ABSTRACTHigh-magnitude (amplitude $\le -3{{\ \rm per\ cent}}$) Forbush decreases (FDs) are generally employed by researchers investigating the solar-terrestrial connection mechanisms. Although it has been observed that small-amplitude FDs are relatively important as they may be the outcome of the response of cosmic ray time-intensity variations to solar ejections that generate interplanetary and solar wind plasma disturbances, empirical relations between weak FDs and solar-terrestrial parameters are rarely tested. In an attempt to analyse the suggested strong connections between weak FDs and solar-terrestrial phenomena, we employed some comparatively more efficient, accurate, and highly sensitive versions of the recently developed computer FD event selection software. Large catalogues of low-amplitude (FD(per cent) ≥−3) Forbush events were selected from Apatity, Moscow, Newark, and Oulu NMs. These catalogues allow us to test, for the first time, the empirical relations between small-amplitude FDs and solar wind data. We find significant negative correlations between solar wind speed (SWS) and the small FDs at OULU, NWRK, and MOSC stations. While the relation at OULU is strong and statistically significant at 95 per cent confidence level, the weak correlation at NWRK and MOSC is only significant at a 90 per cent level. The negative correlation between the small events at OULU and the interplanetary magnetic field (IMF) is also significant at the a 95 per cent level. The relation between SWS and IMF and high-amplitude FDs were also tested, and the correlation coefficients were negative, strong, and statistically significant at a 99.9 per cent level of significance.