Exploring the Spectrum Power Fractal Scaling Parameters by Hurst Range Increment - Second Order Moment Generation Techniques

Spectral power analysis was employed to assess the Fractal Dimension (FD) and explore fractal scaling using Hurst increment ranges and second-order moment relations in the context of urban population trends. This research aimed to scrutinize population trends in Karachi over both uneven periods (1729 to 1946) and even periods (1951 to 2020) using non-parametric Mann-Kendall tests and Hurst error accuracy testing. The primary focus was on analyzing spectrum power fractal scaling through Hurst exponent ranges and second-order moment generation.The FD results indicated irregular (1.371) and regular (1.058) intervals within theinequality range of 1< D< 1.5. The log-population trend cumulatively increasedfrom 3.0in1729 to 5.72 in 1946,and from 6.05in1951 to 7.36 in 2020, suggesting that the fractal dimension is more appropriatelyfitted for total regular intervals. The second-order and range exponents wereH2ndM (0.60± 0.09) and H-Range (0.83 ± 0.05) for theuneven period (1729 to 1946), andH2ndM (0.85 ± 0.06) andH-Range (0.93 ± 0.02) for theeven period (1951 to 2020).The study's results demonstrate that the range increment method is suitable and consistent across both long and short intervals. For regular intervals, the Hurst exponents show a linear relationship, indicating stability in the population trend analysis.