Subtle reservoir is a key target of oil and gas exploration in the future. The high similarity of seismic amplitude between the reservoir and surrounding rock presents a challenge to detecting the reservoir boundary and its distribution. We have developed a global measurement of the distance between each sample value of seismic data and the stable fixed point of the data, i.e., the seismic convergence rate defined by the logistic map, and use it to highlight subtle differences in seismic amplitude. The logistic map is a fixed-point iteration system with one control parameter. To establish the relationship between seismic data and the logistic map, we define the stable fixed point of seismic data based on the Banach contraction principle and Cauchy convergence theorem, and we derive an equation for adaptively searching for the optimal control parameter of the logistic map based on the data’s stable fixed point. According to such an equation, we design a workflow to automatically generate seismic convergence rate for an input data. The seismic convergence rate is essentially the stable fixed-point image of seismic data, which is characterized by a fine structure and interpreted as the data’s invariant set. We use numerical experiments to illustrate the characteristics of seismic convergence rate, and we use the Marmousi model experiment to demonstrate the effectiveness of the seismic convergence rate on detecting subtle edges of seismic data. Then, we use real data from two typical carbonate exploration areas in China, the Central Tarim Basin and the Ordos Basin, respectively, to show the abilities of the seismic convergence rate in detecting hidden seismic facies and in detecting subtle edges in high-coherence zones, as well as in extracting the seismic invariant set.