Plot3D[
      (Sqrt[(1/2) (-(x^2) - (y^2)) + Abs[Sin[Pi (((x y))^2)]] + 
          (Sqrt[(x^2) + (y^2)]) -  .1]), {x, -2, 2}, {y, -2, 2}, 
      PlotPoints -> 120, ImageSize -> 600, Mesh -> False]