Hypothesis Distance equation Space-modified time equation Growth rate equation Number of additional parameters
N1 Null NA NA \(r_{t,i}\,=\alpha_{1}+\ \epsilon_{t,i}\) NA
N2 Phase-locked NA NA \(r_{t,i}=\alpha_{1}+f\left(T_{t,i}\right)\ +\ \epsilon_{t,i}\) NA
N3 Static spatial pattern NA NA \(r_{t,i}=\alpha_{1}+f_{p}\left(X_{t,i}Y_{t,i}\right)\ +\ \epsilon_{t,i}\) NA
RE Single expanding radial wave \(D_{i}=-\sqrt{\left(\gamma-X_{i}\right)^{2}+\left(\psi-Y_{i}\right)^{2}}\) \(\rho_{t,i}=T_{t,i}+\left(\frac{1}{\zeta}\right)D_{i}\) \(r_{t,i}=\alpha_{1}+f\left(\rho_{t,i}\right)+\epsilon_{t,i}\) 3
RC Single contracting radial wave \(D_{i}=\sqrt{\left(\gamma-X_{i}\right)^{2}+\left(\psi-Y_{i}\right)^{2}}\) \(\rho_{t,i}=T_{t,i}+\left(\frac{1}{\zeta}\right)D_{i}\) \(r_{t,i}=\alpha_{1}+f\left(\rho_{t,i}\right)+\epsilon_{t,i}\) 3
P Single planar wave \(D_{i}=\sin\left(\theta\right)X_{i}+\cos\left(\theta\right)Y_{i}\) \(\rho_{t,i}=T_{t,i}+\left(\frac{1}{\zeta}\right)D_{i}\) \(r_{t,i}=\alpha_{1}+f\left(\rho_{t,i}\right)+\epsilon_{t,i}\) 2
RFE Two expanding radial waves separated by river \(D_{N,i}=-\sqrt{\left(\gamma_{N}-X_{i}\right)^{2}+\left(\psi_{N}-Y_{i}\right)^{2}},\ \ \ \ \ \ if\text{\ Y}_{i}\geq 5,068m\) \(\rho_{N,t,i}=T_{t,i}+\left(\frac{1}{\zeta_{N}}\right)D_{N,i}\) \(r_{t,i}=\alpha_{1}+f_{\text{North}}\left(\rho_{N,t,i}\right)+f_{\text{South}}\left(\rho_{S,t,i}\right)+\epsilon_{t,i}\) 6
\(D_{S,i}=-\sqrt{\left(\gamma_{S}-X_{i}\right)^{2}+\left(\psi_{S}-Y_{i}\right)^{2}},\ \ \ \ \ \ if\ Y_{i}<5,068m\) \(\rho_{S,t,i}=T_{t,i}+\left(\frac{1}{\zeta_{S}}\right)D_{S,i}\)
RDE Dual overlapping expanding radial waves \(D_{1,i}=-\sqrt{\left(\gamma_{1}-X_{i}\right)^{2}+\left(\psi_{1}-Y_{i}\right)^{2}}\) \(\rho_{A,t,i}=T_{t,i}+\left(\frac{1}{\zeta_{1}}\right)D_{1,i}\) \(r_{t,i}=\alpha_{1}+f_{1}\left(\rho_{1,t,i}\right)+f_{2}\left(\rho_{2,i,t}\right)+\epsilon_{i,t}\) 6
\(D_{2,i}=-\sqrt{\left(\gamma_{2}-X_{i}\right)^{2}+\left(\psi_{2}-Y_{i}\right)^{2}}\) \(\rho_{I,t,i}=T_{t,i}+\left(\frac{1}{\zeta_{2}}\right)D_{2,i}\)
RFC Two contracting radial waves separated by river \(D_{N,i}=\sqrt{\left(\gamma_{N}-X_{i}\right)^{2}+\left(\psi_{N}-Y_{i}\right)^{2}},\ \ \ \ \ \ if\text{\ Y}_{i}\geq 5,068m\) \(\rho_{N,t,i}=T_{t,i}+\left(\frac{1}{\zeta_{N}}\right)D_{N,i}\) \(r_{t,i}=\alpha_{1}+f_{\text{North}}\left(\rho_{N,t,i}\right)+f_{\text{South}}\left(\rho_{S,t,i}\right)+\epsilon_{t,i}\) 6
\(D_{S,i}=\sqrt{\left(\gamma_{S}-X_{i}\right)^{2}+\left(\psi_{S}-Y_{i}\right)^{2}},\ \ \ \ \ \ if\ Y_{i}<5,068m\) \(\rho_{S,t,i}=T_{t,i}+\left(\frac{1}{\zeta_{S}}\right)D_{S,i}\)
RDC Dual overlapping contracting radial waves \(D_{1,i}=\sqrt{\left(\gamma_{1}-X_{i}\right)^{2}+\left(\psi_{1}-Y_{i}\right)^{2}}\) \(\rho_{1,t,i}=T_{t,i}+\left(\frac{1}{\zeta_{1}}\right)D_{1,i}\) \(r_{t,i}=\alpha_{1}+f_{1}\left(\rho_{1,t,i}\right)+f_{2}\left(\rho_{2,i,t}\right)+\epsilon_{i,t}\) 6
\(D_{2,i}=\sqrt{\left(\gamma_{2}-X_{i}\right)^{2}+\left(\psi_{2}-Y_{i}\right)^{2}}\) \(\rho_{2,t,i}=T_{t,i}+\left(\frac{1}{\zeta_{2}}\right)D_{2,i}\)
PF Two planar waves separated by river \(D_{N,i}=\sin\left(\theta_{N}\right)X_{i}+\cos\left(\theta_{N}\right)Y_{i},\ \ \ \ \ \ if\text{\ Y}_{i}\geq 5,068m\) \(\rho_{N,t,i}=T_{t,i}+\left(\frac{1}{\zeta_{N}}\right)D_{N,i}\) \(r_{t,i}=\alpha_{1}+f_{\text{North}}\left(\rho_{N,t,i}\right)+f_{\text{South}}\left(\rho_{S,t,i}\right)+\epsilon_{t,i}\) 4
\(D_{S,i}=\sin\left(\theta_{S}\right)X_{i}+\cos\left(\theta_{S}\right)Y_{i},\ \ \ \ \ \ if\ Y_{i}<5,068m\) \(\rho_{S,t,i}=T_{t,i}+\left(\frac{1}{\zeta_{S}}\right)D_{S,i}\)
PD Dual overlapping planar waves \(D_{1,i}=\sin\left(\theta_{1}\right)X_{i}+\cos\left(\theta_{1}\right)Y_{i}\) \(\rho_{A,t,i}=T_{t,i}+\left(\frac{1}{\zeta_{1}}\right)D_{1,i}\) \(r_{t,i}=\alpha_{1}+f_{1}\left(\rho_{1,t,i}\right)+f_{2}\left(\rho_{2,t,i}\right)+\epsilon_{i,t}\) 4
\(D_{2,i}=\sin\left(\theta_{2}\right)X_{i}+\cos\left(\theta_{2}\right)Y_{i}\) \(\rho_{I,t,i}=T_{t,i}+\left(\frac{1}{\zeta_{2}}\right)D_{2,i}\)