Prior to performing any geostatistical analysis, we fit the bed elevation data to a 1 km resolution grid using the GStatSim grid_data function which averages values within each grid cell. We do this because the interpolation examples shown in this paper use a 1 km resolution. In this way, the spatial statistics of the data will be compatible with the interpolation grid. This gridded data set has 3650 data points. In order to satisfy the Gaussian assumptions of our interpolation functions, we convert our data to a standard Gaussian distribution using the scikit-learn QuantileTransformer function. This step converts the data to a standard Gaussian distribution (mean =0, variance =1), shown in Fig. . It is not strictly necessary for data to be Gaussian when using GStatSim. However, the interpolation results will be more robust if a Gaussian distribution is used.

The interpolation functions in GStatSim rely on the variogram, which describes spatial statistics by quantifying the covariance between data points as a function of their separation distance or lag . The experimental variogram γ(h) is defined as the averaged squared differences of data points separated by a lag distance h: 1γ(h)=12N(h)∑α=1N(Z(uα)-Z(uα+h))2, where Z is a variable, u is a spatial location, and N(h) is the number of pairs of data points for a lag h. Typically, the variance increases with lag distance until a threshold is reached where measurements are no longer spatially correlated. This threshold is called the sill, which is, in theory, equal to 1 for normalized data sets. The lag distance where the variogram reaches the sill is called the range.

GStatSim does not include variogram estimation tools as there are already existing resources (e.g., SciKit-GStat). Our interpolation functions are designed to be integrated with outside variogram modeling tools. We perform our variogram analysis with the SciKit-GStat functions, which offer robust tools for variogram estimation and modeling. We compute both the isotropic and anisotropic variograms. The anisotropic variogram computes the variogram for different angles from horizontal. Figure  shows that the data do not contain significant anisotropy. While the anisotropic variograms do exhibit some disagreement at lags greater than ∼35 km, these differences can be attributed to regional trends in the topography. Additionally, the first part of the variogram with small lags is more important than the tail distributions when it comes to calculating the kriging weights used for interpolation. As such, only the isotropic variogram is used for the variogram modeling analysis.

In order to use the variogram for interpolation, a variogram model must be fit to the experimental variogram. GStatSim can currently accommodate exponential, Gaussian, and spherical variogram model types. The exponential model is as follows: 2γexp⁡(h)=0ifh=0b+(C0-b)1-e-3haifh>0, where a is the range parameter, C0 is the sill, and b is the nugget. The nugget describes the variance at near-zero lags, which is often the result of measurement error. In most cases, the nugget is zero. The Gaussian model is 3γGauss(h)=0ifh=0b+(C0-b)1-e-3h2a2ifh>0, and the spherical model is 4γsph(h)=0ifh=0b+(C0-b)1.5⋅ha-0.5⋅h3a3if0<h≤aC0ifh>a.

See for a detailed description of these models. In practice, variogram model parameters are often selected manually, though SciKit-GStat also offers automatic variogram modeling tools. Using these tools, we fit each of these model types to the experimental variogram, shown in Fig. a. A visual inspection of Fig. a shows that the exponential variogram model produces the best fit. As such, we use this model type for the topography interpolations shown in this paper. The exponential model has a nugget of 0, a range of 31.9 km, and a sill of 0.7. Geostatistical interpolations can be sensitive to the methods and parameters used to compute and model variograms. As such, we recommend referring to the SciKit-GStat documentation for best practices and considerations when conducting variogram analysis. The effect of different variogram models on interpolations is described in Sect. .

Kriging is used to produce deterministic interpolations of a spatial variable, where the goal is to minimize estimation variance (e.g., ). For a spatial variable Z, kriging models the residuals Z* from a mean m: 5Z*(uα)=Z(uα)-m(uα).

