I've never thought about it much other than as the generic statement in textbooks in front of headache inducing equations: 'assume the soil is linear, homogenous, continous, linear elastic half space'
In a half-space of sedimentary granular soil under a geostatic state of initial stress, the density and the Poisson's ratio do not vary considerably with depth. In such an Earth body, the dynamic shear modulus is the parameter that mainly affects the dispersion of propagating waves.
My interpretation is that "half-space" is defined as the space to the boundaries in which an specific solution is not affected.
For example, if you look at the settlement of a footing on a soil layer (say a uniform sand), and the footing is circular with a radius r, you can construct the analysis model with increasing distances from the center of the footing to the lateral and depth boundaries. With the boundaries set at a distance of 2r you will get one value for the settlement. Then you can do the analysis again with the boundaries at 3r and you will get a different value for the settlement (it will be bigger). As you subsequently increase the distances to the boundaries to 4r, 5r, etc. the solution will converge to a given value (although you will probably still see changes of smaller and smaller magnitudes, 8.01 mm, 8.012 mm, etc.). So, for practical purposes, when the distance to the boundary is 4-5 times the diameter of the footing, the result is essentially correct (that is, a larger domain is not needed to approximate the "half space").
BigH gave you a nice answer... half space is considered when a load is applied on the surface. If you are "buried", the you are not applying the load on the surface of the half space but you are "within the space". For example, the load at the tip of a pile may be considered as a "within the space" condition.
Not agreeing with that definition, but if it suits your purposes, why not. There may be no need to analyze the space to such extent that there is no apparent effect of loading. In fact spaces do not have to, or don't always physically extend to a no-effect region.
My idea is that density and Poissin ratio remain constant within the space being analyzed, thus stress is transmitted between grid points in an easily predictable manner. As such wave speeds are constant and will pass through the material unhindered by velocity changes, refraction and reflection due to changes in density and that the stress-strain relationships remain constant as they do.
