  This paper studies the local structure of continuous random fields on
$\mathbb R^d$ taking values in a complete separable linear metric space
${\mathbb V}$. Extending seminal work of Falconer, we show that the generalized
$(1+k)$-th order increment tangent fields are self-similar and almost
everywhere intrinsically stationary in the sense of Matheron. These results
motivate the further study of the structure of ${\mathbb V}$-valued intrinsic
random functions of order $k$ (IRF$_k$,\ $k=0,1,\cdots$). To this end, we focus
on the special case where ${\mathbb V}$ is a Hilbert space. Building on the
work of Sasvari and Berschneider, we establish the spectral characterization of
all second order ${\mathbb V}$-valued IRF$_k$'s, extending the classical
Matheron theory. Using these results, we further characterize the class of
Gaussian, operator self-similar ${\mathbb V}$-valued IRF$_k$'s, generalizing
results of Dobrushin and Didier, Meerschaert and Pipiras, among others. These
processes are the Hilbert-space-valued versions of the general $k$-th order
operator fractional Brownian fields and are characterized by their
self-similarity operator exponent as well as a finite trace class operator
valued spectral measure. We conclude with several examples motivating future
applications to probability and statistics.
  In a technical Supplement of independent interest, we provide a unified
treatment of the Matheron spectral theory for second-order stationary and
intrinsically stationary processes taking values in a separable Hilbert space.
We give the proofs of the Bochner-Neeb and Bochner-Schwartz theorems.
