## 4 Discussion  
From the above numerical results, we can see how each of the four variables (\(\gamma\), \(d/D\), \(F_{N}\), and \(R\)) influence the curve shapes of PWFs, which is helpful in the creation of a desired PWF using the linear combination method (Equation (2)). When we focused on the PWFs for different \(F_{N}\) values without changing the other variables, we found that the effect of diffraction on the covariance of the tilt phase or piston amplitude is far smaller than that on the variance. Note that the PWFs investigated here only considered the common case where one source is used and the receive apertures are unobscured circles. When the shape of the receive aperture is changed or two sources are used, the curve shapes of the related PWFs will also change, as in the case of MASS [13] (Multi-Aperture Scintillation Sensor), where the receive aperture is annulus, or SloDAR [14] (Slope Detection And Ranging) and SciDAR [15,16] (Scintillation Detection And Ranging), where two sources are used. Thus, there are potentially a large number of various \(W_{i}(u)\) values that we can obtain, which would make the linear combination method more powerful. In some applications, the PWFs of \(P_{d}\) (the desired turbulence integral parameter) have complex analytic expressions, most of which can only be evaluated numerically, and \(P_{d}\) is often difficult to measure at present. For example, in the angular or focal anisoplanatic problems in adaptive optics, the effective phase variance \(\sigma^{2}{}_{Eff}\) (the total phase variance \(\sigma^{2}{}_{\phi}\) with the piston removed or both the piston \(\sigma^{2}{}_{P}\) and the tilt \(\sigma^{2}{}_{TA}\) removed, i.e., \(\sigma^{2}{}_{Eff}=\sigma^{2}{}_{\phi}-\sigma^{2}{}_{P}\) (\(-\sigma^{2}{}_{TA}\))) is such a \(P_{d}\)[3, 7, 8, 9]. For natural guide star adaptive optics (NGS AO), it is well known that \(\sigma^{2}{}_{\phi}=(\theta/\theta_{0})^{5/3}\), and one can easily derive \(\sigma^{2}{}_{P}=4.25\)\(\sigma^{2}{}_{TA}=4.25\)\((\theta/\theta_{TA})^{2}\) for small angles, where \(\theta_{TA}\) is the tilt isoplanatic angle [1, 3, 17]. Therefore, if \(\theta_{TA}\) can be measured along with \(\theta_{0}\), one can obtain the effective phase variance in NGS AO using the relationship \(\sigma^{2}{}_{Eff}=(\theta/\theta_{0})^{5/3}-4.25(\theta/\theta_{TA})^{2}\) (\(-(\theta/\theta_{TA})^{2}\)). As far as we know, the direct measurement of \(\theta_{TA}\) has not been reported. Based on the linear combination method proposed in this paper, we have proposed one direct measurement scheme of \(\theta_{TA}\) in our other accompanying manuscript [18]. In some other applications, though expressions for the PWF of \(P_{d}\) are simple, it cannot be measured under non-ideal conditions. For example, \(r_{0,p}\) and \(\theta_{0}\) in a finite distance cannot be measured using the conventional method (DIMM and stellar scintillation isoplanometer) since the light wave propagating in the turbulence path is not a plane wave. Generally, the light wave is a spherical wave for a point beacon in a finite distance. Then, the parameter measured using DIMM is actually \(r_{0,s}\) rather than \(r_{0,p}\), and the parameter measured using the conventional isoplanometer has a PWF with a curve shape similar to the red dashed line in Figure 5a rather than the red dashed line in Figure 3a (i.e., the well-known \(u^{5/3}\)). The linear combination method shows promise of estimating the real-time \(P_{d}\) with high accuracy if the suitable \(W_{i}(u)\) can be found. As an application example of this method, one can measure the isoplanatic angle in a finite distance through the combination of one spherical wave scintillation and two covariances of intensity in three receive apertures [10], and the validity of this method was proven by real data from the validation experiment [19]. One can also roughly estimate \(P_{d}\) through the optical turbulence profile obtained from those turbulence profilers [12, 13, 14, 15, 16].