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The Stochastically Perturbed Parametrization Tendencies (SPPT) scheme is used at weather and climate forecasting centres worldwide to represent model uncertainty that arises from simplifications involved in the parametrization process. It uses spatio-temporally correlated multiplicative noise to perturb the sum of the parametrized tendencies. However, SPPT does not distinguish between different parametrization schemes, which do not necessarily have the same error characteristics. A generalization to SPPT is proposed, whereby the tendency from each parametrization scheme can be perturbed using an independent stochastic pattern. This acknowledges that the forecast errors arising from different parametrizations are not perfectly correlated. Two variations of this 'independent SPPT' (iSPPT) approach are tested in the Integrated Forecasting System (IFS). The first perturbs all parametrized tendencies independently, while the second groups tendencies before perturbation. The iSPPT schemes lead to statistically significant improvements in forecast reliability in the Tropics in medium-range weather forecasts. This improvement can be attributed to a large, beneficial increase in ensemble spread in regions with significant convective activity. The iSPPT schemes also lead to improved forecast skill in the extratropics for a set of cases in which the synoptic initial conditions were more likely to result in European 'forecast busts'. Longer 13 month simulations are also considered to indicate the effect of iSPPT on the mean climate of the IFS. [ABSTRACT FROM AUTHOR]