A method to characterize macroscopically homogeneous rigid frame porous media from impedance tube measurements by deterministic and statistical inversion is presented. Equivalent density and bulk modulus of the samples are reconstructed with the scattering matrix formalism, and are then linked to its physical parameters via the Johnson-Champoux-Allard-Lafarge model. The model includes six parameters, namely the porosity, tortuosity, viscous and characteristic lengths, and static flow and thermal permeabilities. The parameters are estimated from the measurements in two ways. The first one is a deterministic procedure that finds the model parameters by minimizing a cost function in the least squares sense. The second approach is based on statistical inversion. It can be used to assess the validity of the least squares estimate, but also presents several advantages since it provides valuable information on the uncertainty and correlation between the parameters. Five porous samples with a range of pore properties are tested, and the pore parameter estimates given by the proposed inversion processes are compared to those given by other characterization methods. Joint parameter distributions are shown to demonstrate the correlations. Results show that the proposed methods find reliable parameter and uncertainty estimates to the six pore parameters quickly with minimal user input.
Individual tree detection methods leave smaller trees hiding below larger ones undetected. This is a problem for remote sensing forest inventories, leading e.g. to severe underestimation of stand density. We develop new methods of formulating the probability of detecting individual trees-the detectability-based on stochastic geometry, and use them to derive estimators of stand density. We assume that a tree remains undetected if the center point of the crown falls within an erosion set based on the larger tree crowns. These estimators allow the tree to be undetected even if a portion of its crown would be visible, taking into account possible differences in accuracy of remote sensing data and detection algorithms. The behaviour of these estimators is quantified using 36 field plots, and compared to a previously proposed estimator. The best estimator according to the data used, allowing trees to be undetected when 40 percent or more of crown radius is hidden, performs well compared to the estimator formed directly from the number of algorithmically detected trees. It produces a 54 percent reduction in RMSE and shifts the mean of errors notably towards zero in the modelling data. Small variations in allowed visible crown radius do not seem to impact the accuracy of the estimates. Generalization of the results remains as a topic of future research.
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