Adjoint modeling

Finlay JC, Foster TH. Recovery of hemoglobin oxygen saturation and intrinsic fluorescence with a forward-adjoint model. Applied Optics 2005, 44, 1917-1933. 

In IFS tracer data assimilation mode, the IFS tracer forecast mode is applied in the outer loops of ECMWF data assimilation system, i.e. the calculation of the trajectories runs of the complete model of the 4D VAR (Mahfouf and Rabier 2000) The iimer loops used in the minimisation step with the tangent linear and adjoint model are currently run uncoupled, i.e. without the application of the source and sink tendencies from the CTM. 

The adjoint modelling technique is a good instrument to improve on the climatological emissions that are currently used for many species. 

In many cases due to unknown or uncertain parameters of the release, the estimation of source term characteristics, based on environmental pollution monitoring, is a very important issue for emergency response systems. A combination of the forward and inverse (adjoint) modelling approaches allows to solve such environmental risk and emergency management problems (e.g., source-term estimation) more effectively compared with the traditional ways based on only the forward modelling. 

Kaminski, T., Heimann, M., and Gicring, R. (1999). A coarse grid three-dimensional global inverse model of the atmospheric transport. 1. Adjoint model and Jacobian matrix. J. Geophys. Res. 104, 18,535-18,55.3. 

Houweling S, Kaminski T, Dentener F, Lelieveld I, Heimann M. 1999. Inverse modeling of methane sources and sinks using the adjoint of a global transport model. Journal of Geophysical Research-Atmospheres 104 26137-26160. 

These methods are efficient for problems with initial-value ODE models without state variable and final time constraints. Here solutions have been reported that require from several dozen to several hundred model (and adjoint equation) evaluations (Jones and Finch, 1984). Moreover, any additional constraints in this problem require a search for their appropriate multiplier values (Bryson and Ho, 1975). Usually, this imposes an additional outer loop in the solution algorithm, which can easily require a prohibitive number of model evaluations, even for small systems. Consequently, control vector iteration methods are effective only when limited to the simplest optimal control problems. 

Gradient calculations for the x variables are obtained from implicit reformulations of the DAE model. Clearly the easiest, but least accurate, way is simply to re-solve the model for each perturbation of the parameters. Sargent and Sullivan (1977, 1979) derived these gradients using an adjoint formulation. In addition, they were able to accelerate the adjoint computations by retaining the information from the model solution (the forward step) for the adjoint solution in the backward step. This approach was later refined for variable stepsize methods by Morison (1984). The adjoint approach to parameterized optimal control was also used by Jang et al. (1987) and Goh and Teo (1988). 

Gear s type method is used for solving the process model which provides the value of the objective function and constraints. Gradient information of the objective function and constraints with respect to the decision variables is evaluated in an efficient way using adjoint variable approach. Then, a NLP solver determines a new set of control parameters and sends it back to the model solver. This procedure is repeated until the optimal value is found satisfying a specified accuracy. 

The adjoint system approach requires integration of the model equations forward in time before integration of the adjoint system equations backward. 

Similar to those presented in section 5.5 the adjoint equations for the model represented by Equation 5.4 can be written as ... 

With the results of the forward integration of the model Equation 5.4 and then by integrating backward the adjoint equations (Equation 5.18) the gradients can be determined from ... 

Results from this model were verified by Neupauer and Wilson [51] using the adjoint method. In this method, the forward governing equation, with concentration as the dependent variable, is replaced by the adjoint equation, with the adjoint state as the dependent variable. They showed that backward-in-time location and travel time probabilities are adjoint states of the forward-in-time resident concentration. In this and the follow-up paper, Neupauer and Wilson [51,52] presented the adjoint method as a formal framework for obtaining the backward-in-time probabilities for multidimensional problems and more complex domain geometries. 

Heberton, C.I., Russel, T.F., Konikow, L.F., and Hornberger, G.Z., A Three-Dimensional Finite-Volume Eulerian-Lagrangian Localized Adjoint Method (ELLAM) for Solute-Transport Modeling, Water Resources Investigations Report 00-4087, U.S. Geological Survey, Reston, VA, 2000. 

They are easier to use for the inverse modelling and adjoint problem... 

In the second mode, a consistent treatment of the emission injection and vertical transport would be achieved. In particular, the adjoint formulation of diffusion and convection in data assimilation would be consistent with the forward model. However, dislocation of the chemistry tendencies is more likely than in case 1 because the IPS concentration fields tend to differ more from the CTM fields. 

First, the SILAM model (Sofiev et al. 2006) was applied in adjoint mode to identify the potential sources of pollution. It was found that the aerosol peak of May 2-3 most probably originated from the Nikel metallurgy factory (Kola Peninsula, Russia) located about 200 km north from Varrio (Kaasik et al. 2007). Then the SILAM model was applied in a forward mode comparatively with the ECMWF and HIRLAM (FMI) meteorological datasets EMEP emission data on sulphate and PM, sea salt emissions calculated by SILAM, emission model based on... 

We can see that practical implementation of this algorithm requires computing the adjoint of the Frechet derivative Pg Hn corresponding forward modeling... 

Time domain electromagnetic (EM) migration is based on downward extrapolation of the residual field in reverse time. In this section I will show that electromagnetic migration, as the solution of the boundary value problem for the adjoint Maxwell s equation, can be clearly associated with solution of the inverse problem in the time domain. In particular, I will demonstrate that the gradient of the residual field energy flow functional with respect to the perturbation of the model conductivity is equal to the vector cross-correlation function between the predicted field for the given... 

We can get a better understanding of the physical meaning of the elastic adjoint FV chet operator if we consider the first iteration in the inversion scheme (15.238). Let us assume that the initial distribution of Lame velocities in the model is given by some background parameters Cpt, (r) and Csb (r) ... 

A combination of the forward and inverse modelling approaches allows to solve some environmental and nuclear risk problems more effectively compared with the traditional ways based on the forward modelling. For the inverse modelling problem, most of the western scientists (Persson et al., 1987 [491] Prahm et al., 1980 [509] Seibert, 2001 [569]) use the common back- trajectory techniques, suitable only for the Lagrangian models. The Novosibirsk scientific school established by G.I. Marchuk in Russia has suggested a fruitful theoretical method for inverse modelling, based on adjoint equations (Marchuk, 1982 [391], 1995 [392] Penenko, 1981 [486]) and suitable for the Eulerian models. This approach has further been used and improved by several authors (Baklanov, 1986 [20], 2000 [25] Pudykiewicz, 1998 [512] Robertson and Lange, 1998 [538]) for estimation of source-term parameters in the atmospheric pollution problems. 

Figure 9.21 Example of source term estimation, based on the forward (a) and inverse (b) local-scale modelling in a case of accidental contamination from an unknown release (Baklanov, 2000 [25]). Simulated wind fields and isolines of a - air pollution concentration in surface layer based on forward modelling b - probability density function base on the adjoint problem.

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