Evaluating a K-Means Cluster Model Grouping Atmospheric Modified Refractivity

Sharp vertical gradients in humidity and temperature in the atmosphere refract electromagnetic (EM) energy incident at sufficiently low grazing angles, which can impact the performance of radio frequency (RF) systems operating at microwave frequencies. The magnitude of the refractive effect is typically summarized using modified refractivity, defined by Bean and Dutton, (1968):

𝑀 = 77.6/𝑇 [𝑝 + 4810𝑒/𝑇] + (𝑧/𝑎) × 10⁶ (1)

Where 𝑇 is temperature in Kelvin, 𝑝 is pressure in millibars, 𝑒 is partial vapor pressure also in millibars, 𝑎 is the radius of the earth in meters, and 𝑧 is altitude in meters. Negative modified refractivity gradients result in trapping or ducting of the RF energy down towards the earth's surface, which can enhance effective ranges for RF systems operating in certain configurations. Alternatively, sufficiently positive modified refractivity gradients deflect the energy away from the Earth, typically leading to reduced ranges. To estimate refractivity and the subsequent trapping layers, numerical weather prediction (NWP) models are implemented to forecast refractivity over large domains spanning environments. Due to the large domain sizes, forecasts of refractivity can vary spatially but over the relatively homogenous marine environment, negligible variability in refractivity can exist within a forecasted space. Thus, grouping like refractive environments could be an effective method for limiting the number of refractive profiles needed to describe a specific environment relative to RF propagation.

To this end, this study aims to compute singular profiles of refractivity that accurately represent NWP based forecasts of refractivity using a machine learning technique to group the data. Furthermore, the singular profiles representing grouped refractivity are evaluated, relative to grouped refractivity, to highlight differences in both refractive and propagation space. Large scale refractivity forecasts from the Coupled Ocean Atmosphere Mesoscale Prediction System (COAMPS®; Karimian et al., 2013) necessitate correspondingly large number of EM propagation simulations for use with sensor performance assessments. To reduce redundancy while preserving fidelity of these simulations, a technique to group relevant meteorological parameters into effectively similar classes was developed using a K-means++ clustering technique. To represent grouped refractivity, mean profiles are computed for each of the refractivity clusters and are evaluated to ensure proper representation in refractive space. Error between mean and grouped refractivity profiles, capturing variations in vertical gradients of refractivity between profiles, is computed. Normality of these error distributions is assessed via Kolmogorov-Smirnov test to ensure the assumption that the mean profile is an appropriate estimation of a refractive group. For cases where error distributions are found to be non-normal, outlier profiles are removed, and the mean profile is recomputed.

Further investigation into adequacy of the mean profile to represent grouped refractivity is performed via comparison of S-Band propagation simulations. Propagation simulations are performed using the Tropospheric Electromagnetic Parabolic Wave Equation Routine (TEMPER; Dockery, 1998) at a frequency of 3 GHz, for a vertically polarized wave, a transmitter height of 10m, and over a smooth sea surface. Propagation is compared between representative refractivity profiles and randomly sampled refractive profiles for all groups of refractivity (~ 40,000 groups) to determine similarity between propagation predations. Results illustrate that a RF propagation from mean representative profiles is very similar to grouped refractivity profiles, ultimately indicating that a mean profile is an adequate representation of K-means grouped refractivity.