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This paper describes the correction or recovery of the original ground motion acceleration time histories from accelerometer digital records. It deals specifically with the situation where the recording accelerometer instrument is unknown. This is the case with some older (legacy) records. The term instrument implies all processes that have modified the time history in some way, such as the accelerometer response transfer function, anti-alias filters used in A/D conversion, digital quantisation, etc. The total least squares (TLS) method is used to identify the unknown system (instrument) that must be used to de-convolute the recorded time histories. This approach is compared and contrasted with the recursive least squares method (QR-RLS) and a standard second order, single-degree-of-freedom, idealised instrument de-convolution. A range of seismic events from Iceland and Taiwan (SMART-1 array) are considered. These data sets include a number of different strong motion accelerometers, from Iceland: the SMA-1, DCA-333, A-700, SSA-1 instruments and the SA-3000 used in the SMART-1 array. Without any assumed information about the instrument the TLS is shown to provide a reasonable estimate of its characteristics from just the recorded time history data.