Optimal linearization trajectories for tangent linear models

We examine differential equations where nonlinearity is a result of the advection part of the total derivative or the use of quadratic algebraic constraints between state variables (such as the ideal gas law). We show that these types of nonlinearity can be accounted for in the tangent linear model by a suitable choice of the linearization trajectory. Using this optimal linearization trajectory, we show that the tangent linear model can be used to reproduce the exact nonlinear error growth of perturbations for more than 200 days in a quasi-geostrophic model and more than (the equivalent of) 150 days in the Lorenz 96 model. We introduce an iterative method, purely based on tangent linear integrations, that converges to this optimal linearization trajectory. The main conclusion from this article is that this iterative method can be used to account for nonlinearity in estimation problems without using the nonlinear model. We demonstrate this by performing forecast sensitivity experiments in the Lorenz 96 model and show that we are able to estimate analysis increments that improve the two-day forecast using only four backward integrations with the tangent linear model. Copyright © 2011 Royal Meteorological Society

Evaluation of water and energy budgets in regional climate models applied over Europe

This study presents a model intercomparison of four regional climate models (RCMs) and one variable resolution atmospheric general circulation model (AGCM) applied over Europe with special focus on the hydrological cycle and the surface energy budget. The models simulated the 15 years from 1979 to 1993 by using quasi-observed boundary conditions derived from ECMWF re-analyses (ERA). The model intercomparison focuses on two large atchments representing two different climate conditions covering two areas of major research interest within Europe. The first is the Danube catchment which represents a continental climate dominated by advection from the surrounding land areas. It is used to analyse the common model error of a too dry and too warm simulation of the summertime climate of southeastern Europe. This summer warming and drying problem is seen in many RCMs, and to a less extent in GCMs. The second area is the Baltic Sea catchment which represents maritime climate dominated by advection from the ocean and from the Baltic Sea. This catchment is a research area of many studies within Europe and also covered by the BALTEX program. The observed data used are monthly mean surface air temperature, precipitation and river discharge. For all models, these are used to estimate mean monthly biases of all components of the hydrological cycle over land. In addition, the mean monthly deviations of the surface energy fluxes from ERA data are computed. Atmospheric moisture fluxes from ERA are compared with those of one model to provide an independent estimate of the convergence bias derived from the observed data. These help to add weight to some of the inferred estimates and explain some of the discrepancies between them. An evaluation of these biases and deviations suggests possible sources of error in each of the models. For the Danube catchment, systematic errors in the dynamics cause the prominent summer drying problem for three of the RCMs, while for the fourth RCM this is related to deficiencies in the land surface parametrization. The AGCM does not show this drying problem. For the Baltic Sea catchment, all models similarily overestimate the precipitation throughout the year except during the summer. This model deficit is probably caused by the internal model parametrizations, such as the large-scale condensation and the convection schemes.

A comparison of models for estimating the riverine export of nitrogen from large watersheds

We evaluated the accuracy of six watershed models of nitrogen export in streams (kg km2 yr−1) developed for use in large watersheds and representing various empirical and quasi-empirical approaches described in the literature. These models differ in their methods of calibration and have varying levels of spatial resolution and process complexity, which potentially affect the accuracy (bias and precision) of the model predictions of nitrogen export and source contributions to export. Using stream monitoring data and detailed estimates of the natural and cultural sources of nitrogen for 16 watersheds in the northeastern United States (drainage sizes = 475 to 70,000 km2), we assessed the accuracy of the model predictions of total nitrogen and nitrate-nitrogen export. The model validation included the use of an error modeling technique to identify biases caused by model deficiencies in quantifying nitrogen sources and biogeochemical processes affecting the transport of nitrogen in watersheds. Most models predicted stream nitrogen export to within 50% of the measured export in a majority of the watersheds. Prediction errors were negatively correlated with cultivated land area, indicating that the watershed models tended to over predict export in less agricultural and more forested watersheds and under predict in more agricultural basins. The magnitude of these biases differed appreciably among the models. Those models having more detailed descriptions of nitrogen sources, land and water attenuation of nitrogen, and water flow paths were found to have considerably lower bias and higher precision in their predictions of nitrogen export.

