SPECPR Users’ Manual Page 8-f12.1 F12: Cubic Spline Interpolation Alias: cspl[in] F12: Derivative This routine fits a cubic spline to an input data set, and then computes a new data set based on a new wavelength set specified by the user. This allows you to smoothly interpolate one data set to another (not this is not a convolution). Errors, if specified, are also interpolated. The input data set and its associated wavelengths are used to derive a cubic spline describing the curve. The curve may then be recomputed at other wavelengths, or because the curve is described by a set of polynomials, the derivative of the curve may be computed. The input wavelength set can be the same as the output wavelength set, then with selected parts of the spectrum deleted, a continuum can be computed. For example, in this mode, you would simply delete all the channels comprising a particular absorption band. The computed spectrum is then the input spectrum without the absorption band. See Clark, R.N., Water Frost and Ice: The Near-Infrared Spectral Reflectance 0.65-2.5 μm, J. Geophys. Res., 3087-3096, 1981. or Clark, R.N. and T.L. Roush, Reflectance Spectroscopy: Quantitative Analysis Techniques for Remote Sensing Applications, J. Geophys. Res., 6329-6340, 1984. for examples of continua and a discussion of the theoretical aspects of continua. The program requires that the input data set and its error bars, if any, be specified when called from Math Operations. For example; v23f14e or v23csplin e

Upon entering the routine, the title to the data to be splined is displayed and you are asked select normal spline or derivative mode:

<return> for cubic spline mode, d for derivative mode, e or x to exit.

Next, you will be asked to input two wavelength data sets, the first for the input data set, the second for the output interpolation. The wavelength sets should be entered with capital letter file IDs. Example:

V24 U46 or V15 V15

where the first interpolates to a new wavelength set, and the second example interpolates to the same wavelength set, presumably with the intent of deleting some channels to use the result as a continuum. Finally, you are asked if you want to delete points. If you do, just type in the channels to be deleted, separated by spaces, or ranges separated by a "t", and then terminated by a "c". For example:

1 2 59t63 120 c |

deletes channels 1, 2, 59, 60, 61, 62, 63, and 120. You can use more than one line, and the deletion routine is not finished until it gets the "c". WARNING: Cubic splining can be tricky. For example, glitches in data could lead to erroneous results, so you should delete those channels. The cubic spline routine used requires each wavelength set to be sorted in increasing order, and the interpolated wavelength to be within the bounds of the other wavelength set. F12 handles this, minimizing errors that could occur. However, it seems common to have problems. If you get one of the following "F12 errors," note the number and other pertinent information (e.g. data sets in use) and submit a bug report if you can’t figure out a cause. NOTE, HOWEVER that a usual cause is bad input, most likely due to two wavelengths that have the same value (causing an infinite slope), and an extremely high data value (causing a near infinite slope), or some channel or wavelength not what you think it is (like not in the proper overlap range). Check your data carefully before reporting bugs. If you have problems, you should carefully check all your data by both plotting it and examining the values with the data editor, f14. For example, if you find deleted channels in the input wavelength data set, you should include those channels in the list of channels to be deleted in the spline routine (f12).

Error codes from the cubic spline routine:

33: Found value(s) in interpolated wavelength set less than input wavelength set. 34: Found value(s) in interpolated wavelength set greater than input wavelength set. 129: The dimension of spline coefficient matrix is less than one less the number of channels. 130: The number of channels is less than 2 (user should check wavelength data sets in use). 131: Wavelength files are not ordered. (Since the routine sorts the data, this should not happen. It may indicate fundamental problem with the input data). |

The spline function is very useful for interpolating data to new wavelength values and also for generating continua. When interpolating to a new wavelength set, the resolution of the spectra should be equal (use the smoothing function of F17 to degrade a high resolution spectrum to that of the low resolution spectrum). Spurious data points should be deleted. For generating a continuum to a spectrum, specify the same wavelength file for the input and output spectra, then delete data points which are not on the "continuum." This can be tricky depending on the data and what you are trying to do so talk to people who have used the routine first (e.g. Roger Clark or Bob Singer).

How the Cubic Spline Interpolating Function Works

The cubic spline interpolating function may be visualized as follows. Bend a flexible strip (like a plastic ruler) so that it passes through each of the data points in the spectrum to be interpolated

[f(x1), f(x2), . . . f(xn)]. |

The physics of the bent strip shows that the equation of the strip can be represented as a series of cubic polynomials with appropriate boundary conditions. A different cubic polynomial is calculated for each interval in the spectrum

[i.e. S1(x) on the interval (x1, x2), S2(x) on (x2, x3) . . . S{n-1}(x) on (x{n-1}, xn) |

where Sn represents a cubic polynomial

Anx3 + Bnx2 + Cnx + Dn = Sn(x)]. |

The boundary conditions are that the spline must go through each function value

[S1 (x1) = f(x1), S1 (x2) = f(x2) = S2(x2) = etc.], |

and the first and second derivatives of the cubic polynomials are continuous

[S{m-1}’(xm) = Sm’(xm), S{m-1}"(xm) = Sm"(xm) |

for m on the interval 2 to (n-1) (inclusive)]. Finally, the curvature is forced to zero at the endpoints

[S1"(x1) = 0, S{n-1}"(xn) = 0]. |

The spline used here is NOT a smoothing function. Therefore, data with large noise spikes present should be pre-smoothed before fitting with the interpolating spline (or the data points with anomalous values should be deleted). For a more complete discussion see Carnahan and J. O. Wilkes, "Digital Computing and Numerical Methods," John Wiley & Sons, New York, New York. p 307. (1973).