Editorial Type:
Article
Article Type:
Research Article

Calculating State-Dependent Noise in a Linear Inverse Model Framework

Cristian Martinez-Villalobos Department of Atmospheric and Oceanic Sciences, University of Wisconsin–Madison, Madison, Wisconsin, and Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles, Los Angeles, California

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Daniel J. Vimont Department of Atmospheric and Oceanic Sciences, University of Wisconsin–Madison, Madison, Wisconsin

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Cécile Penland Physical Sciences Division, NOAA/Earth System Research Laboratory, Boulder, Colorado

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Matthew Newman Cooperative Institute for Research in Environmental Sciences, University of Colorado Boulder, and NOAA/Earth System Research Laboratory, Boulder, Colorado

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J. David Neelin Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles, Los Angeles, California

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Print Publication:
01 Feb 2018
DOI:
https://doi.org/10.1175/JAS-D-17-0235.1
Page(s):
479–496
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Abstract

The most commonly used version of a linear inverse model (LIM) is forced by state-independent noise. Although having several desirable qualities, this formulation can only generate long-term Gaussian statistics. LIM-like systems forced by correlated additive–multiplicative (CAM) noise have been shown to generate deviations from Gaussianity, but parameter estimation methods are only known in the univariate case, limiting their use for the study of coupled variability. This paper presents a methodology to calculate the parameters of the simplest multivariate LIM extension that can generate long-term deviations from Gaussianity. This model (CAM-LIM) consists of a linear deterministic part forced by a diagonal CAM noise formulation, plus an independent additive noise term. This allows for the possibility of representing asymmetric distributions with heavier- or lighter-than-Gaussian tails. The usefulness of this methodology is illustrated in a locally coupled two-variable ocean–atmosphere model of midlatitude variability. Here, a CAM-LIM is calculated from ocean weather station data. Although the time-resolved dynamics is very close to linear at a time scale of a couple of days, significant deviations from Gaussianity are found. In particular, individual probability density functions are skewed with both heavy and light tails. It is shown that these deviations from Gaussianity are well accounted for by the CAM-LIM formulation, without invoking nonlinearity in the time-resolved operator. Estimation methods using knowledge of the CAM-LIM statistical constraints provide robust estimation of the parameters with data lengths typical of geophysical time series, for example, 31 winters for the ocean weather station here.

Supplemental information related to this paper is available at the Journals Online website: https://doi.org/10.1175/JAS-D-17-0235.s1.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Publisher’s Note: This article was revised on 29 January 2018 to include a funding source in the Acknowledgments that was missing when originally published.

Corresponding author: Cristian Martinez-Villalobos, cmartinezvil@atmos.ucla.edu

Abstract

The most commonly used version of a linear inverse model (LIM) is forced by state-independent noise. Although having several desirable qualities, this formulation can only generate long-term Gaussian statistics. LIM-like systems forced by correlated additive–multiplicative (CAM) noise have been shown to generate deviations from Gaussianity, but parameter estimation methods are only known in the univariate case, limiting their use for the study of coupled variability. This paper presents a methodology to calculate the parameters of the simplest multivariate LIM extension that can generate long-term deviations from Gaussianity. This model (CAM-LIM) consists of a linear deterministic part forced by a diagonal CAM noise formulation, plus an independent additive noise term. This allows for the possibility of representing asymmetric distributions with heavier- or lighter-than-Gaussian tails. The usefulness of this methodology is illustrated in a locally coupled two-variable ocean–atmosphere model of midlatitude variability. Here, a CAM-LIM is calculated from ocean weather station data. Although the time-resolved dynamics is very close to linear at a time scale of a couple of days, significant deviations from Gaussianity are found. In particular, individual probability density functions are skewed with both heavy and light tails. It is shown that these deviations from Gaussianity are well accounted for by the CAM-LIM formulation, without invoking nonlinearity in the time-resolved operator. Estimation methods using knowledge of the CAM-LIM statistical constraints provide robust estimation of the parameters with data lengths typical of geophysical time series, for example, 31 winters for the ocean weather station here.

Supplemental information related to this paper is available at the Journals Online website: https://doi.org/10.1175/JAS-D-17-0235.s1.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Publisher’s Note: This article was revised on 29 January 2018 to include a funding source in the Acknowledgments that was missing when originally published.

Corresponding author: Cristian Martinez-Villalobos, cmartinezvil@atmos.ucla.edu

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