Evaluation of water and energy budgets in regional climate models applied over Europe

Abstract This study presents a model intercomparison of four regional climate models (RCMs) and one variable resolution atmospheric general circulation model (AGCM) applied over Europe with special focus on the hydrological cycle and the surface energy budget. The models simulated the 15 years from 1979 to 1993 by using quasi-observed boundary conditions derived from ECMWF re-analyses (ERA). The model intercomparison focuses on two large atchments representing two different climate conditions covering two areas of major research interest within Europe. The first is the Danube catchment which represents a continental climate dominated by advection from the surrounding land areas. It is used to analyse the common model error of a too dry and too warm simulation of the summertime climate of southeastern Europe. This summer warming and drying problem is seen in many RCMs, and to a less extent in GCMs. The second area is the Baltic Sea catchment which represents maritime climate dominated by advection from the ocean and from the Baltic Sea. This catchment is a research area of many studies within Europe and also covered by the BALTEX program. The observed data used are monthly mean surface air temperature, precipitation and river discharge. For all models, these are used to estimate mean monthly biases of all components of the hydrological cycle over land. In addition, the mean monthly deviations of the surface energy fluxes from ERA data are computed. Atmospheric moisture fluxes from ERA are compared with those of one model to provide an independent estimate of the convergence bias derived from the observed data. These help to add weight to some of the inferred estimates and explain some of the discrepancies between them. An evaluation of these biases and deviations suggests possible sources of error in each of the models. For the Danube catchment, systematic errors in the dynamics cause the prominent summer drying problem for three of the RCMs, while for the fourth RCM this is related to deficiencies in the land surface parametrization. The AGCM does not show this drying problem. For the Baltic Sea catchment, all models similarily overestimate the precipitation throughout the year except during the summer. This model deficit is probably caused by the internal model parametrizations, such as the large-scale condensation and the convection schemes.

Comparison of polar ozone loss rates simulated by one-dimensional and three-dimensional models with Match observations in recent Antarctic and Arctic winters

Simulations of ozone loss rates using a three-dimensional chemical transport model and a box model during recent Antarctic and Arctic winters are compared with experimental loss rates. The study focused on the Antarctic winter 2003, during which the first Antarctic Match campaign was organized, and on Arctic winters 1999/2000, 2002/2003. The maximum ozone loss rates retrieved by the Match technique for the winters and levels studied reached 6 ppbv/sunlit hour and both types of simulations could generally reproduce the observations at 2-sigma error bar level. In some cases, for example, for the Arctic winter 2002/2003 at 475 K level, an excellent agreement within 1-sigma standard deviation level was obtained. An overestimation was also found with the box model simulation at some isentropic levels for the Antarctic winter and the Arctic winter 1999/2000, indicating an overestimation of chlorine activation in the model. Loss rates in the Antarctic show signs of saturation in September, which have to be considered in the comparison. Sensitivity tests were performed with the box model in order to assess the impact of kinetic parameters of the ClO-Cl2O2 catalytic cycle and total bromine content on the ozone loss rate. These tests resulted in a maximum change in ozone loss rates of 1.2 ppbv/sunlit hour, generally in high solar zenith angle conditions. In some cases, a better agreement was achieved with fastest photolysis of Cl2O2 and additional source of total inorganic bromine but at the expense of overestimation of smaller ozone loss rates derived later in the winter.

A 27 day persistence model of near-Earth solar wind conditions: a long lead-time forecast and a benchmark for dynamical models

Geomagnetic activity has long been known to exhibit approximately 27 day periodicity, resulting from solar wind structures repeating each solar rotation. Thus a very simple near-Earth solar wind forecast is 27 day persistence, wherein the near-Earth solar wind conditions today are assumed to be identical to those 27 days previously. Effective use of such a persistence model as a forecast tool, however, requires the performance and uncertainty to be fully characterized. The first half of this study determines which solar wind parameters can be reliably forecast by persistence and how the forecast skill varies with the solar cycle. The second half of the study shows how persistence can provide a useful benchmark for more sophisticated forecast schemes, namely physics-based numerical models. Point-by-point assessment methods, such as correlation and mean-square error, find persistence skill comparable to numerical models during solar minimum, despite the 27 day lead time of persistence forecasts, versus 2–5 days for numerical schemes. At solar maximum, however, the dynamic nature of the corona means 27 day persistence is no longer a good approximation and skill scores suggest persistence is out-performed by numerical models for almost all solar wind parameters. But point-by-point assessment techniques are not always a reliable indicator of usefulness as a forecast tool. An event-based assessment method, which focusses key solar wind structures, finds persistence to be the most valuable forecast throughout the solar cycle. This reiterates the fact that the means of assessing the “best” forecast model must be specifically tailored to its intended use.

