Studies of trace gas fluxes have advanced the understanding of bulk interactions between the atmosphere and ecosystems. Micrometeorological instrumentation is currently unable to resolve vertical scalar sources and sinks within plant canopies. Inverted analytical Lagrangian equations provide a non-intrusive method to calculate source distributions. These equations are based on Taylor's (1921) description of scalar dispersion, which requires a measure of the degree of correlation between turbulent motions, defined by the Lagrangian length scale (L). Inverse Lagrangian (IL) analyses can be unstable, and the uncertainty in L leads to uncertainty in source predictions. A review of the literature on studies using IL analysis with various scalars in a multitude of canopy types found that parameterizations where L reduces to zero at the ground produce better results in the IL analysis than those that increase closer to the ground, but no individual L parameterization gives better results than any other does. The review also found that the relationship between L and the measurable Eulerian length scale (Le) may be more complex in plant canopies than the linear scaling investigated in boundary layer flows. The magnitude and profile shape of L was investigated within a corn and a forest canopy using field measurements to constrain an analytical Lagrangian equation. Measurements of net CO2 flux, soil-to-atmosphere CO2 flux, and in-canopy profiles of CO2 concentrations provided the information required to solve for L in a global optimization algorithm for half hour intervals. For dates when the corn was a strong CO2 sink, and for the majority of dates for the forest, the optimization frequently located L profiles that follow a convex shape. A constrained optimization then smoothed the profile shape to a sigmoidal equation. Inputting the optimized L profiles in the forward and inverse Lagrangian equations leads to strong correlations between measured and calculated concentrations (corn canopy: C_{calc} = 1.00C_{meas} +52.41 mumol m^{-3}, r^2 = 0.996; forest canopy: C_{calc} = 0.98C_{meas} +276.5 mumol m^{-3}, r^2 = 0.99) and fluxes (corn canopy: F_{soil} = 0.67F_{calc} - 0.12 mumol m^{-2}s^{-1}, r^2 = 0.71, F_{net} = 1.17F_{calc} + 1.97mumol m^{-2}s^{-1}, r^2 = 0.85; forest canopy: F_{soil} = 0.72F_{calc} - 1.92 mumol m^{-2}s^{-1}, r^2 = 0.18, F_{net} = 1.24F_{calc} + 0.65 mumol m^{-2}s^{-1}, r^2 = 0.88). In the corn canopy, coefficients of the sigmoidal equation were specific to each half hour and did not scale with any measured variable. Coefficients of the optimized L equation in the forest canopy scaled weakly with variables related to the stability above the canopy. Plausible L profiles for both canopies were associated with negative bulk Richardson number values. / Funding from NSERC.

Trial and error and the use of highly time-consuming methods are often necessary for modeling, simulating and characterizing nonlinear dynamical systems. However, for the rather common special case when a nonlinear system has linear relations between many of its degrees of freedom there are particularly interesting opportunities for more efficient approaches. The aim of this thesis is to develop and validate new efficient methods for the theoretical and experimental study of mechanical systems that include significant zero-memory or hysteretic nonlinearities related to only small parts of the whole system. The basic idea is to take advantage of the fact that most of the system is linear and to use much of the linear theories behind forced response simulations. This is made possible by modeling the nonlinearities as external forces acting on the underlying linear system. The result is very fast simulation routines where the model is based on the residues and poles of the underlying linear system. These residues and poles can be obtained analytically, from finite element models or from experimental measurements, making these forced response routines very versatile. Using this approach, a complete nonlinear model contains both linear and nonlinear parts. Thus, it is also important to have robust and accurate methods for estimating both the linear and nonlinear system parameters from experimental data. The results of this work include robust and user-friendly routines based on sinusoidal and random noise excitation signals for characterization and description of nonlinearities from experimental measurements. These routines are used to create models of the studied systems. When combined with efficient simulation routines, complete tools are created which are both versatile and computationally inexpensive. The developed methods have been tested both by simulations and with experimental test rigs with promising results. This indicates that they are useful in practice and can provide a basis for future research and development of methods capable of handling more complex nonlinear systems.

