## 2 Materials and Methods  
The turbulence integral parameter noted as \(P\) is written as follows:  
\[P=CL\int_{0}^{1}C_{n}^{2}(uL)W(u)du \tag{1}\]  
where \(C\) is a constant related to \(P\), \(L\) is the propagation path length through turbulence, \({C_{n}}^{2}\) is the turbulence strength, \(W(u)\) is the PWF, and \(u=z/L\) denotes the normalized position along the path with propagation from \(u=1\) to \(u=0\) at the receiver. The principle of the linear combination method is that any linear combination of turbulence integral parameters corresponds to the linear combination of their PWFs. The desired integral parameter \(P_{d}\) can be estimated from measurable integral parameters \(P_{i}\) if only the PWF of \(P_{d}\), noted as \(W_{d}(u)\), can be approximated by linear combination of the PWFs of \(P_{i}\), noted as \(W_{i}(u)\), i.e.,  
\[W_{d}(u)\approx\sum_{i=1}^{N}a_{i}W_{i}(u) \tag{2}\]  
where \(a_{i}\) are the coefficients of \(W_{i}(u)\), and \(N\) is the number of measurable integral parameters used in this method. In practical applications, one should first select a suitable \(W_{i}(u)\) based on the shape of \(W_{d}(u)\) and then fit \(W_{i}(u)\) to \(W_{d}(u)\) using the least square fitting method. Equation (2) shows that the more measurable integral parameters and their PWFs we know, the more desired PWFs we can approximate using this method and, thus, the more integral parameters we can estimate. Therefore, it is important to investigate more measurable quantities and their PWFs. Atmospheric turbulence causes phase and amplitude perturbations, and at present, only the tilt phase and piston amplitude, corresponding to the angle of arrival and the intensity of the light collected with an aperture, can be measured directly and relatively easily. In practice, the variance of the angle of arrival and the variance of intensity are used to obtain turbulence information through their PWFs, which is exactly the case in the DIMM and the stellar scintillation isoplanometer, respectively. However, the covariance functions for the angle of arrival or intensity provide more turbulence information due to their potentially large number of PWFs. The covariance of the tilt phase can be measured from the covariance of the angle of arrival [3; 4], and the covariance of the piston amplitude can be measured from the covariance of intensity [6; 10]. Assuming the Rytov approximation and Kolmogorov turbulence, using the analytic approach developed by [PERSON] [3], the covariance functions for the phase and the log-amplitude related quantities can be derived as follows:  
\[\left[ \begin{array}{c}C_{\phi}(d)\\ C_{\chi}(d)\end{array} \right]=0.132\pi^{2}k^{2}L\int_{0}^{1}C_{n}^{2}(uL)\left[ \begin{array}{c}W_{C,\phi}(u)\\ W_{C,\chi}(u)\end{array} \right]du \tag{3}\]  
where \(\phi\) and \(\alpha\) denote the phase and the log-amplitude related quantities, respectively; \(d\) is the center-to-center distance of two receive apertures; the optical wavenumber \(k=2\pi/\lambda\)where \(\lambda\) is the wavelength of the observed beacon; and \(W_{C,\phi}(u)\) and \(W_{C,\chi}(u)\) are path-weighting functions which are given by the following:  
\[\left[ \begin{array}{c}W_{C,\phi}(u)\\ W_{C,\chi}(u)\end{array} \right]=\int_{0}^{\infty}\kappa F(\gamma\kappa)J_{0}( \gamma\kappa d)\kappa^{-11/3}\!\left[ \begin{array}{c}\cos^{2}(\frac{x^{2} \gamma uL}{2k})\\ \sin^{2}(\frac{x^{2}\gamma uL}{2k})\end{array} \right]dx \tag{4}\]  
where \(\kappa\) is the spatial wavenumber transverse to the \(z\) direction; \(\gamma\) is the propagation parameter, which depends on the geometric divergence of the light wave and has the simple value of \(\gamma=1\) for plane waves and \(\gamma=1-z/L\) for spherical waves; \(J_{n}\) is the \(n\)th order Bessel function of the first kind; and \(F(\gamma\kappa)\) is the filter function, which depends on propagation geometry, such as the size of the receive aperture and the distributed source, and measured related quantities, such as the tilt phase or piston amplitude.