Bailey–Borwein–Plouffe formula: Difference between revisions

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'''Barnes interpolation''', named after Stanley L. Barnes, is the [[interpolation]] of unstructured data points from a set of measurements of an unknown function in two dimensions into an [[analytic function]] of two variables. An example of a situation where the Barnes scheme is important is in [[weather forecasting]]<ref>{{cite web|url=http://www.bom.gov.au/bmrc/mdev/expt/rainanal/rainanal.shtml|title= Objective Rainfall Analysis System |accessdate=6 May 2009}}</ref><ref>{{cite journal | title=Thunderstorm distribution and frequency in Australia|series=Australian Meteorological Magazine|year=2002|pages=145–154|author=Y.Kuleshov, G. de Hoedt, W.Wright and A.Brewster}}</ref> where measurements are made wherever monitoring stations may be located, the positions of which are constrained by [[topography]]. Such interpolation is essential in data visualisation, e.g. in the construction of [[contour plot]]s or other representations of analytic surfaces.
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== Introduction ==
Barnes proposed an objective scheme for the interpolation of two dimensional data using a multi-pass scheme.<ref>{{cite journal|first=S. L | last =Barnes |title=A technique for maximizing details in numerical weather-map analysis|journal=Journal of Applied Meteorology|volume=3|issue=4|pages=396&ndash;409|year=1964|url=http://ams.allenpress.com/perlserv/?request=get-abstract&doi=10.1175%2F1520-0450(1964)003%3C0396:ATFMDI%3E2.0.CO%3B2|doi=10.1175/1520-0450(1964)003<0396:ATFMDI>2.0.CO;2|bibcode = 1964JApMe...3..396B }}</ref><ref>{{cite journal | last=Barnes | title=Mesoscale objective analysis using weighted time-series observations|series=NOAA Technical Memorandum|publisher=National Severe Storms laboratory|year=1964|first=S.L}}</ref>  This provided a method to interpolating sea-level pressures across the entire United States of America, producing a [[synoptic chart]] across the country using dispersed monitoring stations. Researchers have subsequently improved the Barnes method to reduce the number of parameters required for calculation of the interpolated result, increasing the objectivity of the method.<ref name="Koch_et_al">{{citation |author=Koch, S. E.,DesJardins, M and Kocin, P|title=An interactive Barnes Objective Map Analysis Scheme for Use with Satelite and Conventional Data|year=1983|journal=Journal of Climate and Applied Meteorology}}</ref>
 
The method constructs a grid of size determined by the distribution of the two dimensional data points. Using this grid, the function values are calculated at each grid point. To do this the method utilises a series of [[Gaussian function]]s, given a [[distance weighting]] in order to determine the relative importance of any given measurement on the determination of the function values. Correction passes are then made to optimise the function values, by accounting for the [[spectral response]] of the interpolated points.
 
== Method ==
Here we describe the method of interpolation used in a multi-pass Barnes interpolation.
 
=== First pass ===
For a given grid point ''i'',&nbsp;''j'' the interpolated function ''g''(''x''<sub>''i''</sub>,&nbsp;''y''<sub>''i''</sub>) is first approximated by the inverse weighting of the data points. To do this as weighting values is assigned to each Gaussian for each grid point, such that
 
: <math>w_{ij} = \exp\left(-\frac{r_m^2}{\kappa}\right), \, </math>
 
where <math>\kappa</math> is a falloff parameter that controls the width of the Gaussian function. This parameter is controlled by the characteristic data spacing, for a fixed Gaussian cutoff radius ''w''<sub>''ij''</sub>&nbsp;=&nbsp;''e''<sup>&minus;1</sup> giving Δ''n'' such that:
 
: <math>\kappa = 5.052\,\frac{2\, \Delta n}{\pi}^2. \, </math>
 
The initial interpolation for the function from the measured values <math>f_k(x,y)</math> then becomes:
 
: <math> g_0(x_i,y_j) = \frac{\displaystyle \sum_k w_{ij} f_k(x,y)}{\displaystyle \sum_k w_{ij}}.</math>
 
=== Second pass ===
The correction for the next pass then utilises the difference between the observed field and the interpolated values at the measurement points to optimise the result:
 
: <math>g_1(x_i,y_j) = g_0(x_i,y_j)  +  \sum( f(x,y) - g_0(x,y)) \exp\left(-\gamma \kappa \frac{\pi}{\lambda}^2\right). \,</math>
 
== Parameter selection ==
Although described as an objective method, there are many parameters which control the interpolated field. The choice of Δ''n'', grid spacing Δ''x'' and <math>\gamma</math> as well influence the final result. Guidelines for the selection of these parameters have been suggested,<ref name="Koch_et_al"/> however the final values used are free to be chosen within these guidelines.
 
The data spacing used in the analysis, Δ''n'' may be chosen either by calculating the true experimental data inter-point spacing, or by the use of a [[complete spatial randomness]] assumption, depending upon the degree of [[Cluster analysis|clustering]] in the observed data. The smoothing parameter <math>\gamma</math> is constrained to be between 0.2 and&nbsp;1.0. For reasons of interpolation integrity, Δ''x'' is argued to be constrained between 0.3 and&nbsp;0.5.
 
== Notes ==
<references/>
 
[[Category:Interpolation]]
[[Category:Spatial data analysis]]

Latest revision as of 01:57, 19 May 2014

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