Fields are assessed at 1 m above ground
The preferred height above ground for assessing fields, particularly electric fields, under power lines is 1 m.
This is partly because where standards specify a height they specify 1 m, e.g. the ANSI/IEEE standard. Some other standards, for example the IEC standard, do not specify a height. But 1 m has become a widespread practical standard.
However, there is also a scientific reason why 1 m is the right height when you are considering induced currents and compliance with exposure limits.
The reason why
The derivation of an external electric field from the basic restriction (the induced current density) is performed by numerical modelling. This can be performed for any field required, but for convenience, is usually performed for uniform fields. Specifically, the value of 9 kV/m which is the limit in the UK is derived from modelling by Dimbylow that used uniform fields.
Under power lines, the field is not uniform, increasing by typically 10% or so over the height of a person. The question is therefore, at what height does this non-uniform field induce the same current as a uniform field of 9 kV/m?
The answer is, to a first approximation, at the half-way point up the body, making 1 m a reasonable approximation. The reason is as follows. The electric field meets the surface of the body all over the body. At each point of the surface of the body, a current is induced. The total current induced, built up from all over the surface, flows through the body and distributes itself amongst the various organs according to their conductivity. So what matters is the total current induced. To a first approximation, this is proportional to the space potential at the top of the head. The space potential at the top of the head is the integral of the field over a vertical line from ground to that point, and for slowly varying fields, this is given by the equivalent uniform field at the midpoint.
This approach needs modifying in detail, in particular to allow for the central nervous system - the part of the body the limits apply to - not being in the path of total current flow. But to a first approximation, the field at the centre of the body is a more reasonable approximation to the equivalent uniform field than the field at the top of the body.
For magnetic fields, currents are induced mainly in the head and trunk, so a similar argument might lead to a slightly higher height, perhaps the centre of the chest. But we usually standardise on 1 m for magnetic fields too.
There are two counter arguments, both are which are based on misunderstandings.
One counter argument is that the highest field (which, underneath a power line, will be at the top of the body) should be assessed. This would be valid if the electric field were itself the limit; then it would clearly be correct to find the highest field and compare it to the limit. But the electric field is not the limit. The limit is the basic restriction, and the electric field is relevant only in relation to the size induced current it produces. As discussed above, this depends more on the average field than the maximum field.
The other counter argument is that if the organ of most interest is the neck, the field should be assessed at the neck. This is wrong because the current flowing through any given organ is not produced by the field at that point. The current is produced by the field over the surface of the body, which then flows through the inside of the body and the organs in it.
Clearly, 1 m won't be exactly the right height for all circumstances. Children are shorter than adults so the best height may be smaller - but that in turn means that calculating at 1 m for a child is usually erring on the safe side. For a person riding on a horse the height might be greater. But in terms of compliance with exposure limits in the UK, compliance with the public limits is required when exposure is for a "significant period of time". A person is unlikely to spend a significant period of time on horseback directly underneath a power line.