Thomas Kaminski

Assimilation and Network Design

Information on the carbon cycle comes from a variety of sources. The methods described in this chapter provide a formalism for combining this information. Without such a formalism we are left making ad hoc choices about how to improve our understanding in the light of disagreements among various streams of information. The introduction of such methods into carbon cycle research, principally via the atmospheric studies of Enting et al. (1993, 1995), revolutionised the field and laid the groundwork for most of the subsequent investigations. The methods in question are fundamentally statistical. They hence provide estimates of the confidence we should have in quantitative statements about the carbon cycle. These statements are usually couched as spreads of probability distributions or as confidence intervals. We refer to them generally as posterior uncertainties. These posterior uncertainties depend on the prior uncertainties of the various data streams that feed the estimation process, the method for combining these data streams (usually some kind of model) and the particular state of the system. Of course, an important aim of measurements is to reduce the posterior uncertainty. The present chapter is concerned with quantitative network design, by which we understand the optimisation of a measurement strategy via minimisation of this posterior uncertainty for target quantities of particular interest. Examples of such target quantities are the long-term global mean terrestrial flux to the atmosphere over a period in the past or in the future. The computational tool that transforms the information provided by an observational network of the carbon cycle into an estimate of posterior uncertainty is a Carbon Cycle Data Assimilation System (CCDAS). Hence, network design is closely linked to assimilation both conceptually and computationally. Much of the work reviewed in this chapter lies in a small subset of possible network design applications for the carbon cycle. In particular, it uses a limited set of types of observations. This is not an inherent limitation of the approach but rather a limitation in modelling approaches that can combine many streams of measurements. This is changing now. Hence, much of the chapter looks forward to applications that combine different measurement approaches. It is useful, therefore, to describe the problem in general even if most cited examples are from simpler cases.

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.

Recomputations in reverse mode AD

The main challenge of the reverse (or adjoint) mode of automatic differentiation (AD) is providing the accurate values of required variables to the derivative code. We discuss different strategies to tackle this challenge. The ability to generate efficient adjoint code is crucial for handling large scale applications. For challenging applications, efficient adjoint code must provide at least a fraction of the values of required variables through recomputations, but it is essential to avoid unnecessary recomputations. This is achieved by the Efficient Recomputation Algorithm implemented in the Tangent linear and Adjoint Model Compiler and in Transformation of Algorithms in Fortran, which are source-to-source translation AD tools for Fortran programs. We describe the algorithm and discuss possible improvements.

Efficient Sensitivities for the Spin-Up Phase

In geosciences, it is common to spin up models by integrating with anually repeated boundary conditions. AD-generated code for evaluating sensitivities of the final cyclo-stationary state with respect to model parameters or boundary conditions usually includes a similar iteration for the derivative statements, possibly with a reduced number of iterations. We evaluate an alternative strategy that first carries out the spin-up, then evaluates the full Jacobian for the final iteration and from there applies the implicit function theorem to solve for the sensitivites of the cyclo-stationary state. We demonstrate the benefit of the strategy for the spin-up of a simple box-model of the atmospheric transport. We derive a heuristic inequality for this benefit, which increases with the number of iterations and decreases with the size of the state space.

Tangent Linear and Adjoint Versions of NASA/GMAO’s Fortran 90 Global Weather Forecast Model

The NASA finite-volume General Circulation Model (fvGCM) is a three-dimensional Navier-Stokes solver that is being used for quasi-operational weather forecasting at NASA/GMAO. By means of the automatic differentiation tool TAF, efficient tangent linear and adjoint versions are generated from the Fortran-90 source code of fvGCM’s dynamical core. fvGCM’s parallelisation capabilities based on OpenMP and MPI have been transferred to the tangent linear and adjoint codes. For OpenMP, TAF automatically inserts corresponding OpenMP directives in the derivative code. For MPI, TAF generates interfaces to hand-written tangent linear and adjoint wrapper routines. TAF also generates a scheme that allows the tangent linear and adjoint models to linearise around an external trajectory of the model state. The generation procedure is set up in an automated way, allowing quick updates of the derivative codes after modifications of fvGCM.