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We present a novel technique for synthesizing textures over dynamically changing fluid surfaces. We use both image textures, as well as bump maps as example inputs. Image textures can enhance the rendering of the fluid by either imparting a realistic appearance to it or by stylizing it, whereas bump maps enable the generation of complex microstructures on the surface of the fluid that may be very difficult to synthesize using simulation. To generate temporally coherent textures over a fluid sequence, we transport texture information, that is, color and local orientation, between free surfaces of the fluid from one time step to the next. This is accomplished by extending the texture information from the first fluid surface to the 3D fluid domain, advecting this information within the fluid domain along the fluid velocity field for one time step and interpolating it back onto the second surface-this operation, in part, uses a novel vector advection technique for transporting orientation vectors. We then refine the transported texture by performing texture synthesis over the second surface using our "surface texture optimization" algorithm, which keeps the synthesized texture visually similar to the input texture and temporally coherent with the transported one. We demonstrate our novel algorithm for texture synthesis on dynamically evolving fluid surfaces in several challenging scenarios.