On the Nusselt number for frazil ice growth—a correction to ”Frazil evolution in channels“ by Lars Hammar and Hung-Tao Shen

The growth (melt) rate of frazil ice is governed by heat transfer away from (towards) the ice crystal, which can be represented by the Nusselt number. We discuss choices for the Nusselt number and turbulent length scale appropriate for frazil ice and note an inaccuracy in the study ”Frazil evolution in channels“ by Lars Hammar and Hung-Tao Shen, which has also led to potentially significant errors in several other papers. We correct this error and suggest an appropriate strategy for determining the Nusselt number applicable to frazil ice growth and melting.

Forecasting US output growth with non-linear models in the presence of data uncertainty

We consider the impact of data revisions on the forecast performance of a SETAR regime-switching model of U.S. output growth. The impact of data uncertainty in real-time forecasting will affect a model's forecast performance via the effect on the model parameter estimates as well as via the forecast being conditioned on data measured with error. We find that benchmark revisions do affect the performance of the non-linear model of the growth rate, and that the performance relative to a linear comparator deteriorates in real-time compared to a pseudo out-of-sample forecasting exercise.

Real-time forecasting of inflation and output growth with autoregressive models in the presence of data revisions

We examine how the accuracy of real-time forecasts from models that include autoregressive terms can be improved by estimating the models on ‘lightly revised’ data instead of using data from the latest-available vintage. The benefits of estimating autoregressive models on lightly revised data are related to the nature of the data revision process and the underlying process for the true values. Empirically, we find improvements in root mean square forecasting error of 2–4% when forecasting output growth and inflation with univariate models, and of 8% with multivariate models. We show that multiple-vintage models, which explicitly model data revisions, require large estimation samples to deliver competitive forecasts. Copyright © 2012 John Wiley & Sons, Ltd.

Representation of model error in a convective-scale ensemble prediction system

In this paper ensembles of forecasts (of up to six hours) are studied from a convection-permitting model with a representation of model error due to unresolved processes. The ensemble prediction system (EPS) used is an experimental convection-permitting version of the UK Met Office’s 24- member Global and Regional Ensemble Prediction System (MOGREPS). The method of representing model error variability, which perturbs parameters within the model’s parameterisation schemes, has been modified and we investigate the impact of applying this scheme in different ways. These are: a control ensemble where all ensemble members have the same parameter values; an ensemble where the parameters are different between members, but fixed in time; and ensembles where the parameters are updated randomly every 30 or 60 min. The choice of parameters and their ranges of variability have been determined from expert opinion and parameter sensitivity tests. A case of frontal rain over the southern UK has been chosen, which has a multi-banded rainfall structure. The consequences of including model error variability in the case studied are mixed and are summarised as follows. The multiple banding, evident in the radar, is not captured for any single member. However, the single band is positioned in some members where a secondary band is present in the radar. This is found for all ensembles studied. Adding model error variability with fixed parameters in time does increase the ensemble spread for near-surface variables like wind and temperature, but can actually decrease the spread of the rainfall. Perturbing the parameters periodically throughout the forecast does not further increase the spread and exhibits “jumpiness” in the spread at times when the parameters are perturbed. Adding model error variability gives an improvement in forecast skill after the first 2–3 h of the forecast for near-surface temperature and relative humidity. For precipitation skill scores, adding model error variability has the effect of improving the skill in the first 1–2 h of the forecast, but then of reducing the skill after that. Complementary experiments were performed where the only difference between members was the set of parameter values (i.e. no initial condition variability). The resulting spread was found to be significantly less than the spread from initial condition variability alone.

Forecasting exchange rate volatility using conditional variance models selected by information criteria

This paper uses appropriately modified information criteria to select models from the GARCH family, which are subsequently used for predicting US dollar exchange rate return volatility. The out of sample forecast accuracy of models chosen in this manner compares favourably on mean absolute error grounds, although less favourably on mean squared error grounds, with those generated by the commonly used GARCH(1, 1) model. An examination of the orders of models selected by the criteria reveals that (1, 1) models are typically selected less than 20% of the time.

