Uncertainty Analysis of Spatial Accessibility to General Hospitals in Isfahan City Based on the Three-Step Floating Catchment Area (3SFCA) Method

ABSTRACT
The Three-Step Floating Catchment Area (3SFCA) method is a widely used approach for evaluating spatial accessibility to services, incorporating two critical parameters: catchment size and distance decay function. However, variations in these parameters can introduce significant uncertainty into accessibility estimates. This study investigates the extent of uncertainty in spatial accessibility to general hospitals in Isfahan by applying the 3SFCA method. Accessibility uncertainty was assessed using the coefficient of variation (CV), derived from the mean and standard deviation of accessibility scores. To evaluate the impact of parameter variability, catchment sizes were randomly varied, from a minimum of 5 minutes to the maximum observed travel time between each census block and the farthest hospital, and different distance decay functions were applied. Using 100 Monte Carlo simulations, uncertainty was calculated for each census block under four types of distance decay functions as kernel, cumulative, exponential, and Gaussian. A Hot Spot Analysis was conducted to identify statistically significant clusters of high and low uncertainty. The kernel function produced the most stable results. Spatially, clusters of high uncertainty were concentrated in the eastern and southeastern districts of Isfahan, while areas with low uncertainty were primarily located in the central and southwestern parts of the city. Overall, regions with lower accessibility to general hospitals were found to exhibit greater levels of uncertainty.
Extended Abstract
Introduction
Assessing spatial accessibility to healthcare services is a fundamental concern in urban and regional planning, particularly where equitable access is directly tied to public health outcomes and social justice. Among various approaches developed to evaluate accessibility, the Three-Step Floating Catchment Area (3SFCA) method has emerged as one of the most comprehensive and widely applied techniques. The 3SFCA method relies on two key parameters as catchment size and distance decay function. While these inputs provide a more realistic representation of how people access healthcare services compared with traditional methods, they are inherently uncertain. Variability in catchment size, which indicates the travel time or distance that individuals are willing or able to cover, along with the selection of various distance decay functions, illustrates how service utility diminishes with distance and can significantly affect the resulting accessibility estimates. Consequently, results derived from the 3SFCA method may be subject to uncertainty, which, if left unexamined, can reduce the reliability of research findings and their usefulness for evidence-based policymaking.
This study quantifies and analyzes the uncertainty associated with spatial accessibility to general hospitals in Isfahan City, Iran,  using the 3SFCA method. Monte Carlo simulations were employed to introduce random variations in catchment size and to apply multiple distance decay functions, capturing a realistic spectrum of possible conditions. By using the coefficient of variation (CV) as a statistical measure, the research not only evaluates accessibility but also assesses the degree of confidence in those estimates. The novelty of this study lies in explicitly quantifying uncertainty, an aspect often overlooked in similar research despite its significant implications for urban health equity and spatial planning.
 
Methodology
This research integrates demographic, infrastructural, and spatial datasets. Population distribution data at the census block-level were obtained for Isfahan city, while information on 23 public hospitals, including the number of active beds represented the supply side. The street network was derived from OpenStreetMap (OSM) to calculate travel times between demand and supply points, ensuring that accessibility estimates reflect real transportation conditions rather than simple Euclidean distances. The 3SFCA method was implemented through three sequential steps. First, catchment areas were delineated around each census block based on travel time thresholds. To account for variability, catchment sizes were randomly varied from a minimum of five minutes, representing the shortest plausible travel time, to the maximum observed travel time between each block and the farthest hospital. Second, supply-to-demand ratios were computed for each hospital within its catchment area by dividing available active beds by the population of surrounding census blocks. Third, accessibility scores for each demand block were calculated by aggregating these supply-to-demand ratios while applying different distance decay functions: Gaussian, kernel, exponential, and cumulative, each representing distinct assumptions about how accessibility decreases with distance. Specifically, the Gaussian function models a nonlinear decline, the kernel function follows a parabolic trend, the exponential function decreases rapidly at short distances, and the cumulative function represents a linear decline.
Monte Carlo simulations were then conducted to capture uncertainty systematically. A total of 100 simulation runs were performed, each generating a unique configuration of catchment sizes and decay function. For each block, the coefficient of variation was computed by dividing the standard deviation of accessibility scores by their mean, quantifying the relative variability of accessibility estimates. In addition, Hot Spot analysis using the Getis-Ord Gi* statistic was applied to the CV values to identify spatial clusters of significantly high or low uncertainty across the city.
 
 
Results and discussion
Among the four tested distance decay functions, the kernel function demonstrated the lowest uncertainty, with an average CV of 60.68, indicating relatively stable accessibility estimates.  In contrast, the exponential function showed the highest level of uncertainty. The central areas of Isfahan exhibited higher accessibility and lower uncertainty, whereas peripheral zones, particularly in the eastern and southeastern districts, showed lower accessibility and higher uncertainty. Hot Spot analysis revealed clusters of high uncertainty in the eastern and southeastern parts of the city, mainly due to the sparse distribution of hospitals.  Conversely,  clusters of low uncertainty were observed in the central and southwestern areas, where hospital density is greater. Correlation analysis confirmed an inverse relationship between accessibility and uncertainty, indicating that areas with lower accessibility tend to exhibit greater variability.
 
Conclusion
This study demonstrates the significant influence of input parameters on spatial accessibility analysis using the 3SFCA method. Among the tested distance decay functions, the kernel function proved to be the most reliable in minimizing uncertainty. Areas characterized by high uncertainty in Isfahan corresponded to lower hospital density and limited accessibility. Future research should evaluate the impact of establishing new hospitals on both accessibility and uncertainty, particularly in underserved districts.
 
Funding
This research did not receive any specific grant from funding agencies in the public, commercial, or not for profit sectors.
 
Author Contribution
Hassan Atashgah: Data curation, Visualization, Methodology, Drafting.
Babak Mirbagheri: Methodology, Editing, Supervision, Validation.
Alireza Shakiba: Methodology, Editing, Supervision, Validation,
Ali Akbar Matkan: Methodology, Editing, Supervision, Validation.

Conflict of Interest
The authors declare no conflict of interest.
 
Acknowledgments
The authors gratefully acknowledge the valuable guidance and construvtive feedback provided by scientific consultants who contributed to the completion of this study.