Piecewise constant aquifer parameter identification recovery

For many inverse problems which arise in contributing to real-world decision-making, such as formulating policy objectives for freshwater fish health, only an indicative understanding of the global structure of the solution is all that is required for the associated decision-support. Examples include: (i) Situations where only some linear functional of the solution is required. As noted in various publications, this occurs in quite independent situations. They include: the recovery of moments of the spheres distribution from observations of the corresponding circles distribution on random plane sections through the spheres (Jakeman and Anderssen (1975)); the evaluation of local solutions in geodesy (Bauer et al. (2007)); a strategy for sparse recovery (Lu and Pereverzev (2009)), where the required information corresponds to some substructure in a complex solution such as the wavelength vibration of a target protein molecule in an NIR spectrum (Anderssen et al. (2003)); the evaluation of linear functionals of the molecular weight distributions of polymers (Anderssen et al. (1997); Anderssen (1999)). (ii) Where it is only necessary to recover some feature of the solutions such as whether the solution, as a function of the independent variable, is increasing, decreasing, convex or concave. Such situations arise in: the foliage angle distribution problem (Anderssen et al. (1985, 1984)); the location of some peak in the data as arises in resolution enhancement and derivative spectroscopy (Hegland and Anderssen (2005); Anderssen and Hegland (2010)); some of the situations in (i) where the required feature can be defined as a linear functional of the solution. In aquifer parameter identification, a similar situation arises. It is often only the regional structure of and connectivity within the aquifer that is required (Blakers et al. (2011)). For such situations, a piecewiseconstant approximation of the solution is all that is required to highlight the global features of the solution. Here, we examine the utilization of such approximations for the recovery of information about the regionalized structure of an aquifer. Here, using simulations, we examine the numerical performance of an anzatz proposed by Chow and Anderssen (1991) for the recovery of acquifer transmissivity from observational data. The clear advantage of their localization approach is that it removes the need to know the precise extent of the aquifer and the corresponding boundary conditions, and it allows the structure of the zonation within an aquifer to be explored in an iterative manner. Simulations, with synthetic data, confirm the utility of the proposed method to determine the zonation structure within an aquifer.