The pilot was maneuvering the airplane at a low altitude over open, snow-covered terrain, and noticed that the engine was beginning to lose power and running roughly. The airplane was starting to lose altitude and in response, he applied the carburetor heat and added full engine power. Just as the engine started to regain power, the main landing gear broke through the crusty top layer of the snow, slowing the airplane. The propeller contacted the snow and the pilot decreased the engine power and attempted to transition to a landing. The airplane nosed over inverted, resulting in substantial damage to the vertical stabilizer and both left wing lift struts. According to the pilot, with the exception of the accumulation of carburetor ice, there were no mechanical malfunctions or failures with the airframe or engine that would have precluded normal operation.