Ralf Giering

Recipes for adjoint code construction

Adjoint models are increasingly being developed for use in meteorology and oceanography. Typical applications are data assimilation, model tuning, sensitivity analysis, and determination of singular vectors. The adjoint model computes the gradient of a cost function with respect to control variables. Generation of adjoint code may be seen as the special case of differentiation of algorithms in reverse mode, where the dependent function is a scalar. The described method for adjoint code generation is based on a few basic principles, which permits the establishment of simple construction rules for adjoint statements and complete adjoint subprograms. These rules are presented and illustrated with some examples. Conflicts that occur due to loops and redefinition of variables are also discussed. Direct coding of the adjoint of a more sophisticated model is extremely time consuming and subject to errors. Hence, automatic generation of adjoint code represents a distinct advantage. An implementation of the method, described in this article, is the tangent linear and adjoint model compiler (TAMC).

Two decades of terrestrial carbon fluxes from a carbon cycle data assimilation system (CCDAS)

This paper presents the space-time distribution of terrestrial carbon fluxes for the period 1979-1999 generated by a terrestrial carbon cycle data assimilation system (CCDAS). CCDAS is based around the Biosphere Energy Transfer Hydrology model. We assimilate satellite observations of photosynthetically active radiation and atmospheric CO2 concentration observations in a two-step process. The control variables for the assimilation are the parameters of the carbon cycle model. The optimized model produces a moderate fit to the seasonal cycle of atmospheric CO2 concentration and a good fit to its interannual variability. Long-term mean fluxes show large uptakes over the northern midlatitudes and uptakes over tropical continents partly offsetting the prescribed efflux due to land use change. Interannual variability is dominated by the tropics. On interannual timescales the controlling process is net primary productivity (NPP) while for decadal changes the main driver is changes in soil respiration. An adjoint sensitivity analysis reveals that uncertainty in long-term storage efficiency of soil carbon is the largest contributor to uncertainty in net flux. (Less)

A coarse grid three-dimensional global inverse model of the atmospheric transport. 2. Inversion of the transport of CO2 in the 1980s

Models of atmospheric transport can be used to interpret spatiotemporal differences in the observed concentrations of CO2 in terms of its surface exchange fluxes. Inversion of the atmospheric transport is the systematic search for both a flux field that yields an optimal match between modeled and observed concentrations and, equally importantly, the uncertainties in this inferred flux field. The present inversion study combines observations of the CO2 concentration at the global station network of the NOAA/CMDL in the 1980s with a representation of the atmospheric transport model TM2 by its Jacobian matrix, which has been previously computed by the adjoint model of TM2. This Jacobian matrix maps monthly fluxes on the approximately 8° latitude by 10° longitude horizontal model grid onto the resulting changes in the monthly CO2 concentration at every station. Since the number of observational sites is much smaller than the number of grid cells, the inverse problem is highly underdetermined. A unique solution is determined by including a priori information on the surface exchange fluxes derived from output of high-resolution models of both the terrestrial biosphere and the ocean, combined with statistics of fossil fuel burning and land use change. Performing a Bayesian synthesis inversion, for a target period in the 1980s, the average seasonal cycle and the mean annual magnitude of CO2 surface fluxes on the TM2 grid are inferred. The resulting simulated concentration compares well with independent observations. On a global scale, an oceanic sink of 1.5±0.4 gigatons of carbon (GtC) is estimated. This sink is stronger in the northern than in the southern hemisphere. On a regional scale, however, the inferred exchange fluxes exhibit high uncertainty, indicating a low capacity of the global observational network to monitor regional trace gas emissions. These findings are relatively insensitive to the year of meteorological driving data, suggesting interannual changes in concentration should primarily result from source not transport changes.