Implications of model error for numerical climate prediction

Numerical climate models constitute the best available tools to tackle the problem of climate prediction. Two assumptions lie at the heart of their suitability: (1) a climate attractor exists, and (2) the numerical climate model's attractor lies on the actual climate attractor, or at least on the projection of the climate attractor on the model's phase space. In this contribution, the Lorenz '63 system is used both as a prototype system and as an imperfect model to investigate the implications of the second assumption. By comparing results drawn from the Lorenz '63 system and from numerical weather and climate models, the implications of using imperfect models for the prediction of weather and climate are discussed. It is shown that the imperfect model's orbit and the system's orbit are essentially different, purely due to model error and not to sensitivity to initial conditions. Furthermore, if a model is a perfect model, then the attractor, reconstructed by sampling a collection of initialised model orbits (forecast orbits), will be invariant to forecast lead time. This conclusion provides an alternative method for the assessment of climate models.

A systematic approach to identify the sources of tropical SST errors in coupled models using the adjustment of initialised experiments

Understanding the sources of systematic errors in climate models is challenging because of coupled feedbacks and errors compensation. The developing seamless approach proposes that the identification and the correction of short term climate model errors have the potential to improve the modeled climate on longer time scales. In previous studies, initialised atmospheric simulations of a few days have been used to compare fast physics processes (convection, cloud processes) among models. The present study explores how initialised seasonal to decadal hindcasts (re-forecasts) relate transient week-to-month errors of the ocean and atmospheric components to the coupled model long-term pervasive SST errors. A protocol is designed to attribute the SST biases to the source processes. It includes five steps: (1) identify and describe biases in a coupled stabilized simulation, (2) determine the time scale of the advent of the bias and its propagation, (3) find the geographical origin of the bias, (4) evaluate the degree of coupling in the development of the bias, (5) find the field responsible for the bias. This strategy has been implemented with a set of experiments based on the initial adjustment of initialised simulations and exploring various degrees of coupling. In particular, hindcasts give the time scale of biases advent, regionally restored experiments show the geographical origin and ocean-only simulations isolate the field responsible for the bias and evaluate the degree of coupling in the bias development. This strategy is applied to four prominent SST biases of the IPSLCM5A-LR coupled model in the tropical Pacific, that are largely shared by other coupled models, including the Southeast Pacific warm bias and the equatorial cold tongue bias. Using the proposed protocol, we demonstrate that the East Pacific warm bias appears in a few months and is caused by a lack of upwelling due to too weak meridional coastal winds off Peru. The cold equatorial bias, which surprisingly takes 30 years to develop, is the result of an equatorward advection of midlatitude cold SST errors. Despite large development efforts, the current generation of coupled models shows only little improvement. The strategy proposed in this study is a further step to move from the current random ad hoc approach, to a bias-targeted, priority setting, systematic model development approach.

Systematic model forecast error in Rossby wave structure

Diabatic processes can alter Rossby wave structure; consequently errors arising from model processes propagate downstream. However, the chaotic spread of forecasts from initial condition uncertainty renders it difficult to trace back from root mean square forecast errors to model errors. Here diagnostics unaffected by phase errors are used, enabling investigation of systematic errors in Rossby waves in winter-season forecasts from three operational centers. Tropopause sharpness adjacent to ridges decreases with forecast lead time. It depends strongly on model resolution, even though models are examined on a common grid. Rossby wave amplitude reduces with lead time up to about five days, consistent with under-representation of diabatic modification and transport of air from the lower troposphere into upper-tropospheric ridges, and with too weak humidity gradients across the tropopause. However, amplitude also decreases when resolution is decreased. Further work is necessary to isolate the contribution from errors in the representation of diabatic processes.

Estimating correlated observation error statistics using an ensemble transform Kalman filter

For certain observing types, such as those that are remotely sensed, the observation errors are correlated and these correlations are state- and time-dependent. In this work, we develop a method for diagnosing and incorporating spatially correlated and time-dependent observation error in an ensemble data assimilation system. The method combines an ensemble transform Kalman filter with a method that uses statistical averages of background and analysis innovations to provide an estimate of the observation error covariance matrix. To evaluate the performance of the method, we perform identical twin experiments using the Lorenz ’96 and Kuramoto-Sivashinsky models. Using our approach, a good approximation to the true observation error covariance can be recovered in cases where the initial estimate of the error covariance is incorrect. Spatial observation error covariances where the length scale of the true covariance changes slowly in time can also be captured. We find that using the estimated correlated observation error in the assimilation improves the analysis.