## 2 Dynamic Lognormal Shadowing Framework
### Random Field of Shadowing
In this case (5) is transformed to:  
\[C(t_{1},t_{2})=e^{-\frac{|\mathbf{y}|_{2}-\mathbf{x}_{1}|}{r_{c}}}\sigma^{2}= e^{-al[t_{2}-t_{1}]}\sigma^{2} \tag{6}\]  
where \(a=-\frac{|\mathbf{y}|}{r_{c}}\), introducing thus the temporal autocorrelation as:  
\[\rho(t_{1},t_{2})=e^{-al[t_{2}-t_{1}]} \tag{7}\]  
However, movement is a relative phenomenon and the influence of moving obstacles to shadowing for fixed users can also be taken into account in a similar manner. We may thus attribute movement to obstacles and consider a fixed user. We introduce for a fixed point \(\mathbf{X}\) the temporal autocorrelation of (7), which assuming a location specific variance \(\sigma(\mathbf{x})\) and dynamic parameter \(a(\mathbf{x})\) results in the following temporal autocovariance:  
\[C_{\mathbf{x}}(t_{1},t_{2})=e^{-a(\mathbf{x})|t_{2}-t_{1}|}\sigma(\mathbf{x} )^{2} \tag{8}\]  
In order to express the temporal variability along with the spatial for two fixed locations \(\mathbf{x}_{1},\mathbf{x}_{2}\) in two different time instants \(t_{1},t_{2}\) we notice that the choice of the exponential temporal autocorrelationfor each location implies that the shadowing process is Gaussian and Markov. Thus, without loss of generality, assuming \(t_{1}<t_{2}\), we can express the shadowing r.v. \(S(\mathbf{x}_{2},t_{2})\) in terms of \(S(\mathbf{x}_{2},t_{1})\) as:  
\[S(\mathbf{x}_{2},t_{2})=e^{-a(\mathbf{x}_{2})|t_{2}-t_{1}|}S(\mathbf{x}_{2},t_ {1})+\sigma(\mathbf{x}_{2})\sqrt{1-e^{-2a(\mathbf{x}_{2})|t_{2}-t_{1}|}}N \tag{9}\]  
where \(N\) is a standard normal r.v. independent of the shadowing parameters. By using (9) we find that the covariance function of the shadowing random field \(S(\mathbf{x},t)\) is expressed as  
\[C(\mathbf{x}_{1},t_{1},\mathbf{x}_{2},t_{2})=e^{-a(\mathbf{x}_{2})|t_{2}-t_{1 }|}\rho(\mathbf{x}_{1},t_{1},\mathbf{x}_{2},t_{1})\sigma(\mathbf{x}_{1})\sigma (\mathbf{x}_{2}) \tag{10}\]  
where stationarity is assumed. The correlation coefficient can be chosen as:  
\[p(\gamma)=\frac{\xi}{\gamma\sqrt{2\pi}\sigma}e^{-\frac{(10\log\gamma-\mu)}{2 \sigma^{2}}} \tag{2}\]0  
and in this case (10) describes the temporal evolution of auto-correlation of shadowing for a fixed user. A similar reasoning has been used in [35] for the case of cross-correlation of shadowing affecting a mobile user. With the selection of an appropriate correlation coefficient, the covariance function (10) can be easily modified to include cross-correlation of shadowing and any correlation exhibited by adjacent links. We assume \(m\) base stations and the vector shadowing random field \(\mathbf{S}(\mathbf{x},t)=[S_{1}(\mathbf{x},t),\dots,S_{m}(\mathbf{x},t)]\), where \(S_{i}(\mathbf{x},t)\), \(i=1,\dots,m\) is the shadowing process affecting a station located at \(\mathbf{x}\) receiving the signal from base station \(i\), \(i=1,\dots,m\). The vector random field is completely described by the cross-covariances:  
\[p(\gamma)=\frac{\xi}{\gamma\sqrt{2\pi}\sigma}e^{-\frac{(10\log\gamma-\mu)}{2 \sigma^{2}}} \tag{2}\]1  
where \(\rho_{ij}(\mathbf{x}_{1},t_{1},\mathbf{x}_{2},t_{2})\) is the correlation coefficient between the shadowing processes at points \(\mathbf{x}_{1},\mathbf{x}_{2}\) for the time instants \(t_{1},t_{2}\) respectively and for the signals from base stations \(i,j\). It may be chosen in a general setting to express autocorrelation, cross-correlation of shadowing, as well as correlation between adjacent links without a common node. The model proposed here assumes the exponential temporal correlation model:  
\[p(\gamma)=\frac{\xi}{\gamma\sqrt{2\pi}\sigma}e^{-\frac{(10\log\gamma-\mu)}{2 \sigma^{2}}} \tag{2}\]2  
Stationarity is also assumed by considering \(\rho_{ij}(\mathbf{x}_{1},t_{1},\mathbf{x}_{2},t_{1})\) independent of time. Except for these prerequisites any consistent correlation model may be used.