerical approximation techniques, we derive certain simple formulae which permit to predict the magnitude of errors in tunnel alignment and in breakthrough in three dimensions. These formulae are for simple horizontal tunnels, constrained by geodetic observations at one portal and indicate that errors broadly follow a cantilever-type pattern. In addition, hints on how these formulae can be easily extended to all other cases of curved, inclined, etc. tunnels are provided. In the case, however, of complex tunnels (for instance double tunnels connected with transversal segments) errors are smaller for their propagation is controlled by redundant observations.