The first and second laws of geography have been applied to the simulation of urban growth in many studies. However, by focusing on the spatial complexity of urban growth, these studies have the shared problem of ignoring the temporal complexity of urban growth, which can be solved by incorporating historical information into the simulation of urban growth. In this paper, we describe how we constructed a Logistic-CA model using smoothed transition rules (the SM-Logistic-CA model). Specifically, in this paper, we: 1) propose an expansion similarity index to measure the similarity of the urban expansion processes in two periods; 2) use linear smoothing and exponential smoothing to integrate the historical transition rules; 3) assign smoothing weights to each period based on the expansion similarity index; and 4) compare the SM-Logistic-CA model with the standard Logistic-CA model. The results show that the SM-Logistic-CA model can exhibit good control of urban growth, and can avoid the problem of new urban land expanding blindly along the original urban land when smoothing is performed using the transition rules of appropriate historical periods. The similarity of the expansion processes between the historical period and the target period and the temporal distance of the historical period from the target period affect the simulation accuracy of the SM-Logistic-CA model, and the neighborhood size changes the relative importance of these two factors on the simulation results.

在许多研究中，地理学第一和第二定律已应用于模拟城市增长。然而，这些研究关注城市增长的空间复杂性，存在忽略城市增长时间复杂性的共同问题，可以通过将历史信息纳入城市增长模拟来解决。在本文中，我们描述了我们如何使用平滑转换规则（SM-Logistic-CA 模型）构建 Logistic-CA 模型。具体而言，在本文中，我们： 1）提出了一个扩张相似性指数来衡量两个时期城市扩张过程的相似性；2）使用线性平滑和指数平滑来整合历史过渡规则；3）根据扩展相似度指数为每个周期分配平滑权重；4) 将 SM-Logistic-CA 模型与标准 Logistic-CA 模型进行比较。结果表明，SM-Logistic-CA模型对城市增长表现出较好的控制能力，在采用适当历史时期的过渡规则进行平滑时，可以避免新城市用地沿原有城市用地盲目扩张的问题。历史时期与目标时期扩张过程的相似性以及历史时期与目标时期的时间距离影响SM-Logistic-CA模型的模拟精度，邻域大小改变了这两者的相对重要性影响仿真结果的因素。利用适当的历史时期的过渡规则进行平滑，可以避免新的城市用地沿原有城市用地盲目扩张的问题。历史时期与目标时期扩张过程的相似性以及历史时期与目标时期的时间距离影响SM-Logistic-CA模型的模拟精度，邻域大小改变了这两者的相对重要性影响仿真结果的因素。利用适当的历史时期的过渡规则进行平滑，可以避免新的城市用地沿原有城市用地盲目扩张的问题。历史时期与目标时期扩张过程的相似性以及历史时期与目标时期的时间距离影响SM-Logistic-CA模型的模拟精度，邻域大小改变了这两者的相对重要性影响仿真结果的因素。