Forward and Inverse Modeling

In most cases models describing biogeochemical cycles are used to estimate the concentration (or total mass) in the various reservoirs based on information about source and sink processes, as in the examples given in Section 4.4. This is often called forward modeling. If direct measurements of the concentration are available, they can be compared to the model estimates. This process is referred to as model testing. If there are significant differences between observations and model simulations, improvements in the model are necessary. A natural step is then to reconsider the specification of the sources and/or the sinks and perform additional simulations. 

---

Kettle A. J., Kuhn U., von Hobe M., Kesselmeier J., Liss P. S., and Andreae M. O. (2002) Comparing forward and inverse models to estimate the seasonal variation of hemisphere-integrated fluxes of carbonyl sulfide. Atmos. Chem. Phys. 2, 343-361. 

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. 

Geochemists observe patterns of element and mineral distribution in nature and then use forward and inverse models of unit processes to understand how these patterns developed. The patterns result from the... 

The general problems of forward and inverse modelling were discussed in Section 5 and the problem of integrating production data is, in principle, of the same t3rpe. However, in practice the large computer times needed to perform flow simulations and the strongly time dependent natme of the data to be processed makes the problem very difficult, compared to the inversion of small scale measurements. 

Forward and Inverse Modelling Many inverse problems in the geosciences involve sparse data used to constrain functions representing three-dimensional heterogeneous property fields, such as porosity or permeability. It is not that clear which parameters are to be determined in the inversion. Is it properties as functions of position, or is it the parameters in the correlation functions summarising the heterogeneity It may be some mixture of the two, with the parameters in the correlation functions displaying the most sensitivity to the measurements. There is room for much research here model problems should be studied so that our intuitions can be further developed. 

Plummer L. N. (1984) Geochemical modelling a comparison of forward and inverse methods, In First Canadian/American Conf Hydrogeol. In Practical Applications of Ground Water Geochemistry (eds. B. Hitchon and E. I. Wallick). National Well Water Association, Worthington, OH, pp. 149-177. 

In this introductory section, I will give a mathematical formulation of several forward and inverse problems typical for geophysical methods. The definition of general forward and inverse problems can be described schematically by the following chart FORWARD PROBLEM model model parameters m data d. 

The mathematical formulation of seismic forward and inverse problems in the simplest case of an acoustic model in the frequency domain is given by equation (1.25), which we will repeat here for convenience ... 

In practice we usually solve forward and inverse problems in the space of discrete data and model parameters. For a numerical formulation of QA inversion we can use the matrix formula for QA approximation (9.241), reproduced here for convenience ... 

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. 

FIGURE 6.22 Comparison of the forward- and inverse-dynamics methods for determining muscle forces during movement. Top Body motions are the inputs and muscle forces are the outputs in inverse dynamics. Thus, measurements of body motions are used to calculate the net muscle torques exerted about the joints, from which muscle forces are determined using static optimization. Bottom Muscle excitations are the inputs and body motions are the outputs in forward dynamics. Muscle force (F ) is an intermediate product (i.e., output of the model for musculotendon dynamics). 

Forward and Inverse Multiphase Flow Modeling 419 Flow... 

In this section, we will consider immiscible radial flows with capillary pressure and prescribed mudcake growth. In particular, we will derive the relevant governing equations, develop the numerical finite difference algorithm and the Fortran implementation, and proceed to demonstrate the computational model in both forward and inverse modes. 

---