What is a half-space?

In geophysics we often refer to half-space” when talking about survey methods and theory. But what is it? 

Consider that we, as humans, exist in a “full-space”: we are completely surrounded by material which we can interact with – the ground beneath us and the air around and above us. But for almost all surface geophysical methods, they can only interact with the ground beneath them, the air above acts as a barrier to measurement signals: electrical currents can only flow within the conductive earth, not up into the air; seismic vibrations are transmitted through the solid ground; the transient field in a TEM survey can only be generated in the earth beneath the loop; GPR signals are (by antenna design) transmitted predominantly into the material beneath the antenna. So, when considering the theory of what happens to geophysical signals, we often simplify the case by defining a half-space: an infinite volume of material, beneath a flat surface, above which no signals can travel. 

There are special instances where the half-space consideration does not hold true, for example doing underwater resistivity measurements. where the cable is sat beneath the water column and current can now flow both beneath and above the cable. In these instances, we need to make additional special theoretical considerations and apply different processing routines.