Abstract
A detailed development of the principal component proxy method of dynamical tracer reconstruction is presented, including error analysis. The method works by correlating the largest principal components of a matrix representation of the transport dynamics with a set of sparse measurements. The Lyapunov spectrum was measured and used to quantify the lifetime of each principal component. The method was tested on the 500 K isentropic surface with stratospheric ozone concentration measurements from the Polar Aerosol and Ozone Measurement III satellite instrument during October and November 1998 and compared with the older proxy tracer method which works by correlating measurements with a single other tracer or proxy. Using a 60 day integration time and five (5) principal components, cross validation of globally reconstructed ozone and comparison with ozone sondes returned root-mean-square errors of 0.16 and 0.36 ppmv, respectively. This compares favourably with the classic proxy tracer method in which a passive tracer equivalent latitude field was used for the proxy and which returned RMS errors of 0.22 and 0.59 ppmv for cross-validation and sonde validation respectively. The method seems especially effective for shorter lived tracers and was far more accurate than the classic method at predicting ozone concentration in the Southern hemisphere over the same time period. It is also more effective when reconstruction is performed over the entire Earth rather than a single hemisphere allowing for seamless reconstruction of global fields.
Similar content being viewed by others
References
Allen DR, Nakamura N (2003) Tracer equivalent latitude: a diagnostic tool for isentropic transport studies. J Atmos Sci 60:287–303
Butchart N, Remsberg EE (1986) The area of the stratospheric polar vortex as a diagnostic for tracer transport on an isentropic surface. J Atmos Sci 43:1319–1339
Golub GH, van Loan CF (1996) Matrix computations. Johns Hopkins University Press, Baltimore
Hare EW, Carty EJ, Wardle DI (2000) Guide to the WMO/GAW world ozone data centre. Technical Report, Meteorological Service of Canada, Environment Canada
Hoskins BJ, McIntyre ME, Robertson AW (1985) On the use and significance of isentropic potential vorticity maps. Q J R Meteorol Soc 111:877–946
Kalnay E, Kanamitsu M, Kistler R, Collins W, Deaven D, Gandin L, Iredell M, Saha S, White G, Woollen J, Zhu Y, Chelliah M, Ebisuzaki W, Higgins W, Janowiak J, Mo K, Ropelewski C, Wang J, Leetmaa A, Reynolds R, Jenne R, Joseph D (1996) The NCEP/NCAR 40-year reanalysis project. Bull Am Meteorol Soc 77:437–470
Lehoucq RB, Scott JA (1996) An evaluation of software for computing eigenvalues of sparse nonsymmetric matrices. Technical Report MCS-P547-1195, Argonne National Laboratory
Lucke RL, Korwan DR, Bevilacqua RM, Hornstein JS, Shettle EP, Chen DT, Daehler M, Lumpe JD, Fromm MD, Debrestian D, Neff B, Squire M, Knig-Langlo GJ, Davies J (1999) The Polar Ozone and Aerosol Measurement (POAM) III instrument and early validation results. J Geophys Res 104(D15):18785–18799
Lumpe JD, Bevilacqua RM, Hoppel KW, Randall CE (2002) POAM III retrieval algorithm and error analysis. J Geophys Res 107(D21):ACH5.1–ACH5.32
Mills P (2004) Following the Vapour trail: a study of chaotic mixing of water vapour in the upper troposphere. Master’s thesis, University of Bremen
Mills P (2009) Isoline retrieval: an optimal method for validation of advected contours. Comput Geosci 35(20):2020–2031
Ott E (1993) Chaos in dynamical systems. Cambridge University Press, Cambridge
Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical recipes in C, 2nd edn. Cambridge University Press, Cambridge
Randall CE, Lumpe JD, Bevilacqua RM, Hoppel KW, Fromm MD, Salawitch RJ, Swartz WH, Lloyd SA, Kyro E, von der Gathen P, Claude H, Davies J, DeBacker H, Dier H, Molyneux MJ, Sanchoi J (2002) Reconstruction of three-dimensional ozone fields using POAM III during SOLVE. J Geophys Res 107(D20):8299–8312
Rodgers CD (2000) Inverse methods for atmospheric sounding: theory and practice. World Scientific, Singapore
Tang YR, Kleeman R, Miller S (2006) ENSO predictability of a fully coupled GCM model using singular vector analysis. J Clim 19(14):3361–3377
Acknowledgements
Thanks to the National Center for Environmental Prediction and the National Center for Atmospheric Research for the reanalysis data used in the simulations. Thanks also to World Ozone and Ultraviolet Data Center and Environment Canada for ozone sonde data. And thanks especially to my former colleagues at the Naval Research Laboratory for POAM III ozone data. Contour maps were created with Generic Mapping Tools (GMT) while scatter plots and historgrams were done in Open Office.
Author information
Authors and Affiliations
Corresponding author
Appendices
Appendix 1: Model properties
1.1 Appendix 1.1: Mass conservation
Suppose that:
This will be true for non-divergent flows on equal area grids. Then:
Therefore:
If (41) is true, then:
is also be true. Continuing:
which shows the second part of (26) and (27):
1.2 Appendix 1.2: Diffusion and the Lyapunov spectrum
A discrete tracer mapping will always require some amount of diffusion. This means that the tracer configuration will tend towards a uniform distribution over time, that is, it will “flatten out.” We can show that, given the constraint in (41), a tracer field with all the same values has the smallest magnitude. Suppose there are only two elements in the tracer vector, \(\mathbf {q}=\lbrace q,~q \rbrace\). The magnitude of the vector is:
Now we introduce a separation between the elements, \(2{\varDelta }q\), that nonetheless keeps the sum of the elements constant:
This will generalize to higher-dimensional vectors. In general, we can say that:
Implying that for the eigenvalue problem,
Therefore the Lyapunov exponents are all either zero or negative. Note however that this does not constitute a proof; the actual proof is more involved.
To prove (54) from (53), we first expand \(\mathbf {q}\) in terms of the right singular vectors, \(\lbrace \mathbf {v}_i \rbrace\):
where \(\lbrace c_i \rbrace\) are a set of coefficients. Substituting this into the left-hand-side of (53):
where \(\delta\) is the Kronecker delta. Similarly, we can show that:
If we assume that \(s_i \le 1\) for every i, then:
since each term on the left side is less-than-or-equal-to the corresponding term on the right side. Note that in order for the inequality in (61) to be broken, at least one singular value must be greater-than one. Therefore (53) is true for every \(\mathbf {q}\) if-and-only-if (54) is true for every s. In the language of set theory and first-order logic:
Appendix 2: Deviation from equal area
Here we calculate the ratio between the largest and smallest grid boxes in the azimuthal equidistant coordinate system. First we show that there is no distortion at the pole:
hence the ratio between projected and unprojected areas is 1. Grid areas become progressively smaller the further from the pole you get. Since the projection is hemi-spherical, r takes on a maximum value at the equator:
Hence the largest possible values for x and y are:
which represents a point on the equator along a diagonal from the origin in the projected coordinate system. The metric coefficients can be calculated:
Rights and permissions
About this article
Cite this article
Mills, P. PC proxy: a method for dynamical tracer reconstruction. Environ Fluid Mech 18, 1533–1558 (2018). https://doi.org/10.1007/s10652-018-9615-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10652-018-9615-7