A toy model for monthly river flow forecasting

River flow forecasting depends on land-atmosphere coupled processes, and is relevant to hydrological applications and land-ocean coupling. A toy model is developed here for monthly river flow forecasting using the river flow and river basin averaged precipitation in prior month. Model coefficients are calibrated for each month using historical data. The toy model is based on water balance, easy to use and reproduce, and robust to calibrate with a short period of data. For five major rivers in the world, its results agree with observations very well. Its prediction uncertainty can be quantified using the model's error statistics or using a dynamic approach, but not by the dispersion of 10,000 ensemble members with different sets of coefficients in the model. Its results are much better than those from a physically based land model even after the mean bias correction. The toy model and a standard neural network available from the MATLAB give similar results, but the latter is more sensitive to the length of calibration period. For the monthly prediction of river flow with a strong seasonal cycle, a modified Nash-Sutcliffe coefficient of efficiency is introduced and is found to be more reliable in model evaluations than the original coefficient of efficiency or the correlation coefficient.