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To mitigate two major environmental concerns of global warming and air pollution, natural gas power generation is growing in power supply, which makes power and natural gas system increasingly integrated. In this work, an environmental-economic dispatch incorporating diversified emission control is established for integrated power and gas system (IPGS). The traditional emission amount control for carbon emissions is adopted in IPGS, while a novel spatial and temporal diffusion control concerning ground concentration and spatial environmental protection requirements is proposed for air pollutant emissions. Furthermore, in view of information sharing difficulties between power and gas system, this paper proposes a novel two-layer convex decentralized optimization for the issue. The upper layer handles the nonconvex gas flow transmission with penalty convex-concave procedure based convexification. The lower layer solves each step of upper layer convexification with alternating direction method of multipliers based decentralized optimization, such that the information privacy of these two sub-systems is protected. Finally, simulation studies on two test systems validate the superiority of spatial and temporal diffusion control in reducing the air pollutant concentration, as well as the nearly global optimality and convergence efficiency of the two-layer method.