# Development of a GPS Forest Signal Absorption Coefficient Index
## 3 Results  
The models presented below are the results of regression modeling. This process consisted of initial data exploration of many different variables other than those presented. During the analysis, we identified ways to linearize the relationships within each model. Four models are presented and include an atmospheric model, a model optimized using HSOPs, and two models using dummy variables for each forest to generate an absorption index associated with the different forests. The first analysis conducted used all the open receiver GPS observations to create an overall GPS SNR atmospheric model. In this work, we built on the previous work where the natural log of the angle from the horizon (_lnel_) of the GPS SV was the key parameter in the modeling process [21]. The resulting overall GPS L-band SNR atmospheric model (SAM) has a Root Mean Square Error (RMSE) of 2.01 dB and an adjusted \(R^{2}\) of 0.81. The SAM equation applies to all observations where no vegetation is present over the GPS receiver. The SAM equation is:  
\[SNR_{open}=7.79\ (lnel)+18.85 \tag{2}\]  
When exploring how different forests influence GPS signal, we incorporated the use of the CCPM. The CCPM approach to model SNR incorporates GPS observations from all forest study sites. The CCPM uses two variables. The first variable is \(lnel\), as in the SAM equation, which linearizes the problem and is also vital in modeling the atmospheric component of the observed SNR. The other variable in the CCPM is the Beer's Law component, the product of CC and the sine of the zenith angle. Table 2 shows the results of the CCPM for each individual forest, with the equations taking the following form:  
\[SNR=a+B_{1}\ (lnel)+B_{2}\ CC\ \sin(ZA) \tag{3}\]  
where \(lnel\) is the natural log of the angle of the SV above the horizon, \(ZA\) is the zenith angle, and \(SNR\) units are in decibels. The last model we developed incorporated all aspects of the CCPM and also included dummy variables associated with each different forest site. The resulting dummy variables simply provide a Y-intercept shift for the expected SNR value based on a particular forest. The resulting dummy variable coefficients for each specific forest provide absorption indexes that help establish an understanding of how each forest affects the reception of GPS signals. The resulting model is termed the SNR forest \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**ID** & **City Vicinity** & **a** & **B\({}_{1}\)** & **B\({}_{2}\)** & **RMSE** & **Adj \(R^{2}\)** \\ \hline
1 & West Point, NY & 18.85 & 7.79 & \(-5.53\) & 3.28 & 0.60 \\
2 & IMPAC & 19.32 & 7.79 & \(-5.49\) & 2.78 & 0.71 \\
3 & Hogtown Forest, Gainesville, FL & 25.05 & 5.26 & \(-6.02\) & 3.02 & 0.66 \\
4 & Charleston, SC & 27.87 & 4.25 & \(-7.00\) & 3.10 & 0.64 \\
5 & Alexandria, LA & 25.77 & 4.87 & \(-5.24\) & 3.03 & 0.60 \\
6 & Cold Spring, TX & 26.89 & 5.61 & \(-16.35\) & 2.80 & 0.59 \\
7 & Georgetown, TX & 23.94 & 6.25 & \(-8.02\) & 3.71 & 0.61 \\
8 & Cloudcroft, NM & 25.71 & 5.99 & \(-6.99\) & 3.77 & 0.60 \\
9 & Flagstaff, AZ & 21.70 & 6.70 & \(-0.50\) & 3.33 & 0.57 \\
10 & Guadalupe, CA & 28.83 & 5.08 & \(-8.81\) & 2.66 & 0.66 \\
11 & San Luis Obispo & 26.26 & 5.79 & \(-29.39\) & 3.75 & 0.60 \\
12 & Davenport, CA & 27.50 & 5.46 & \(-6.03\) & 2.83 & 0.72 \\
13 & Davenport, CA & 27.50 & 3.15 & \(-14.49\) & 3.02 & 0.70 \\
14 & Tahoe NF & 30.03 & 4.56 & \(-8.59\) & 3.24 & 0.70 \\
15 & Nederland, CO & 31.04 & 4.79 & \(-12.99\) & 2.83 & 0.74 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Canopy Closure Predictive Model Results with coefficients \(a,\ B_{1}\), and \(B_{2}\) are in reference to Equation (3) and the root Mean Square Error (RMSE). index model (SFIM). The SFIM had an RMSE of 3.24 dB and an adjusted \(R^{2}\) of 0.65, with the SFIM equation is as follows:  
\[SNR=7.79\ ( Intel)-0.26 CC\ \sin(ZA)+18.85+I \tag{4}\]  
where \(I\) is the index value. Inspection of the SFIM equation shows that the Beer's Law component has a very small influence on the model. This result was unexpected initially.