Tracer inverse modeling

Enting, I. G., Trudinger, C. M., Francey, R. J. and Granek, H. Synthesis inversion of atmospheric CO2 using the GISS tracer transport model. Tech. Pap. 29, Div. of Atmos. Res., Comm. Sci. and Ind. Res. Org., Melbourne, Victoria, Australia. 

Within the framework of the lumped-parameter models, it is relatively simple to derive information about the mixing regime from multi-tracer data. As discussed above for the H- Kr tracer pair, the predicted concentrations for any tracer can be calculated for different lumped-parameter models with various parameter values, and the model that best fits the data can be found. This inverse modeling procedure can be illustrated by plotting the predicted output concentrations for any pair of tracers versus each other (Fig. 20). By plotting the measured concentrations in such a figure containing a series of curves that represent different models and parameter values, the curve that best fits the... 

Of course, inverse-modeling approaches can also be used to determine the parameters of numerical ground-water transport models from fits to observed tracer data. Estimates of residence times for a number of locations in an aquifer provide a powerful calibration target for numerical ground-water transport models. If enough data are available to constrain the numerous unknowns in such models, this is presumably the most effective way to extract useful information from tracer data, in particular because the numerical models can be used to make predictions for the future development of the investigated system. Such predictions may become much more constrained and trustworthy if the model has been calibrated against tracer data. 

Peelers F, Kipfer R, Hofer M, Imboden DM, Domysheva VM (2000b) Vertical turbulent diffusion and upwelling in Lake Baikal estimated by inverse modeling of transient tracers. J Geophys Res 105 14283 and 3451-3464... 

Inverse models , in contrast, follow a seemingly more intuitive approach, treating the measured tracer concentrations and other auxiliary knowledge formally as knowns, whereas the physical and biogeochemical parameters, to be determined on the basis of the tracer data, are treated as unknowns. 

Tracer behavior is different in each of the ideal chemical reactors, so the first step in developing a tracer kinetics model is to decide which reactor type best simulates the real situation and which kind of tracer dosing has occurred. If the amount and timing of tracer introduction is known, a forward model can be developed. If the amount and timing of tracer detected is known, an inverse model can be produced. The equations derived in the following section use concentration imits of mg/L because these units are typical of field studies. For laboratory experiments the models would use molal concentration units. The mass (M) variables would be replaced by mole quantities ( ). The volume variables (V) would be replaced by mass of water (M) and the flow rate (0 would have units of kg/sec. 

The objective of this study is to develop an analytical model for a soil column s response to a sinusoidally varying tracer loading function by applying the familiar Laplace transform method in which the convolution integral is used to obtain the inverse transformation. The solution methodology will use Laplace transforms and their inverses that are available in most introductory texts on Laplace transforms to develop both the quasi steady-state and unsteady-state solutions. Applications of the solutions will be listed and explained. 

When the necessary condition that the chemistry at point A does not change with time is met, inverse mass balance models are still applicable when the chemistry at point B changes with time. An example is a laboratory column study, in which the chemistry of influent is maintained in the experiments while the effluent chemistry continues to change. In this case, we are assured that the effluent is chemically evolved from the influent. The variation of chemistry with time in the effluent does not violate the steady-state assumption. Another example is field injection of reactive tracers, during which the injectate chemistry is constant. Actually, laboratory titration experiments would also fit into this category because we know the initial solution chemistry from which the final solution evolves. Inverse mass balance modeling should find applications in these situations. 

Chloride is usually a major constituent in groundwater and is widely considered a conservative tracer. In the N aquifer, Cl- concentrations are considerably higher in Holocene water than in late Pleistocene water (Figure 9.3). The groundwater ages indicated on the horizontal axis are results from inverse mass balance modeling and age corrections (Zhu, 2000). These age data are also supported by the SD and S lsO data of the same samples. It is generally known that Pleistocene water has depleted H and O stable isotope values with respect to recent water because of a cooler and more humid climate in the late Pleistocene (Merlivat and Jouzel, 1979). 

As described in more detail below, the inverse approach generally leads to underdetermined mathematical systems that are much harder to solve than the systems encountered in forward models. Error and resolution analysis are two issues of particular importance when solving underdetermined inverse problems. First, the solved-for physical and biogeochemical parameters depend directly on the tracer data, and errors in the data propagate into errors in the solution. Second, owing to the incompleteness of information in underdetermined systems, the unknowns are usually not fully resolved. Instead, only specific linear combinations of unknowns may be well constrained by the data, while individual unknowns or other combinations of unknowns may remain poorly determined. Both, error and resolution analysis are essential for a quality assessment of the solution of underdetermined systems. 

The interchange coefficient, kc, has been estimated from the response to pulse inputs of tracer gas [8,13], Patience and Chaouki [8] used sand in a 0.083-m riser and reported values (their k) that ranged from 0.03 to 0.08 m/sec. The values increased with gas velocity, but with considerable scatter in the data. White and coworkers [13] used sand and FCC catalyst in a 0.09-m riser and found k (their k values of 0.05-0.02 m/sec that decreased with gas velocity. The interchange coefficient kc is based on a unit coreannulus area and can be converted to a volumetric coefficient K for comparison with other models. For kc = 0.03 m/sec, AT 1.2 sec which seems the right order of magnitude based on the values shown in Figure 9.12. Flowever, if kc is independent of diameter, D, the volumetric coefficient K will vary inversely with D if rJR is constant. 

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