  Improving the performance of superconducting qubits and resonators generally
results from a combination of materials and fabrication process improvements
and design modifications that reduce device sensitivity to residual losses. One
instance of this approach is to use trenching into the device substrate in
combination with superconductors and dielectrics with low intrinsic losses to
improve quality factors and coherence times. Here we demonstrate titanium
nitride coplanar waveguide resonators with mean quality factors exceeding two
million and controlled trenching reaching 2.2 $\mu$m into the silicon
substrate. Additionally, we measure sets of resonators with a range of sizes
and trench depths and compare these results with finite-element simulations to
demonstrate quantitative agreement with a model of interface dielectric loss.
We then apply this analysis to determine the extent to which trenching can
improve resonator performance.
