  We discuss the evolution of the local quasiparticle spectral density and the
related tunneling conductance measurable by the scanning tunneling microscope,
as a function of distance r and angle \theta from the vortex core in a
d_{x^2-y^2} superconductor. We consider the effects of electronic disorder and
of a strongly anisotropic tunneling matrix element, and show that in real
materials they will likely obscure the ~1/r asymptotic tail in the zero-bias
tunneling conductance expected from the straightforward semiclassical analysis.
We also give a prediction for the tunneling conductance anisotropy around the
vortex core and establish a connection to the structure of the tunneling matrix
element.
