# Introduction

An interesting paper (Smith and Wessel, 1990) points out a weakness in using splines in cases with data gaps. Their illustration of the problem with isobaths was particularly compelling. I cannot reproduce their Fig 1b here, owing to copyright problems, but I digitized the data so I could test two R functions for splines. Readers can follow along with the code given in this post.

# Methods

The first step is to load the data. (Ignore the extra digits, which simply result from naive copying of digitized values. I suppose the error is of order 2 percent or so, given the scanning of the diagram and my ability to position the pointer on my computer screen.)

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distance <- c(23.17400, 25.09559, 31.15092, 40.75568,
46.42938, 124.68628, 132.20556, 136.81277,
141.37564, 145.08263, 149.38977)
topography <- c(-98.99976, -97.44730, -100.15198, -98.66016,
-98.66016, -193.94266, -296.89045, -392.63991,
-493.85330, -591.21586, -692.82951)

Next, plot the data.

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if (!interactive()) png("splines.png", width=7, height=4, unit="in", res=150, pointsize=8)
par(mar=c(3, 3, 1, 1), mgp=c(2, 0.7, 0))
plot(distance, topography, ylim=c(-700, 0), pch=16)
grid()

Now, set up a grid, and show the results of `smooth.spline()`

.

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d <- seq(min(distance), max(distance), length.out=100)
s <- smooth.spline(topography ~ distance)
ps <- predict(s, d)
lines(ps$x, ps$y)

Now, load the Akima package to get `aspline()`

, which presumably stems from Akima (1970).

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library(akima)
akima <- aspline(distance, topography, d)
lines(akima$x, akima$y, col='red')

Finally, draw a legend and finish up.

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legend("bottomleft", lwd=par('lwd'), bg='white',
col=c("black", "red"),
legend=c("smooth.spline", "aspline"))
if (!interactive()) dev.off()

# Results

At least on this problem, `aspline()`

does a much better job than `smooth.spline()`

.

# Citations

W. H. F. Smith and P. Wessel, 1990. Gridding with continuous curvature splines in tension. Geophysics, 55(3): 293-305.

Hiroshi Akima, 1970. A new method of interpolation and smooth curve fitting based on local procedures. Journal of the Association for Computing Machinery, 17(4): 589-602.