Understanding the flow created by particle motion at interfaces is a critical step toward understanding hydrodynamic interactions and colloidal self organization. We have developed correlated displacement velocimetry to measure flow fields around interfacially trapped Brownian particles. These flow fields can be decomposed into interfacial hydrodynamic multipoles, including force monopole and dipole flows. These structures provide key insights essential to understanding the interface's mechanical response. Importantly, the flow structure shows that the interface is incompressible for scant surfactant near the ideal gaseous state and contains information about interfacial properties and hydrodynamic coupling with the bulk fluid. The same dataset can be used to predict the response of the interface to applied, complex forces, enabling virtual experiments that produce higher order interfacial multipoles.