A coarse grid three‐dimensional global inverse model of the atmospheric transport: 1. Adjoint model and Jacobian matrix

TM2 is a global three-dimensional model of the atmospheric transport of passive tracers. The adjoint of TM2 is a model that allows the efficient evaluation of derivatives of the simulated tracer concentration at obs ervational locations with respect to the tracers sources and sinks. We describe the ge neration of the adjoint model by applying the Tangent linear and Adjoint Model Compiler in the reverse mode of automatic differentiation to the code of TM2. Using CO as an example of a chemically inert tracer, the simulated concentration a t observational locations is linear in the surface exchange fluxes, and thus the transpo rt can be represented by the models Jacobian matrix. In many current inverse mode ling studies, such a matrix has been computed by multiple runs of a transport model for a set of prescribed surface flux patterns. The computational cost ha s been proportional to the number of patterns. In contrast, for differentiation in reverse mode, the cost is independent of the number of flux components. Hence, by a si ngle run of the adjoint model, the Jacobian for the approximately 8 latitude by 10 longitude horizontal resolution of TM2 could be computed efficiently. We quantify this efficiency by comparison with the conventional forward mode ling approach. For some prominent observational sites, we present visualizat ions of the Jacobian matrix by series of illustrative global maps quantifying th e impact of potential emissions on the concentration in particular months. Furth ermore, we demonstrate how the Jacobian matrix is employed to completely analyze a transport model run: A simulated monthly mean value at a particular station is decomposed into the contributions to this value by all flux components, i.e., the fluxes into every surface model grid cell and month. This technique also results in a series of global maps.

Generating efficient derivative code with TAF adjoint and tangent linear Euler flow around an airfoil

FastOpts new automatic differentiation tool TAF is applied to the two-dimensional Navier-Stokes solver NSC2KE. For a configuration that simulates the Euler flow around an NACA airfoil, TAF has generated the tangent linear and adjoint models as well as the second derivative (Hessian) code. Owing to TAFs capability of generating efficient adjoints of iterative solvers, the derivative code has a high performance: running both the solver and its adjoint requires 3.4 times as long as running the solver only. Further examples of highly efficient tangent linear, adjoint, and Hessian codes for large and complex three-dimensional Fortran 77-90 climate models are listed. These examples suggest that the performance of the NSC2KE adjoint may well be generalised to more complex three-dimensional CFD codes. We also sketch how TAF can improve the adjoints performance by exploiting self-adjointness, which is a common feature of CFD codes.

Carbon cycle data assimilation with a generic phenology model

Photosynthesis by terrestrial plants is the main driver of the global carbon cycle, and the presence of actively photosynthesizing vegetation can now be observed from space. However, challenges remain when translating remotely sensed data into carbon fluxes. One reason is that the Fraction of Absorbed Photosynthetically Active Radiation (FAPAR), which documents the presence of photosynthetically active vegetation, relates more directly to leaf development and leaf phenology than to photosynthetic rates. Here, we present a new approach for linking FAPAR and vegetation-to-atmosphere carbon fluxes through variational data assimilation. The scheme extends the Carbon Cycle Data Assimilation System (CCDAS) by a newly developed, globally applicable and generic leaf phenology model, which includes both temperature and water-driven leaf development. CCDAS is run for seven sites, six of them included in the FLUXNET network. Optimization is carried out simultaneously for all sites against 20 months of daily FAPAR from the Medium Resolution Imaging Spectrometer on board the European Space Agencys ENVISAT platform. Fourteen parameters related to phenology and 24 related to photosynthesis are optimized simultaneously, and their posterior uncertainties are computed. We find that with one parameter set for all sites, the model is able to reproduce the observed FAPAR spanning boreal, temperate, humid-tropical, and semiarid climates. Assimilation of FAPAR has led to reduced uncertainty (by >10%) of 10 of the 38 parameters, including one parameter related to photosynthesis, and a moderate reduction in net primary productivity uncertainty. The approach can easily be extended to regional or global studies and to the assimilation of further remotely sensed data. (Less)

Sensitivity of the seasonal cycle of CO2 at remote monitoring stations with respect to seasonal surface exchange fluxes determined with the adjoint of an atmospheric transport model

The adjoint model to a global three-dimensional atmospheric transport model can be used to efficiently perform a sensitivity analysis, i.e. the computation of the partial derivatives of a particular model output feature with respect to many control variables. We demonstrate this approach by investigating the dependence of the magnitude of the modeled seasonal cycle of CO2 at remote monitoring stations with respect to the magnitudes of the seasonal cycle of the net CO2 surface fluxes prescribed from a simple diagnostic terrestrial biosphere model. The technique results in global maps of those source regions that predominately influence the magnitude of the seasonal cycle at the different monitoring stations.

