Abstract
This article presents a current research application of robust parameter estimation for inverse problems in air–sea gas exchange. The first part illustrates the interaction of measurements, analysis, and parameter estimation in the concept of the modeling cycle: Typical measurement techniques are classified and discussed with respect to application areas and reproducibility. Statistical analysis of the assumed error distributions is presented as the link to parameter estimation via choosing a suitable parameter estimator. In the case that the error distribution cannot be determined definitely and appears to be highly non-Gaussian the use of robust estimators is advised instead of the standard least-squares approach. Huber’s M-estimator is used here as robust method. Reliable parameter reconstruction by the chosen estimator closes the modeling cycle by validating model against experiment. In the second part all previously presented theoretical aspects are applied to the spectral reconstruction approach (SPERA), a special measurement technique for determining water-sided gas concentration fields via Laser-induced fluorescence. In this application the use of a robust estimator allows the reconstruction of local gas concentration as the model parameters of an inverse problem, for which least-squares methods failed. The article finishes with a brief outlook of applying optimal experimental design techniques to measurements in environmental physics.
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W. Aeschbach-Hertig, F. Peeters, U. Beyerle, and R. Kipfer. Interpretation of dissolved atmospheric noble gases in natural waters. Water Res. Research, 35, 1999.
R. Ahmadov, C. Gerbig, R. Kretschmer, S. Körner, C. Rödenbeck, P. Bousquet, and M. Ramonet. Comparing high resolution WRF-VPRM simulations and two global CO2 transport models with coastal tower measurements of CO2. Biogeosciences, 6:807–817, May 2009.
J. Aldrich. R. A. Fisher and the making of maximum likelihood 1912-1922. Stat. Sci., 12:162–176, 1997.
J. D. Allan, D. O. Topping, N. Good, M. Irwin, M. Flynn, P. I. Williams, H. Coe, A. R. Baker, M. Martino, N. Niedermeier, A. Wiedensohler, S. Lehmann, K. Mller, H. Herrmann,, and G. McFiggans. Composition and properties of atmospheric particles in the eastern atlantic and impacts on gas phase uptake rates. Atmos. Chem. Phys., 9:9299–9314., 2009.
A. E. Altmann-Dieses, J. P. Schlöder, H. G. Bock, and O. Richter. Optimal experimental design for parameter estimation in column outflow experiments. Water Resour. Res., 38:1186–1197, 2002.
R. Andersen. Modern Methods for Robust Regression. Sage Publications, 2008.
K. Anderson and H. Croft. Remote sensing of soil surface properties. Prog. Phys. Geog., 33(4):457 – 473, 2009.
H. Arellano-Garcia, J. Schöneberger, and S. Körkel. Optimale Versuchsplanung in der chemischen Verfahrenstechnik. Chem. Ing. Tech., 79:1625–1638, 2007.
W. E. Asher and J. F. Pankow. Direct observation of concentration fluctuations close to a gas-liquid interface. Chem. Eng. Sci., 44:1451–1455, 1989.
A. Berthe, D. Kondermann, C. Christensen, l. Goubergrits, C. S. Garbe, K. Affeld, and U. Kertzscher. Three-dimensional, three-component wall-PIV. Exp. Fluids, 48:983–997, 2010.
T. Binder and E. Kostina. Gauss-Newton methods for robust parameter estimation. In H. G. Bock et al., editor, Model Based Parameter Estimation, Contributions in Mathematical and Computational Sciences 4, pages 53–85. Springer, 2012.
C. M. Bishop. Pattern Recognition and Machine Learning, volume 1. Springer, 2007.
N. Bobrowski, G. Hünninger, F. Lohberger, and U. Platt. Idoas: A new monitoring technique to study the 2D distribution of volcanic gas emissions. J. Volcanol. Geotherm. Res., 4:329–338, 2006.
G. E. P. Box, G. M. Jenkins, and G. C. Reinsel. Time Series Analysis: Forecasting and Control. Wiley, 4 edition, 2008.
D. B. Chelton, M. G. Schlax, M. H. Freilich, and R. F. Milliff. Satellite measurements reveal persistent small-scale features in ocean winds. Science, 303:978–983, 2004.
E. R. Davies. Maschine Vision. Theory, Algorithms, Practicalities, volume 3. Academic Press, 2005.
P. L. Davies and U. Gather. Unmasking multivariate outliers and leverage points. Stat. Journ., 5(1):1–17, 2007.
N. Dimarcq, V. Giordano, and P. Cerez. Statistical properties of laser-induced fluorescence signals. Appl. Phys. B Lasers O., 59:135–145, 1994.
M. A. Donelan and R. Wanninkhof. Gas transfer at water surfaces - conepts and issues. In M. A. Donelan, W. M. Drennan, E. S. Saltzman, and R. Wanninkhof, editors, Gas Transfer at Water Surfaces. American Geophysical Union, 2002.