In cases where the mean is constant, m is treated as a global mean. Each interpolated value Z* at a location u is the weighted sum of neighboring measurements: 6Z*(u0)=∑αNλαZ(uα), where λα are the weights on the N data points. These weights account for the variability in the measurements, their proximity to each other and the node being estimated, and the redundancy between nearby measurements. Specifically, the weights are determined by solving the kriging system of equations which, for a system with three data points, would look like 7C(u1-u1)C(u1-u2)C(u1-u3)C(u2-u1)C(u2-u2)C(u2-u3)C(u3-u1)C(u3-u2)C(u3-u3)λ1λ2λ3=C(u0-u1)C(u0-u2)C(u0-u3), which can be abbreviated to 8Cλ=c, where C describes the covariance between pairs of data points, and c describes the covariance between each data point and the location that is being estimated, u0. Note that the covariance is linked to the variogram: 9C(h)=C(0)-γ(h), where C(0) is the variance of the data. We use the modeled variogram parameters determined in Sect. 4.2.

GStatSim contains functions for both simple and ordinary kriging. With simple kriging, it is assumed that the mean is a constant, known parameter defined by the mean of the data. For data that have undergone a normal score transformation, the mean is 0. For ordinary kriging, the mean is estimated implicitly within a search neighborhood around the coordinate being estimated, and simple kriging is applied to the residuals. This makes ordinary kriging more robust to spatial trends. The ordinary kriging weight estimation procedure is modified from Eq. () to 10C(u1-u1)C(u1-u2)C(u1-u3)1C(u2-u1)C(u2-u2)C(u2-u3)1C(u3-u1)C(u3-u2)C(u3-u3)11110λ1λ2λ3μ=C(u0-u1)C(u0-u2)C(u0-u3)1, where μ is a Lagrange multiplier. The covariance function is also used to compute the uncertainty, or variance, at each location. The variance of an estimate at location u0, σE2(u0) is defined as 11σE2(u0)=C(0)-∑αNλαC(u0-uα).

The ability of kriging to quantify uncertainty is a major advantage of this method over other deterministic interpolation methods such as spline interpolation.

In contrast to kriging, sequential Gaussian simulation (SGS) is designed to preserve the variance of observations. SGS is the stochastic alternative to kriging. In contrast to kriging where the objective is to optimize local accuracy, SGS is used to generate multiple realizations that reproduce the variogram statistics (e.g., ). The implementation of SGS, described in Fig. , is the following:

Define a random simulation order in which to visit each node.

GStatSim has two different SGS functions, okrige_sgs and skrige_sgs, which use ordinary kriging and simple kriging for step 2, respectively. The results are unlikely to differ significantly between these two methods, but okrige_sgs may be more robust when large-scale trends are present. The input parameters are the same as those used for the skrige and okrige functions. We simulate four realizations using okrige_sgs using the same input parameters as in Sect. 4.3.3. To evaluate the ability of SGS to reproduce the spatial statistics of observations, we compute the variogram for realization no. 1 (Fig. ). The variogram of the simulated topography matches the variogram of observations. The realizations are shown in Fig. . Each realization matches the conditioning data and represents an equally probable solution.

GStatSim can perform interpolations using the exponential, Gaussian, or spherical variogram models, as described in Sect. . In this section, we demonstrate how interpolations differ when using different variogram model types. We generate a synthetic conditioning data set of 100 points using the unconditional Gaussian field tools in GSTools (Fig. a). We use a variance of 2 and length scale of 8. While the true variogram for the synthetic data is Gaussian, we perform interpolations using different variogram models for demonstration purposes. We compute the experimental variogram and fit Gaussian, exponential, and spherical variogram models (Fig. b). Then we perform both simple kriging interpolation and SGS with each variogram model. The results, shown in Fig. , show that the Gaussian model produces the SGS realization with the smoothest spatial structure and lowest kriging variance. The exponential model produces the roughest SGS topography and highest kriging variance, and the spherical model falls in the middle. These different model types allow GStatSim to accommodate different types of spatial structures.

Geologic phenomena frequently exhibit anisotropy in their spatial statistics. For example, preferential erosion can cause subglacial topography to be smoother in the direction that is parallel to ice flow . The GStatSim interpolation functions can account for this anisotropy by accepting major and minor range parameters. The major range is the range of the variogram in the smoothest (major) direction, and the minor range is the range of the variogram in the minor direction, which is orthogonal to the major direction. Not all pairs of data points are oriented along the major or minor orientations, so the range is treated as the radius of a rotated ellipse where the major and minor ranges are the lengths of each axis, and the degree of rotation is determined by the angle of anisotropy. The major angle orientation is defined as the angle from zero (horizontal) of the major direction. In practice, these parameters can be determined by modeling the variogram in multiple directions and determining the major angle through visual inspection.

