Correcting Prediction Errors Using Historical Data For Numerical Models

Numerical prediction is the main means of current weather and climate predictions, however, this approach is proposed as an initial problem in mathematics and a large amount of historical data cannot be used, which is a fundamental problem compared with the traditional method of synoptic chart and statistics. Chinese scientists identified this problem and proposed the use of historical data in numerical predictions over the temperal scale within extended-range predictions to short-term climate predictions. This study extends the domain to both ends of this temperal scale based on previous work and explores the new approach of correcting numerical predictions by using historical data. We aim at the medium-range weather forecasts with shorter time scales by using the Global/Regional Assimilation and Prediction System(GRAPES) as the test platform, which is a medium-range operational prediction model developed by China, to improve the prediction skill of China by original strategy. Using "error diagnosis-error inversion-error correction" as the mainline, innovative inversion and correction methods are proposed, and the system framework of correcting medium-range numerical predictions with an analog-dynamical method is constructed. Additionally, we focus on long-term range climate projections with the Coupled Model Intercomparison Project Phase 5(CMIP5), which is the main tool of current climate change assessment and prediction, to improve the reliability of the simulation and projection results. A long-term climate projection model constrained by historical observations is established, and the projection capabilities are greatly improved by utilizing "geometric theory-historical simulation-future projection" as the mainline, which provides a more reliable reference for evaluating climate change. The main results of this study are as follows:(1) The temporal and spatial characteristics and the evolvement rules of medium-range model errors are revealed. The prediction errors are separated into systematic and nonsystematic parts. The systematic errors show that the meridional barometric gradient is underestimated, the temperature is generally overestimated and the westerly wind is underestimated as a whole. Linear trends, planetary scales, synoptic scales and diurnal variations are four main modes of temporal variations in the systematic error. Correcting the systematic error by reconstituting the main mode could improve the global period of validity of 500 hPa by 0.5 days. The magnitude of nonsystematic errors is much larger than that of systematic errors and is located in the contour intensive belt, the front of the upper trough, and the isothermal intensive belt, which are the baroclinic regions at mid-latitudes, whereas the magnitude in barotropic tropics is weak. The temporal variation first exhibits exponential growth, followed by linear growth, and finally reaching a saturation value. The above comprehensive diagnosis provides a reference for error inversion and error correction.(2) A model error inverse algorithm that uses multiple historical data is proposed. The inverse theory was elaborated from the view of information theory and set theory. The proposed inverse algorithm based on this theory can avoid the defect of establishing a tangent linear model and adjoint model. The given convergence criterion could ensure fast and robust convergence and save computational resources. The RMSE of each physical quantity could be reduced to 1/10 of the original value by iteration. The ACC and RMSE after iteration kept stable by increasing the prediction time. The iterated model error could reflect the diurnal variation and exhibit increasing amplitudes and eastward-spreading phases. A historical model error dataset is established by this algorithm, which provides a large amount of historical information for future model error estimation.(3) The analog-dynamical error correction method is developed for medium-range numerical predictions. Introducing the continuity theorem of prediction errors converts the method into an interpolation problem of the spatial curved surface in the phase space, and the rationality is proven. An analog criterion that considers the sea surface temperature and atmospheric circulation is proposed. The optimal analog update period is determined by sensitive experiments. The hindcast results show that this method could extend the validity period of the global 500 hPa height field by 0.8 days, and the effect becomes more significant by increasing the prediction time. This improvement is stable on all vertical levels and is superior to the systematic correction results. Comparing the prediction errors of the height field and temperature field and the kinetic energy of the wind field before and after correction indicates that this method could decrease the high prediction errors of each physical quantity. These results indicate the validity of this method in improving medium-range predictions.(4) The geometric theory of correcting climate predictions in dynamical systems is described. From the starting point of dynamical system theory, weather forecasts and climate prediction are distinguished from the perspective of phase spaces. Correcting weather forecasts involves adjusting the phase path on the same attractor, whereas correcting climate prediction involves adjusting the evolution path of the attractor. Because the properties of phase paths and attractors are different, corrections with historical data should also be different. The problem of evolution path adjustment of the attractor is converted into a prediction problem of the projected trajectory on an orthogonal basis. We propose adjusting the attractor’s evolution path by establishing the mapping relationship between the model’s projected trajectory and the observed projected trajectory, which provides a theoretical basis for correcting climate predictions with historical data.(5) The historical simulation capability of climate system models is improved. An evaluation of CMIP5 indicates that the simulations of global dry and wet climatic change are significantly different from the observations. CMIP5 underestimated the decreasing trend of global precipitation and the expansion of drylands. An optimal ensemble correction scheme is proposed to constraint CMIP5 by establishing the mapping relationship between the historical observations and CMIP5. Cross-validation indicates that this method could increase the average ACC of the aridity index from 0.04 to 0.28 and decrease the average RMSE from 0.24 mm/mm to 0.19 mm/mm. The consistency of the spatial patterns is improved, and the linear trend and decadal variability are reflected on the temporal scale. The verification of other observational data indicates that this correction is robust.(6) The correction of future long-term climate projections with historical data is realized. Posterior independent validation indicates that the mapping relationship established in historical periods is effective for improving the prediction skill during future periods. The correction model is used to project future changes in the dryland area. Under representative concentration pathways(RCPs) RCP8.5 and RCP4.5, the dryland area is projected to increase by 23%(11%) relative to the 1961-1990 baseline, equaling 56%(50%) of the total land surface, whereas 78% of the dryland expansion will occur in developing countries. The increasing aridity, enhanced warming and rapidly growing human population will exacerbate the future risk of land degradation and desertification in the drylands of developing countries and increase regional differences in global economic development. These results provide a scientific reference for sustainable development strategies for developing countries.Most importantly, this study investigated the systematic correction of numerical model errors with historical data under the framework of "mathematical principle-technical route- experiment and application," improved the prediction skills in the field of medium-range weather forecasts and long-term climate projections, and showed wide application prospects. The results provide scientific evidence for the combination of numerical predictions and historical data and will play an important role in exploring the Chinese innovation of numerical predictions.