R. O. Duda, P. E. Hart, and D. G. Stork. Pattern Classification, volume 2. John Wiley and Sons, 2000.
L. L. Eberhardt and J. M. Thomas. Designing environmental field studies. Ecol. Mono., 61:53–73, 1991.
B. Eckhardt, A. Schmiegel, H. Faisst, and T. Schneider. Dynamical systems and the transition to turbulence in shear flows. Phil. Trans. R. Soc. A, 366(1868):1297–1315, 2008.
R. A. Fisher. An absolute criterion for fitting frequency curves. Mess. Math., 41:155–160, 1912.
C. Geels, M. Gloor, P. Ciais, P. Bousquet, P. Peylin, A. T. Vermeulen, R. Dargaville, T. Aalto, J. Brandt, J. H. Christensen, L. M. Frohn, L. Haszpra, U. Karstens, C. Rödenbeck, M. Ramonet, G. Carboni, and R. Santaguida. Comparing atmospheric transport models for future regional inversions over Europe. Part 1: Mapping the CO2 atmospheric signals. Atmos. Chem. Phys. Discuss., 6:3709–3756, May 2006.
P. E. Gill, W. Murray, and M.H. Wright. Practical Optimization. Academic Press, INC, 1981.
J. Hadamard. Sur les problèmes aux dérivées partielles et leur signification physique. Princeton University Bulletin, 13:49–52, 1902.
T. Hara, E. VanInwegen, J. Wendelbo, C. S. Garbe, U. Schimpf, B. Jähne, and N. Frew. Estimation of air-sea gas and heat fluxes from infrared imagery based on near surface turbulence models. In C. S. Garbe, R. A. Handler, and B. Jähne, editors, Transport at the Air Sea Interface - Measurements, Models and Parametrizations. Springer-Verlag, 2007.
H. Haußecker, U. Schimpf, C. S. Garbe, and B. Jähne. Physics from IR image sequences: Quantitative analysis of transport models and parameters of air-sea gas transfer. In E. Saltzman, M. Donelan, W. Drennan, and R. Wanninkhof, editors, Gas Transfer at Water Surfaces, volume 127 of Geophysical Monograph. American Geophysical Union, 2002.
I. Herlina and G. H. Jirka. Experiments on gas transfer at the air–water interface induced by oscillating grid turbulence. J. Fluid. Mech., 594:183–208, 2008.
A. G. Herzog, T. Binder, B. Jähne, and E. A. Kostina. Estimating water-sided vertical gas concentration profiles by inverse modeling. In 2. International Conference on Engineering Optimization, Lisbon, September 2010.
P. J. Huber. Robust statistics: A review. Annals of Mathematical Statistics, 43:1041–1067, 1972.
IPCC. Climate change 2007: The physical science basis. Contribution of working group I to the fourth assessment report of the intergovernmental panel on climate change, 2007.
B. Jähne. Digitale Bildverarbeitung, 6. Auflage. Springer, 2005.
B. Jähne, H. Haußecker, and P. Geißler, editors. Handbook of Computer Vision and Applications. Academic Press, 1999.
M. Johansson, B. Galle, C. Rivera, and Y. Zhang. Tomographic reconstruction of gas plumes using scanning DOAS. B. Volcanol., 71:1169–1178, 2009.
J. Kaipio and E. Somersalo. Statistical and Computational Inverse Problems. Applied Mathematical Sciences. Springer, 1 edition, 2004.
T. Kluge, D. F. C. Riechelmann, M. Wieser, C. Spötl, J. Sültenfuß, A. Schröder-Ritzrau, S. Niggemann, and W. Aeschbach-Hertig. Dating cave drip water by tritium. J. Hydrol., 394:396–406, 2010.
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen. PhD thesis, Univ. Heidelberg, 2002.
J. Lippold, J. Grützner, D. Winter, Y. Lahaye, A. Mangini, and M. Christl. Does sedimentary 231Pa ∕ 230Th from the bermuda rise monitor past atlantic meridional overturning circulation? Geophys. Res. Lett., 36:L12601, 2009.
T. Marbach, S. Beirle, U. Platt, P. Hoorand F. Wittrock, A. Richter, M. Vrekoussis, M. Grzegorski, J. P. Burrowsand, and T. Wagner. Satellite measurements of formaldehyde from shipping emissions. Atmos. Chem. Phys., 9, 2009.
D.C. Montgomery, E. A. Peck, and G. G. Vinning. Introduction to Linear Regression Analysis. Wiley InterScience, 4 edition, 2006.
R. R. Nemani, C. D. Keeling, H. Hashimoto, W. M. Jolly, S. C. Piper, C. J. Tucker, R. B. Myneni, and S. W. Running. Climate-driven increases in global terrestrial net primary production from 1982 to 1999. Science, 300:1560–1563, June 2003.
W. L. Oberkampf, T. G. Trucano, and C. Hirsch. Verification, validation, and predictive capability in computational engineering and physics. Appl. Mech. Rev., 57(5):345–385, 2004.
