The Earth is better surveyed than is generally realized. It is probable that geophysical survey techniques would have revealed the existence of an underground megastructure.
Geophysical surveys may use a great variety of sensing instruments, and data may be collected from above or below the Earth's surface or from aerial, orbital, or marine platforms. Geophysical surveys have many applications in geology, archaeology, mineral and energy exploration, oceanography, and engineering. Geophysical surveys are used in industry as well as for academic research.
There is a wide variety and considerable spectrum of instruments used for geophysical surveys.
Seismic methods, such as reflection seismology, seismic refraction, and seismic tomography.
Geodesy and gravity techniques, including gravimetry and gravity gradiometry.
Magnetic techniques, including aeromagnetic surveys and magnetometers.
Electrical techniques, including electrical resistivity tomography, induced polarization, spontaneous potential and marine control source electromagnetic (mCSEM) or EM seabed logging.
Electromagnetic methods, such as magnetotellurics, ground penetrating radar and transient/time-domain electromagnetics, magnetic resonance sounding (MRS).
Borehole geophysics, also called well logging.
Remote sensing techniques, including hyperspectral.
Source: Geophysical survey
Many of these instruments and techniques could reveal a deep subterranean structure depending on the nature and construction of the megastructure.
For example, because of its depth the world-girdling tunnels would need to be made of extremely strong and, possibly, dense materials. Gravimetric surveys might detect its presence. Magnetomtric surveys would the same if its construction material(s) were highly magnetic.
Remote sensing imagery could show a continuous band of disturbance or its remnants if earth rock and soil had to be displaced for the megastructure's construction and installation. Building something on this scale at a depth of kilometres will leave its mark on the planet.
Remote sensing, orbital and aerial imaging are excellent at uncovering changes to the surface and subsurface of planetary terrain. Changes or structures that wouldn't be apparent if you were standing in those places yourself.
The ring megastructure is likely to be detected by variations in mass distribution using geodetic remote sensing.
Geodetic remote sensing can be gravimetric or geometric. Overhead gravity data collection was first used in aerial submarine detection. This data revealed minute perturbations in the Earth’s gravitational field that may be used to determine changes in the mass distribution of the Earth, which in turn may be used for geophysical studies, as in GRACE (satellite). Geometric remote sensing includes position and deformation imaging using InSAR, LIDAR, etc
There is also seismic mapping of the Earth's interior which might determine there was two hundred metre diameter tunnel around the world.
While a mine or a bore hole breaking through or hitting the megastructure is more hit or miss geophysics has an arsenal of methods and instruments to survey what is down below the surface of our planet.
In summary, there a wide range of geophysical survey methods. Many of which, both singly or together, could indicate the presence of a subterranean megastructure. Once its existence was suspected the range of geophysical instruments could be brought to beat to further elucidate it. In principle, puny humans have the techniques to find a deep megastructure and have had them since the twentieth century, therefore, they can only be expected to have gotten better this century. This makes the discovery of the megastructure highly probable.