The atmosphere is a dynamic compressible fluid under rather regularly changing conditions of heat, which affect its temperature and pressure.
The altitude variation of pressure is a major feature, because even in static conditions and fixed temperature, the pressure would decrease exponentially with altitude, and so would the density.
On top of this main feature, the rotation of the Earth introduces organized flows of the fluid (due to e.g. Coriolis Forces). This alone, in a equal-and-fixed-temperature conditions, would distort the altitude dependence of pressure and density. First, because is an altitude dependent phenomenon (the rotation of earth is felt more the further from the center, meaning the higher in altitude) and because at some height, when these forces compress the air laterally, the fluid will tend push towards the perpendicular direction (the altitude) and flow upwards, since usually the pressure is lower in the upwards direction.
Furthermore, temperature profile is changing depending on the heat conditions and the flow of air. When is daylight, the earth heats up faster than water mass, heating up the closest air mass which will tend to rise up (becomes lighter than colder nearby air at same pressure) and currents are produced as the surrounding colder air rushes to fill the space left by the ascending air.
All these are main features, but is not an exhaustive list of relevant phenomena driving pressure distribution in the atmosphere. A special mention should be made: the fast flows of air, in any direction, create zones of low pressure, because fluids in motion have lower pressures in the moving parts, so less-moving nearby fluid will rush towards that pressure decrease too.
Going specifically to your question, in case the answer is not evident from the previous text:
What happens to the displaced air above ground?
Ascending air will be generally hotter and less dense. As it rises it becomes colder, and will stop rising until a point where the air above it is as cold and dense, in which point, being pushed-up by the lower air column rising, it spreads to the sides at that altitude, in a spiral motion (actually it is spinning as it goes up). That is one of the reasons why you see in tornadoes large rotation of air, swirling down as the hot air swirls up in its center, link water falling rapidly to a sink as air rushes up through the center of the swirl.
The problem is not simple to quantify, but if you define a model with some of these features, then you can estimate values of pressure, and relate them to density through the ideal gas state equation.
Update based on comments....
The US model considers the layers static, because the temperature profile of each is fixed and the equations used are consistent with static fluids. It also imposes linear variation of temperature inside each layer, which is consistent with considering the layers exchanging heat at constant rates.
I think these conditions are not common in reality, at least at relatively small localities. But apparently are good for large areas.
I would say, in the assumptions of the model, the more plausible origin of lower/higher than average ground pressure should only be due to the election on temperature distribution.
This could be the case in reality when the layers are almost static layers (even lateral motion creates friction and exchange of matter, causing vertical currents or vortex). In these conditions, with a small but constant source heat (cloudy day, or large amount of people concentrated in a place), you could have a bubble of overheated air.
This is much like small bubbles of air formed on a vase of water being heated. Only when the heat transfer is too high and turbulent motion appears (or by stirring) the bubbles "feel" an unbalance in the forces keeping them stuck to the ground, and Archimedes principle "takes" them upwards. This hot air bubbles might be at lower pressure as they expand, specially because this is not a closed bubble and its molecules will escape to colder areas.
Higher pressures on ground, could be explained similarly by the opposite behavior. The earth, cooling down faster by thermal radiation emission, could cool down further the bottom layer, causing it to compress under the top layers' weight, increasing their pressure.
