Joint estimation of tiltmeter drift and volume variation during reservoir monitoring

Abstract

Borehole tiltmeters are widely used to continuously record small surface deformation of reservoirs and volcanoes. Because these instruments display unknown long-term drift, only short-term tilt signal can be used for monitoring purpose. We propose a method to invert long-term time series of tilt data induced by strain variations at depth. The assumption that tiltmeter drift is linear over time is on its own insufficient to remove the drift and uniquely determine the deformation source parameters. To overcome this problem, we first invert the data with no constrain on the drift to obtain one particular solution among all admissible. Then, using the linearity of the forward model, we use the statistical properties of the drift distributions to restore the uniqueness of the solution. We illustrate our approach with four synthetic cases simulating volume changes of a reservoir. We demonstrate the efficiency of our method and show that the accuracy of estimated volume variation dramatically improves if low drift tiltmeters are used.

Tiltmeter data inversion to characterize a strain tensor source at depth: application to reservoir monitoring

“…The input data is measured using an array of tiltmeters composed by monthly-discrete measurements associated to the variation of the strain tensor components at depth. Following Furst et al (2019), we assume that the tilt signal ⃗⃗⃗⃗ ( ) (2 × ) is the sum of the signal produced by the source ⃗⃗⃗⃗ ( ) (2 × ), a time-linear instrumental drift ⃗⃗⃗⃗ ( ) = (2 × ) ( is the drift rate vector of dimension 2 ) and coloured noise ⃗⃗⃗⃗ ( ) (2 × ) associated to the tiltmeters (Eq. 3).…”

“…The comparison between observed and modelled data is made using the weighted squared error as cost function, integrated over time following the trapezoidal rule (Eq. 5 from Furst et al, 2019). The functional needs to converge below one (ideally to 0) for the optimization to be complete, i.e.…”

“…At the end of the optimization, the parametrization of tilt data leads to a system of linear equations whose solutions are infinite combinations of these parameters producing the same signal (Furst et al, 2019):…”

“…This set of admissible source and instrumental parameters ( ( ) and ⃗⃗⃗⃗ ) is not unique since an infinite number of combinations of these parameters leads to the same value of the functional (Furst et al, 2019). This implies that 10 the solution of the optimization process can notably diverge from the target solution.…”

“…However, the source can have a linear term in time. Furst et al (2019) developed a methodology to invert time series of tilt data induced by linearly time-dependent strain variations of a source at a given position and depth. This method permits to automatically separate the instrumental drift from the source signal through a two-step approach: first, tilt data are inverted to determine a set of admissible parameters, and second, uniqueness of the solution, with regards to the mathematical problem setting, is enforced by assuming the independence between instrumental drift and source parameters.…”