Diese Arbeit stellt die nichtlineare Parameterschätzung als alternative Auswertemethode von Kuchenfiltrationsexperimenten vor. Anhand eines größeren Datensatzes werden die Vorteile dieser Methode gegenüber der verbreiteten Auswertung mittels einer linearisierten Form der Kuchenfiltrationsgleichung für den Fall konstanten Drucks gezeigt. Zur Bewertung der Anpassungsgüte werden Residuenplots erläutert und verwendet. Die Unterschiede der Ergebnisse bewegen sich im Bereich von 5 bis 15% bei der Bestimmung des spezifischen Kuchenfiltrationswiderstands, welcher der wichtigste Parameter bei der Auslegung von Filtrationsapparaten ist. Weitere Möglichkeiten der Auswertung werden aufgezeigt, die durch die nichtlineare Parameterschätzung möglich werden, darunter die Auswertung von Experimenten bei variablem Druck, die Bestimmung des Kuchenwiderstands kompressibler Feststoffsysteme sowie eine Bewertung der anfänglichen Verblockungsvorgänge am Filtermedium.:1 Introduction 2 Cake Filtration Theory 2.1 Historical Development 2.2 Derivation of the Cake Filtration Equation 2.3 Fit Procedures for Cake Filtration Data 2.4 Additional Methods for Finding the Time Offset 3 Materials and Methods 3.1 Materials 3.2 Filter Medium 3.3 Laboratory Pressure Filters 3.4 Example Dataset 3.5 Preparation of Example Dataset 3.6 Residual Plots and Chi-Squares 3.7 Bootstrapped Statistics 4 Proposed Fit Procedure 4.1 Nonlinear Regression 4.2 Region of Best Fit 5 Results and Discussion 5.1 Constant-Pressure Filtration 5.2 Hermans & Bredée Models 5.3 Residual Plots of Fit Results 5.4 Nonconstant Filtration 5.5 Compressibility Effects 5.6 Optimal Parameter Definition 5.7 The Role of the t/V-V-Diagram 6 Conclusions 7 Outlook 7.1 Constant-Flux Filtration 7.2 Inline Resistance Measurements 7.3 Parameter Estimation in Chemical Engineering A Appendix A.1 The Concentration Parameter A.2 Obsolete Fit Methods A.3 Residual Statistics A.4 Bootstrapped Statistics Data A.5 Fit Example in Microsoft Excel A.6 Experimental Data and Metadata B References / This thesis presents nonlinear parameter estimation as an alternative method for the evaluation of cake filtration experiments. A dataset of 225 constant-pressure filtration experiments is used to highlight the advantages of this method compared to the widely used evaluation method which uses a linear transformation of the cake filtration equation. The goodness-of-fit is tested through the means of residual plots, which are introduced and discussed. The difference in results for the two methods for the specific cake resistance parameter, which is the most important parameter in the dimensioning of filtration apparatused, lies between 5 and 15%. Further possibilities of evaluation are presented, which become possible through the use of nonlinear parameter estimation, such as: evaluation of filtration experiments with nonconstant pressure, the determination of cake resistances for compressible systems, and the investigation of the processes present in the beginning stages of cake filtration.:1 Introduction 2 Cake Filtration Theory 2.1 Historical Development 2.2 Derivation of the Cake Filtration Equation 2.3 Fit Procedures for Cake Filtration Data 2.4 Additional Methods for Finding the Time Offset 3 Materials and Methods 3.1 Materials 3.2 Filter Medium 3.3 Laboratory Pressure Filters 3.4 Example Dataset 3.5 Preparation of Example Dataset 3.6 Residual Plots and Chi-Squares 3.7 Bootstrapped Statistics 4 Proposed Fit Procedure 4.1 Nonlinear Regression 4.2 Region of Best Fit 5 Results and Discussion 5.1 Constant-Pressure Filtration 5.2 Hermans & Bredée Models 5.3 Residual Plots of Fit Results 5.4 Nonconstant Filtration 5.5 Compressibility Effects 5.6 Optimal Parameter Definition 5.7 The Role of the t/V-V-Diagram 6 Conclusions 7 Outlook 7.1 Constant-Flux Filtration 7.2 Inline Resistance Measurements 7.3 Parameter Estimation in Chemical Engineering A Appendix A.1 The Concentration Parameter A.2 Obsolete Fit Methods A.3 Residual Statistics A.4 Bootstrapped Statistics Data A.5 Fit Example in Microsoft Excel A.6 Experimental Data and Metadata B References