To demonstrate the implementation of interpolation with anisotropy, we perform simple kriging and SGS with anisotropy for angles of 0 and 60∘. We use the modeled variogram parameters from previous sections (nugget =0, range =31.9 km, sill =0.7). We use 31.9 km as the minor range and set the major range to be 15 km greater than the minor range for exaggerated effect. These interpolations (Fig. ) have clearly visible anisotropy.

Data sets often contain variations in spatial statistics, a condition known as non-stationarity. Mean non-stationarity occurs in the presence of a large-scale trend or systematic changes across a study area. While ordinary kriging can accommodate variations in the mean, to some extent, it does not extrapolate well . For this reason, it is sometimes convenient to detrend the data prior to variogram modeling and interpolation. This approach is known as kriging (or SGS) with a trend or universal kriging. Detrending has the additional advantage of producing residual data that more closely represent a random Gaussian process, which makes the variogram modeling more robust. In contrast, data with a strong trend can yield an experimental variogram that never reaches an obvious sill, making it difficult to fit a variogram model.

Here we demonstrate kriging and SGS with a trend (Fig. ). This is achieved by the following:

Estimate the trend.

While it is common practice to estimate the trend by fitting a linear or polynomial function to the data (e.g., ), determining higher-order polynomial models for complex landscapes can be difficult. Instead, we estimate the trend using a radial basis function (RBF), which is a function that changes with distance from a location, such as a thin-plate spline or Gaussian function. RBFs measure the distance between input values and a center point, assigns weights based on these distances, and combines these weights to produce an output value . The RBF trend estimation was implemented using the GStatSim rbf_trend function, which relies on the scipy RBF implementation. rbf_trend uses a multiquadric kernel as its RBF, which is known for its high accuracy and flexibility . The rbf_trend function input parameters are the conditioning data, the grid cell resolution, and a smoothing factor. The smoothing factor ensures that the trend is not overfitting to measurements. The smoothing parameter should be at least as large as the largest data gap. We use a smoothing factor of 100 km.

To evaluate the performance of the detrended SGS approach, we compute the experimental variogram for residual data (Fig. d) and the residual SGS (Fig. e). The variograms, shown in Fig. , show close agreement between the spatial statistics of the residual measurements and simulation. Note that both variograms reach a sill of 1 in contrast to the variogram for the data without detrending (Fig. ). According to the theoretical definition of the variogram (Eq. ), data with a standard Gaussian distribution should reach a sill of 1. As such, the detrended data can often be better suited to variogram-based interpolation.

Variogram non-stationarity describes the condition when the variogram statistics vary throughout a domain. For example, the topographic roughness varies throughout Greenland . To handle these cases, GStatSim includes a cluster_sgs function where the user can assign different variograms to different areas. It is implemented by partitioning the data into different clusters and then computing the variogram for each cluster. To simulate a grid cell,

choose a random grid cell;

This process ensures that the boundaries between different clusters vary for each realization. cluster_sgs takes the same inputs as SGS, as well as the cluster numbers for each data point and a data frame with the variogram parameters for each cluster. Users can use any method they like to partition the data into different clusters. In the next two subsections, we outline two approaches.

Secondary information can be integrated into interpolations via co-kriging and co-simulation, where a well-sampled secondary variable is used to constrain the interpolation of a sparsely sampled primary variable. For example, if the ice surface elevation (Fig. a) is found to be correlated with the subglacial bed elevation, then the ice surface can be used to improve the interpolation. We note that the ice surface and subglacial topography are unlikely to be linearly related ; this example is merely intended to demonstrate the usage of our co-kriging and co-simulation tools. In theory, co-kriging and co-simulation are implemented by modeling the variogram for each variable and a cross-variogram that describes the covariation between the variables. In practice, the full co-kriging system of equations is difficult to solve, so approximations are used instead .

