Solutions for Forward, Inverse and Implicit Problems

Having specified the turbulence properties cr and and thence the dispersion matrix Di, the apparatus is now in place to use Ecp (17) to solve three generic kinds of problem the forward problem of determining the scalar concentration profile c(z) from a specified source density profile (z), the inverse problem of determining 4 z) from specified or measured information about c(z), and the implicit or coupled problem of determining both c z) and 4 z) together when cf is a given function of c. 

This is the primary means of obtaining information about the canopy source distribution of a scalar from atmospheric concentration measurements. A formal discrete solution is found by matrix inversion of Et]. (17), choosing the number of source layers (m) to be ecjual to the number of concentration measurements (n) so that D j is a scjuare matrix. However, this solution provides no redundancy in concentration information, and therefore no possibility for smoothing measurement errors in the concentration profile, which can cause large errors in the inferred source profile. A simple means of overcoming this problem is to include redundant concentration information, and then find the sources / , which produce the best fit to the measured concentrations c, by maximum-likelihood estimation. By minimizing the squared error between measured values and concentrations predicted by Eq. (17), 4 j is found (Raupach, 1989b) to be the solution of m linear ec[uations 

This solution uses a nonsquare dispersion matrix in which n m, to obtain the necessary redundancy in concentration information. The solution is valid whether or not 4 is dependent on c, because it determines the values of (f) consistent with the current c field. The result is the source density profile in a (small) number of canopy layers, with the lowest layer including the ground source (Fq). The total flux from the canopy is also obtained as the sum of /) Azj over all layers. 

These illustrative calculations have been carried out with a very simple inversion procedure based on Ecp (30). Rapid recent developments in the application inverse theory offer several possibilities Ibr improvement. First, the use of singular-value decomposition as a formal inversion framework (e. g.. Press et al, 1992) 

---