An example of an automatic differentiation-based modelling system

We present a prototype of a Carbon Cycle Data Assimilation System (CCDAS), which is composed of a terrestrial biosphere model (BETHY) coupled to an atmospheric transport model (TM2), corresponding derivative codes and a derivative-based optimisation routine. In calibration mode, we use first and second derivatives to estimate model parameters and their uncertainties from atmospheric observations and their uncertainties. In prognostic mode, we use first derivatives to map model parameters and their uncertainties onto prognostic quantities and their uncertainties. For the initial version of BETHY the corresponding derivative codes have been generated automatically by FastOpts automatic differentiation (AD) tool Transformation of Algorithms in Fortran (TAF). From this point on, BETHY has been developed further within CCDAS, allowing immediate update of the derivative code by TAF. This yields, at each development step, both sensitivity information and systematic comparison with observational data meaning that CCDAS is supporting model development. The data assimilation activities, in turn, benefit from using the current model version. We describe generation and performance of the various derivative codes in CCDAS, i.e. reverse scalar (adjoint), forward over reverse (Hessian) as well as forward and reverse Jacobian plus detection of the Jacobians sparsity.

Data assimilation by an intermediate coupled ocean-atmosphere model: Application to the 1997–1998 El Niño

Sea surface temperature, sea level, and pseudo wind stress anomaly data from late 1996 to early 1998 are assimilated into an intermediate coupled model of the Tropical Pacific. Model data consistency is examined. Impact of the assimilation on forecast is evaluated. The ocean component of the coupled model consists of a shallow water model with two baroclinic modes, an Ekman shear layer, and a mixed layer temperature equation. The atmospheric model is a statistical one (based on dominant covariance of historical surface temperature and pseudo wind stress anomaly data). The adjoint method is used to fit the coupled model to 6 months of data by optimally adjusting the initial state and model parameters. A forecast is performed using the end state of an assimilation experiment as initial conditions and using parameters estimated during the assimilation period. Thus the model state during the assimilation and that during the forecast belong to the same model trajectory in different periods. Such an initialization procedure is useful in avoiding initial shock during forecast due to inconsistency of an initial state with the coupled model physics. As a result of optimal adjustments of initial state and parameters, the model is able to reproduce observed interannual variability of sea surface temperature and sea level reasonably well. The averaged residual model data misfits over various 6 month periods are 0.5°C and 5 cm, respectively. The model has a limited skill in reproducing much of the off-equatorial wind anomalies. The residual model data misfit in pseudo wind stress anomaly is larger than 10 m 2 s -2 . Forecasts initialized from the assimilation product are overall more realistic than those simply initialized from wind-forced ocean states. Consistent improvement due to optimal initialization is found for sea surface temperature and sea level anomalies in the central-eastern Pacific and zonal pseudo wind stress anomaly in the central Pacific, both in terms of root-mean-squared deviation from and correlation with the data. The adjustments of parameters in addition to initial state in a coupled context is found to be important to improving the model data consistency during the assimilation and the forecast. In particular, the estimated drag and damping coefficients properly regulate the relative strength of forcing and damping of the ocean state so as to fit the three types of observations during the assimilation (initialization) period, which facilitates the development of a large-amplitude warming event during the forecast. The study demonstrates the utility of oceanic and atmospheric data to estimate initial state and model parameters in a coupled context, which is useful to the evaluation, improvement, and initialization of El Nino-Southern Oscillation forecast models.

Development and First Applications of TAC

The paper describes the development of the software tool Transformation of Algorithms in C++ (TAC++) for automatic differentiation (AD) of C(++) codes by source-to-source translation. We have transferred to TAC++ a subset of the algorithms from its well-established Fortran equivalent, Transformation of Algorithms in Fortran (TAF). TAC++ features forward and reverse as well as scalar and vector modes of AD. Efficient higher order derivative code is generated by multiple application of TAC++. High performance of the generated derivate code is demonstrated for five examples from application fields covering remote sensing, computer vision, computational finance, and aeronautics. For instance, the run time of the adjoints for simultaneous evaluation of the function and its gradient is between 1.9 and 3.9 times slower than that of the respective function codes. Options for further enhancement are discussed.