While there are several different ways to implement co-kriging , GStatSim only includes functions for co-located co-kriging under Markov model 1 (MM1) assumptions, as described by . This version was chosen for its simplicity; expert geostatistical knowledge is not needed to use this algorithm. Co-located co-kriging means that an estimate is made using only the secondary data point that is co-located with, or at the same location as, the grid cell being estimated. This significantly reduces the number of conditioning data points that are used to estimate each grid cell. Co-kriging with MM1 approximates the cross-variogram through a Markov assumption of conditional independence. It is assumed that a secondary data variable Z2 at a location u is conditionally independent of the primary variable Z1 at a location of u′, given Z1(u). u′ refers to a spatial location that is not u. This statement of conditional independence is written as 13E(Z2(u)|Z1(u)=z1,Z1(u′)=z1(u′))=E(Z2(u)|Z1(u)=z1), where E refers to the expected value. In the geostatistical community, this Markov assumption is referred to as Markov model 1 . See and for other co-kriging models. In the case where the ice surface elevation is used as a secondary constraint, MM1 means that the expected value of the ice surface at a location u, given knowledge of the bed elevation at location u, is independent of the bed elevation value at other locations. Under MM1 assumptions, the cross-correlogram describing the covariation between the primary and secondary variables is calculated as 14ρ12(h)=ρ12(0)ρ1(h), where h is the lag distance and ρ is the correlogram, calculated by 15ρ(h)=1-γ(h), assuming that γ(h) is the normal scores variogram. While the variogram measures spatial covariance, the correlogram measures spatial correlation. This means that ρ12(0), the correlogram at a lag distance of zero, is simply the correlation coefficient between the primary and secondary variables. Because the cross-covariance between the primary and secondary data relies only on the correlation coefficient and primary variogram, the secondary variogram is not needed. This means that co-kriging and co-simulation with MM1 only require the primary variogram and the correlation coefficient to determine the co-kriging weights. The simple co-located co-kriging estimate Z1* at a location u0 is defined as 16Z1*(u0)=∑αNλαZ1(uα)+λ2Z2(u0), where λα are the weights on the N primary data points, Z2(u0) is the co-located secondary datum, and λ2 is the weight for the secondary datum. This means that the co-kriging estimate of a bed elevation value is the weighted sum of bed elevation data points within a specified search radius and the co-located ice surface elevation datum. The variance is computed using Eq. (). Co-SGS is implemented in the same way as SGS, except that Eq. () is used instead of Eq. ().

To implement co-kriging and co-SGS, we use the variogram model parameters from Sect. 4.2 and compute the correlation coefficient, ρ12(0), between the bed elevation measurements and co-located ice surface elevation. For convenience, we developed a find_colocated function that extracts the co-located data points from the secondary variable (Fig. b). ρ12(0) was determined to be 0.6. The interpolations are performed using the GStatSim cokrige_mm1 and cosim_mm1 functions. These functions require the same input parameters as skrige, okrige, skrige_sfs, and okrige_sfs with the addition of ρ12(0) and the secondary data set.

The co-kriging and co-SGS results are shown in Fig. . Compared to the previous kriging and simulation examples, these interpolations produce visibly higher-elevation topography on the right side of the region. This is due to the correlation with the ice surface. To evaluate the simulation performance, we compute the correlation coefficient between the ice surface elevation (Fig. a) and co-simulated bed elevation (Fig. d). This correlation coefficient is found to be 0.7, which is greater than the correlation coefficient of the co-located data, 0.6. This inflated correlation between the primary and secondary data is a known issue with MM1 . Co-simulation with MM1 is also known to produce interpolation artifacts and artificially high variance . This is because the MM1 assumption screens out the influence of the primary data on the secondary data, except at the location u0, which can lead to the underestimation of the redundancy between the primary and secondary variables. For this reason, MM1 is most effective in situations where the primary variable is smoother than the secondary variable, which is not the case in our example. As such, it is important to check the statistics (variogram and correlation) of simulations produced with cosim_mm1 to ensure that this method is used appropriately.