This paper demonstrates a technique for extracting complex fracture surfaces in three dimensions from microseismic data recorded during fracturing or ambient quiet times. Cumulative semblance is computed on a three-dimensional grid of the reservoir, using microseismic trace data. The semblance volume has local peaks that coincide approximately with fracture surfaces. We extract fracture surfaces from this semblance data. The natural fracture locations and orientations are independent of the grid.Our mathematical method generates surfaces without regard to the grid coordinate system. Grid independence is achieved by defining an analytic semblance function in three-dimensional space that interpolates the grid data. The fracture surfaces, in three dimensions, are ridges of the hypersurface in four dimensions (space plus semblance) defined by the analytic semblance function. Existing methods known in the literature cannot be used to compute these surfaces, because they are defined as ridges that may intersect each other. The surfaces are therefore not manifolds implicitly defined by a smooth map. We present a new method for computing surfaces, using extrema of the three-dimensional second derivatives of the analytic semblance function to locate points on the ridges. In three dimensions, we show how to robustly find second derivative extrema in the appropriate direction for defining the ridge surfaces. The resulting surfaces are grid-independent and provide useful geometric data about the fractures and the stress field.The geometry of the constructed fractures has multiple applications. Cumulative seismic activity may be used as a proxy for the slip-tendency of the fracture, so that the orientation and activity values of all the surfaces can be inverted for the orientations and relative magnitudes of the neostress (the present-day stress). The fracture surfaces are connected and can be exported and used directly in Discrete Fracture Network (DFN) fracture and reservoir simulations. Surface statistics are used for calibrating stochastic DFN fracture and reservoir simulators, interpreting focal mechanism solutions, and understanding the interaction of hydraulic fractures with pre-existing natural fracture networks (Lacazette et al., 2014). Also, the resulting surfaces can be used directly in DFN simulators. A discrete representation of the surfaces allows for visualization of the fractures, helping geologists and reservoir engineers understand and measure the extent and the effectiveness of the network.We show examples of surfaces computed with our new method, demonstrating robustness on real data.