Two Interpolation Methods Using Multiply-Rotated Piecewise Cubic Hermite Interpolating Polynomials

Author:

Barker Paul M.1,McDougall Trevor J.1

Affiliation:

1. School of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales, Australia

Abstract

AbstractTwo interpolation methods are presented, both of which use multiple Piecewise Cubic Hermite Interpolating Polynomials (PCHIPs). The first method is based on performing 16 PCHIPs on 8 rotated versions of the plot of the data versus an independent variable (such as pressure or time). These 16 PCHIPs are then used to form 8 interpolations of the original data, and finally, these 8 are averaged. When the original data are unevenly spaced with respect to the independent variable, we show that it is best to perform the Multiply-Rotated PCHIP (MR-PCHIP) method using the “data index” as the independent variable, and then to subsequently perform one last PCHIP of the data index with respect to the original independent variable. This MR-PCHIP method avoids the flat spots that are a feature of the PCHIP method when the data have multiple values approximately equal to a local extreme value. The MR-PCHIP interpolated data have continuous first derivatives at the data points. This method also avoids the unrealistic overshoots that can occur when using the standard cubic spline interpolation procedure. The second interpolation method is designed specifically for hydrographic data with the aim of minimizing the formation of unrealistic water masses by the interpolation procedure. This is achieved by applying a Piecewise Cubic Hermite Interpolating Polynomial to each of 8 rotations of the salinity versus temperature plot (Multiply-Rotated Salinity–Temperature PCHIP, MRST-PCHIP) with bottle number (that is, data index) as the vertical interpolating coordinate, thereby making the MRST-PCHIP method independent of the heave of a water column. This method is equivalent to interpolating in the salinity–temperature diagram, and MRST-PCHIP proves very effective at avoiding the production of anomalous water masses that otherwise occur when interpolating temperature and salinity separately.

Funder

Australian Research Council

Publisher

American Meteorological Society

Subject

Atmospheric Science,Ocean Engineering

Reference9 articles.

1. Stabilizing hydrographic profiles with minimal change to the water masses;Barker;J. Atmos. Oceanic Technol.,2017

2. Monotone piecewise cubic interpolation;Fritsch;SIAM J. Numer. Anal.,1980

3. IOC, SCOR, and IAPSO, 2010: The International Thermodynamic Equation of Seawater—2010: Calculation and use of thermodynamic properties. Intergovernmental Oceanographic Commission Manuals and Guides 56, 220 pp., http://www.teos-10.org/pubs/TEOS-10_Manual.pdf.

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