#### 3.1. Comparison of Interpolation Schemes

[9] We estimated the error of interpolated δD and δ^{18}O estimates by subsampling the GNIP data set (jackknifing [e.g., *Tichelaar and Ruff*, 1989]) and using the remaining observations to predict the isotopic composition of precipitation at measurement sites that were excluded from the subsampled data set. Jackknifes were performed for multiple subsample sizes *n* = *N* − *j*, where *N* represents the total number of available data and *j* was assigned values between 1 and *N* − 50. For *j* = 1, *N* subsamples were created where each site was excluded from one subsample. For higher values of *j*, we generated a large number of subsamples by excluding *j* randomly selected data from each subsample. At least 100 subsamples were taken at each value of *j* and, where necessary, a greater number were taken to allow at least 4,000 estimates of δD or δ^{18}O at unsampled stations. By bootstrapping the subsample error estimates, we found these criteria to be sufficient to generate a stable approximation of the mean magnitude of interpolation error with a standard deviation of <0.14‰ for δD and <0.02‰ for δ^{18}O.

[10] Four interpolation schemes were evaluated. Triangulation was chosen to represent simple spatial interpolation with reference only to the nearest data stations, as has been common in paleoclimate studies [e.g., *Sharp and Cerling*, 1998]. For this method, isotopic estimates were made using the equation:

where _{x} is our estimate of the isotopic composition (δD or δ^{18}O) at the location of interest, and δ_{i} and *D*_{xi} represent the isotopic composition of precipitation at the *i*th closest measurement site and the distance between the location of interest and the *i*th closest measurement site in arc degrees, respectively.

[11] The second method examined was inverse distance weighting, where estimates of the isotopic composition at a given location were made using all available stations according to:

where β_{1} (°) determines the relative weight assigned to nearby data. This algorithm is a simple representative of a class of spatial interpolation methods commonly used in the generation of contour maps. For large values of β_{1}, regional variations in isotope compositions will be smoothed over large geographic regions. As β_{1} approaches 0, variability at small spatial scales will be highlighted where data are present, but regions lacking data will take on the global average value of δD or δ^{18}O.

[12] Third, using Cressman objective analysis [*Cressman*, 1959], we interpolated data values onto a global 2.5° latitude × 2.5° longitude grid which was used to estimate the isotopic composition of precipitation at excluded data stations. An initial estimate for δD or δ^{18}O at each grid node was calculated by equation 2 using β_{1} = 1.5°. Correction factors over a series of radii of influence were then determined in sequence and used to incrementally modify the initial estimate. The radii used were 25, 17.5, 10, 7.5, 5 and 2.5°, roughly corresponding to the grid cell radii used to make maps of the isotopic composition of precipitation [*Birks et al.*, 2002]. Correction factors were determined from all data lying with a given radius of influence, and were calculated according to:

In this equation *C*_{x,k} is the correction applied at node *x* determined using the *k*th radius of influence, *p*_{i,k−1} is the isotope value predicted at data station *i* by inverse distance interpolation between the four surrounding grid values calculated at the previous step, *r*_{k} is the current radius of influence, *n*_{k} is the number of stations within the current radius of influence, and the summation is for all stations within *r*_{k} degrees of the location of interest. As per *Birks et al.* [2002], no predictions were made for locations more than 10° distant from the nearest data station.

[13] Finally, we examined the method proposed by *Bowen and Wilkinson* [2002] (hereafter referred to as the BW model), which treats the isotopic composition of precipitation as the sum of temperature driven rainout effects and regional patterns of vapor sourcing and delivery. Temperature effects are represented by model parameters relating the isotopic composition of precipitation to the absolute value of station latitude (∣*LAT*∣) and altitude (*ALT*), according to the equation:

where *p*_{x} is an initial estimate of _{x}, and *a*, *b*, and *c* are empirical parameters. To represent the effects of regional variation in atmospheric circulation patterns on the isotopic composition of precipitation, we spatially interpolate the δD and δ^{18}O variability that is not accounted for by the temperature effects described in the equation above. Combining this interpolation with equation (4) gives the composite model equation:

where β_{2} is a distance weighting parameter analogous to β_{1}. In this study, we fit the model parameters simultaneously by nonlinear least squares, rather than using the two step regression technique proposed by *Bowen and Wilkinson* [2002]. This formulation is more mathematically rigorous in that it assigns equal importance to the latitude and altitude relations, it allows β_{2} to be fit to the data, and it ensures a zero mean residual. The frechêt kernels are:

and:

where:

These were implemented in a standard gradient method used to correct a starting model until corrections were small. In most cases, the model converged on a stable solution with less than 20 iterations.

#### 3.4. Deuterium Excess

[16] We estimated deuterium excess (*d* [*Dansgaard*, 1964]) at 20′ × 20′ resolution directly from our δD and δ^{18}O grids according to:

Confidence intervals (95%) were calculated from the standard distribution of deuterium excess values, given by the equation:

were and indicate the variances of the δD and δ^{18}O estimates, respectively, calculated above.