Quantification of optimal choices of parameters in lognormal variational data assimilation and their chaotic behavior. (English) Zbl 1411.62336
Summary: An important property of variational-based data assimilation is the ability to define a functional formulation such that the minimum of that functional can be any state that is desired. Thus, it is possible to define cost functions such that the minimum of the background error component is the mean, median or the mode of a multivariate lognormal distribution, where, unlike the multivariate Gaussian distributions, these statistics are not equivalent. Therefore, for lognormal distributions it is shown here that there are regions where each one of these three statistics are optimal at minimizing the errors, given estimates of an a priori state. Also, as part of this work, a chaotic signal was detected with respect to the first guess to the Newton-Raphson solver that affect the accuracy of the solution to several decimal places.
MSC:
| 62P12 | Applications of statistics to environmental and related topics |
| 86A22 | Inverse problems in geophysics |
| 86A32 | Geostatistics |
Keywords:
lognormal variational data assimilation; mode; median; mean; Newton-Raphson; Newton fractalsReferences:
| [1] | Cohn SE (1997) An introduction to estimation error theory. J Meteorol Soc Jpn 75:257-288 · doi:10.2151/jmsj1965.75.1B_257 |
| [2] | Fletcher SJ (2010) Mixed lognormal-Gaussian four-dimensional data assimilation. Tellus 62:266-287 · doi:10.1111/j.1600-0870.2010.00439.x |
| [3] | Fletcher SJ (2017) Data assimilation for the geosciences: from theory to applications, 1st edn. Elsevier, Boston |
| [4] | Fletcher SJ, Jones AS (2014) Multiplicative and additive incremental variational data assimilation for mixed lognormal and Gaussian errors. Mon Weather Rev 142:2521-2544 · doi:10.1175/MWR-D-13-00136.1 |
| [5] | Fletcher SJ, Zupanski M (2006a) A data assimilation method for log-normally distributed observational errors. Q J Roy Meteorol Soc 132:2505-2519 · doi:10.1256/qj.05.222 |
| [6] | Fletcher SJ, Zupanski M (2006b) A hybrid normal and lognormal distribution for data assimilation. Atmos Sci Lett 7:43-46 · doi:10.1002/asl.128 |
| [7] | Fletcher SJ, Zupanski M (2007) Implications and impacts of transforming lognormal variables into normal variables in VAR. Meteorologische Zeitschrift 16:755-765 · doi:10.1127/0941-2948/2007/0243 |
| [8] | Gauthier P, Tanguay M, Laroche S, Pellering S, Morneau J (2007) Entension of a 3D-var to 4D-var: implementation of 4D-var at the meteorological service of Canada. Mon Weather Rev 135:2339-2354 · doi:10.1175/MWR3394.1 |
| [9] | Gupta B (2013) Newton Raphson fractals: a review. Int J Emerg Trends Technol Comput Sci IJETTCS 2:119-123 |
| [10] | Kleist DT, Parrish DF, Derber JC, Treadon R, Wu W-S, Lord S (2009) Introduction of the GSI into the NCEP global data assimilation system. Weather Forecast 24:1691-1705 · doi:10.1175/2009WAF2222201.1 |
| [11] | Kliewer AJ, Fletcher SJ, Forsythe JM, Jones AS (2016) Application of mixed lognormal-Gaussian data assimilation techniques to synethic data with the CIRA 1-dimensional optimal estimation retrieval system. Q J R Meteorol Soc 142:274-286 · doi:10.1002/qj.2651 |
| [12] | Lewis JM, Derber JC (1985) The use of adjoints equations to solve a variational adjustment problem with advective contstraints. Tellus 73A:309-322 · doi:10.3402/tellusa.v37i4.11675 |
| [13] | Lorenc AC (1986) Analysis methods for numerical weather prediction. Q J R Meteorol Soc 126:1177-1194 · doi:10.1002/qj.49711247414 |
| [14] | Lorenc AC, Ballard SP, Bell RS, Ingleby NB, Andrews PLF, Barker DM, Bray JR, Clayton AM, Dalby T, Li D, Payne TJ, Saunders FW (2000) The Met. Office global three dimensional variational data assimilation scheme. Q J R Meteorol Soc 126:2991-3012 · doi:10.1002/qj.49712657002 |
| [15] | Lorenz EN (1963) Deterministic nonperiodic flow. J Atmos Sci 20:130-141 · Zbl 1417.37129 · doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2 |
| [16] | Rabier F, Jarvinen H, Klinker E, Mahfouf J-F, Simmons A (2000) The ECMWF implementation of four dimensional variational assimilation. Part I: experimental results with simplified physics. Q J R Meteorol Soc 126A:1143-1170 · doi:10.1002/qj.49712656415 |
| [17] | Rawlins F, Ballard SP, Bovis KJ, Clayton AM, Li D, Inverarity GW, Lorenc AC, Payne TJ (2007) The Met Office global four-dimensional variational data assimilation scheme. Q J Roy Meteorol Soc 133:347-362 · doi:10.1002/qj.32 |
| [18] | Rosmond T, Xu L (2006) Development of NAVDAS-AR: non-linear formulation and outer loop test. Tellus 58A:45-58 · doi:10.1111/j.1600-0870.2006.00148.x |
| [19] | Song H, Edwards CA, Moore AM, Fiechter J (2012) Incremental four-dimensional variational data assimilation of positive-definite oceanic variables using a logarithm transformation. Ocean Modell 54:1-17 · doi:10.1016/j.ocemod.2012.06.001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
