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SUMMARY: The accuracy and sampling rate of a coseismic displacement and velocity waveform can be improved by fusing Global Navigation Satellite System (GNSS) and strong motion (SM) sensor data with a multirate Kalman filter. However, many studies have shown that the noise in GNSS coseismic waveforms includes coloured noise, and it is challenging to obtain proper GNSS and SM fusion results if the GNSS coloured noise is not accurately represented. In this paper, we propose a coloured noise multirate Kalman filter, which uses a stochastic model for modelling the GNSS coloured noise to achieve an accurate fusion of data from GNSS and SM for different sampling rates. In addition, we also propose an estimation method for the multirate Kalman filter stochastic model under coloured noise to achieve the optimal adaptive fusion of GNSS and SM data. After the reliability of the proposed method was confirmed using Monte Carlo simulations and earthquake engineering data tests, the proposed method was applied to data collected from the 2019 Mw 7.1 Ridgecrest earthquake and 2016 Mw 7.8 Kaikoura earthquake. The test results show that the proposed method can effectively fuse GNSS and SM data and accurately obtain broadband coseismic displacement and velocity waveforms. [ABSTRACT FROM AUTHOR]