USDA Forest Service

North Central Research Station

Interpolation Techniques for Late-Spring Freeze Data

The freeze maps were made using observations from 424 National Weather Service (NWS) sites.  For locations between the NWS sites, values were interpolated using inverse distance squared weighting (IDSW).  This common interpolation technique computes the value at a specific location from all known values, with closer stations having more influence on the end result.

An alternative way to compute these values is to treat them as a function of latitude, longitude, and elevation, using an approach called linear regression.

Inverse Distance Squared Weighting is a technique that computes a value for some quantity at a location based on the values of the quantity at other locations where it was measured or known.  In IDSW, the influence of each known value depends on its distance from the point for which the value is being determined.

Mathematically, the value is computed according to the formula:

formula for Inverse Distance Squared Weighting (IDSW)
   where N is the number of locations where the quantity is known or measured;  xi is the value of the quantity at location i;  di is the distance between point i and the point for which x is being determined.

For more information on IDSW, the reader is advised to consult a textbook on statistics or spatial analysis of data.

Linear Regression is another way to interpolate the data from known locations.  This assumes the variables depend on latitude, longitude, and elevation at any location.  The exact form of the dependence is determined mathematically in such a way that the error between the regression values and the observed values at all known locations is minimized.  For the freeze variables discussed here, linear regression produces errors comparable to those from IDSW.

The basic form of these regression equations is:

y(lat, lon, elev) = a + b1*lat + b2*lon + b3*elev

Readers who would prefer to compute values of the variables using linear regression may do so, using the information in the following table:

Threshold Variable a b1 b2 b3 r2
50 GDD Dt -231 7.98 0.37 0.02 0.93
Df -134 6.28 0.48 0.03 0.93
Tf -10.2 0.214 0.02 -7.2x10-4 0.68
nf 114 -2.26 0.055 0.0095 0.61
100 GDD Dt -153 6.94 0.55 0.02 0.93
Df -106 5.97 0.53 0.03 0.92
Tf -7.86 0.179 0.019 -7.3x10-4 0.65
nf 65.7 -1.41 -0.0017 0.0071 0.55
150 GDD Dt -117 6.42 0.60 0.021 0.93
Df -82.7 5.71 0.59 0.028 0.91
Tf -6.22 0.152 0.020 -6.1x10-4 0.59
nf 40.6 -0.893 -0.004 0.0051 0.48
200 GDD Dt -91.3 6.08 0.64 0.022 0.93
Df -65.1 5.55 0.65 0.030 0.89
Tf -5.22 0.132 0.019 -4.1x10-4 0.52
nf 25.51 -0.567 -0.0027 0.0037 0.43
250 GDD Dt -72.4 5.83 0.67 0.024 0.93
Df -52.4 5.53 0.73 0.03 0.89
Tf -4.23 0.126 0.024 -7.4x10-4 0.43
nf 16.5 -0.373 -0.0033 0.0023 0.40
300 GDD Dt -57.1 5.67 0.70 0.025 0.93
Df -48.5 5.64 0.77 0.029 0.89
Tf -3.38 0.087 0.014 -4.6x10-4 0.29
nf 10.9 -0.248 -6.7x10-4 0.0017 0.37
   where Dt is the average day on which a threshold is reached;  Df is the average day of all freezes after the threshold;  Tf is the average temperature of freezes after the threshold;  and nf is the average number of freezes after the threshold.


The final column, r2, is the correlation coefficient for the specific threshold variable.  It indicates the percentage of variability in the quantity that is reproduced by the regression equation.  A value of r2=1.0 would mean the regression equation completely reproduces the observed values at all locations, a perfect fit.  A value of r2=0.0 means that there is no relationship between the equation and the observations.

For more information:

USDA Forest Service - North Central Research Station
Last Modified: January 26, 2005