  Corresponding to any $(m-1)$-tuple of semi-spectral measures on the unit
circle, a weighted Dirichlet-type space is introduced and studied. We prove
that the operator of multiplication by the coordinate function on these
weighted Dirichlet-type spaces acts as an analytic $m$-isometry and satisfies a
certain set of operator inequalities. Moreover, it is shown that an analytic
$m$-isometry which satisfies this set of operator inequalities can be
represented as an operator of multiplication by the coordinate function on a
weighted Dirichlet-type space induced from an $(m-1)$-tuple of semi-spectral
measures on the unit circle. This extends a result of Richter as well as of
Olofsson on the class of analytic $2$-isometries. We also prove that all left
invertible $m$-concave operators satisfying the aforementioned operator
inequalities admit a Wold-type decomposition. This result serves as a key
ingredient to our model theorem and also generalizes a result of Shimorin on a
class of $3$-concave operators.
