This paper addresses the problem of designing a global tone mapping operator for rate distortion optimized backward compatible compression of HDR images. We address the problem of tone mapping design for two different use cases leading to two different minimization problems. The first problem considered is the minimization of the distortion on the reconstructed HDR signal under a rate constraint on the SDR layer. The second problem remains the same minimization with an additional constraint to preserve a good quality for the SDR signal. Both the distortion and the rate are expressed as a function of the spatial gradient in the HDR images. Experiments show that the proposed rate and distortion models based on the HDR image gradient accurately predict the real image rate and distortion measures. Experimental results show that for the first minimization, the optimal rate-distortion performances are achieved, and that the second optimization yields the best trade-off between rate-distortion performance and quality preservation of the SDR signal.