The tensor product kernel
General form
The zeromean smooth GP \(\zeta(\mathbf{x})\) is characterized by its covariance kernel \(C_\zeta(\mathbf{x}, \mathbf{x}') := \mathbb{E}[\zeta(\mathbf{x}),\, \zeta(\mathbf{x}')]\). libKriging uses a specific form of covariance kernel \(C_\zeta(\mathbf{x},\,\mathbf{x}')\) on the input space \(\mathbb{R}^d\) which can be called tensorproduct. With \(\mathbf{h} := \mathbf{x}  \mathbf{x}'\) the kernel value expresses as
where \(\kappa(h)\) is a stationary correlation kernel on \(\mathbb{R}\) and \(\boldsymbol{\theta}\) is a vector of \(d\) parameters \(\theta_\ell> 0\) called correlation ranges. See Stein [Ste12] for a discussion on the tensor product kernel a.k.a. separable kernel.
A further constraint used in libKriging is that \(\kappa(h)\) takes only positive values: \(\gamma(h) >0\) for all \(h\). With \(\lambda(h) :=  \log \gamma(h)\) the derivative w.r.t. the correlation range \(\theta_\ell\) can be computed as
Available 1D correlation kernels
The 1D correlation kernels available are listed in the Table below. Remind that in this setting the smoothness of the paths of the GP \(\zeta(\mathbf{x})\) is controlled by the smoothness of the kernel \(C_\zeta(\mathbf{h})\) at \(\mathbf{h} = \mathbf{0}\) hence by the smoothness of the correlation kernel \(\kappa(h)\) for \(h=0\). Note that the 1D exponential kernel is not differentiable at \(h = 0\) and the corresponding paths are continuous but nowhere differentiable. The kernels are given in the table by order of increasing smoothness.
Note The Gaussian kernel is a radial kernel in the sense that it depends on \(\mathbf{h}\) only through its square norm \(\sum_\ell h_\ell^2 / \theta_\ell^2\).
kernel 
Name 
Expression 


Exponential 
\(\kappa(h) = \exp\{\lvert h \rvert \}\) 

Matérn whith shape \(3/2\) 
\(\kappa(h) = [1 + z] \exp\{z\}\), \(z := \sqrt{3} \, \lvert h \rvert\) 

Matérn whith shape \(5/2\) 
\(\kappa(h) = [1 + z + z^2/3] \exp\{z\}\), \(z := \sqrt{5} \, \lvert h \rvert\) 

Gaussian 
\(\kappa(h) = \exp\{h^2/2\}\) 