U. Platt and J. Stutz. Differential Optical Absorption Spectroscopy, Principles and Applications. Physics of Earth and Space Environments. Springer-Verlag, Berlin, Heidelberg, New York, 2000.
F. Rezanezhad, H. J.Vogel, and K.Roth. Experimental study of fingered flow through initially dry sand. Hydr. Earth Syst. Sci. Disc., 3:2595– 2620, 2006.
P. J. Rousseeuw and A. M. Leroy. Robust Regression and Outlier Detection. John Wiley & Sons, 1987.
U. Ruth, D. Wagenbach, R. Mulvaney, H. Oerter, W. Graf, H. Pulz, and G. Littot. Comprehensive 1000 year climatic history from an intermediate depth ice core from the south dome berkner island, antarctica: methodics, dating, and first results. Ann. Glaciol., 39:146–154, 2004.
K. Schneider-Zapp, O. Ippisch, and K. Roth. Numerical study of the evaporation process and parameter estimation analysis of an evaporation experiment. Hydrol. Earth Syst. Sci., 14:765–781, 2010.
Schöneberger, J. C., H. Arellano-Garcia, G. Wozny, S. Körkel, and H.Thielert. Model-based experimental analysis of a fixed-bed reactor for catalytic SO2 oxidation. Ind. & Eng. Chem. Res., 48(11):5165–5176, 2009.
R. A. Schowengerdt. Remote Sensing: Models and Methods for Image Processing, volume 3. Academic Press, 2006.
T. Stocker, Q. Dahe, G.-K. Plattner, M. Tignor, and P. Midgley. IPCC expert meeting on assessing and combining multi model climate projections. Boulder, Colorado, USA, 25-27 January 2010.
T. Takahashi, S. C. Sutherland, C. Sweeney, A. Poisson, N. Metzl, B. Tilbrook, N. Bates, R. Wanninkhof, R. A. Feely, C. Sabine, J. Olafsson, and Y. Nojiri. Global sea-air CO2 flux based on climatological surface ocean pCO2, and seasonal biological and temperature effects. Deep Sea Res. Pt II, 49(9-10):1601 – 1622, 2002.
A. Tarantola. Inverse Problem Theory and Methods for Model Parameter Estimation. SIAM, 2005.
W. Tsai and L. P. Hung. Three-dimensional modeling of small-scale processes in the upper boundary layer bounded by a dynamic ocean surface. J. Geophys. Res., 112:C02019, 2007.
K. J. Ueltzhöffer, V. Bendel, J. Freitag, S. Kipfstuhl, D. Wagenbach, S. H. Faria, and C. S. Garbe. Distribution of air bubbles in the EDML and EDC (antarctica) ice cores, using a new method of automatic image analysis. J. Glaciol., 56(196):339–348, 2010.
J. Šimůnek and S.A. Bradford. Vadose zone modeling: Introduction and importance. Vadose Zone J., 7:581–586, 2008.
A. Wackerbarth, D. Scholz, J. Fohlmeister, and A. Mangini. Modelling the δ18 o value of cave drip water and speleothem calcite. Earth Planet. Sci. Lett., 299:387–397, 2010.
E. J. Walsh, C. W. Wright, M. L. Banner, D. C. Vandemark, B. Chapron, J. Jensen, and S. Lee. The southern ocean waves experiment.Part III: Sea surface slope statistics and near-nadir remote sensing. J. Phys. Oceanogr., 38:670–685, 2008.
R. Wanninkhof, W. E. Asher, D. T. Ho, C. Sweeny, and W. R. McGillis. Advances in Quantifying Air-Sea Gas Exchange and Environmental Forcing. Ann. Rev. Mar. Sci., 1:213–244, 2009.
U. Wollschläger, T. Pfaff, and K. Roth. Field-scale apparent hydraulic parameterisation obtained from tdr time series and inverse modeling. Hydrol. Earth Syst. Sci., 13:1953–1966, 2009.
Acknowledgements
We gratefully acknowledge financial support by the DFG Research Training Group 1114 ’Optical Techniques for Measurement of Interfacial Transport Phenomena’ and the Heidelberg Graduate School of Mathematical and Computational Methods for the Sciences, as well as Synmikro, Marburg. Moreover, we want to thank our colleagues for their support, especially Tanja Binder (Marburg), Pascal Bohleber, Evelyn Böhm, Samuel Hammer, Barbara May, Gabriele Schenk, and Martin Wieser (all Institute of Environmental Physics, Heidelberg). Furthermore, special thanks go to Felix Friedl for performing the measurements shown in Fig. 3 and the group of PD D.P. Herten, BioQuant, Heidelberg, for giving us the opportunity to use their fluorescence spectrometer.
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Herzog, A.G., Vogel, F.R. (2013). An Application of Robust Parameter Estimation in Environmental Physics. In: Bock, H., Carraro, T., Jäger, W., Körkel, S., Rannacher, R., Schlöder, J. (eds) Model Based Parameter Estimation. Contributions in Mathematical and Computational Sciences, vol 4. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30367-8_14
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