When simulating a system (e.g. visualising a 2d surface) it is common to create some form of random sample of a surface as the base artefact, and then run the simulation across that using a second random sampling of the base artefact to simulate system's sampling noise.
It is normally presumed that the initial random sample is an effective sample. However in some cases the system being modelled has an implicit assumption about surface texture, though there is a presumption that, in the limit, the initial random sample is an effective representation of the system being simulated.
Typical systems may include wave-particle and particle-wave interactions.
I feel that there is a potential problem with the assumptions and presumtions regarding the double sampling and the 'in the limit' expectation.
I'm looking to get the right words and phrases for the different 'random sampling' effects and the issues buried within the assumptions.
I suspect that the initial finite sample used as the base artefact, even when extended to infinity, is only a countable infinity of points and not a true representation of the underlying surface, and thus always has large amounts of sampling texture that may be neccessary for some simulation approaches. Meanwhile other approaches, which start with a continuous surface and only sample once, never see that texture and cannot exploit / show the desired simulation (of a reality) effect.
I'm hoping that the issue is well described as a stats phenomena/terminology (as opposed to being a physics or maths question).
What are the appropriate technical terms covering this issue?