First, to modify the classical Egli model, we introduce some constant free adaption parameters into (9), resulting in eq (10):
$$ {Pl}_{Egli}\left(\mathrm{dB}\right)={A}_1+{A}_2{\log}_{10}\left({f}_{t(MHz)}\right){A}_3{\log}_{10}\left({H}_{tr}\right){A}_4{\log}_{10}\left({H}_r\right)+{A}_5{\log}_{10}\left({d}_c\right) $$
(10)
Now, let Y_{m}define the measured signal propagation loss and \( {\hat{Y}}_m \) be the Egli model with A_{1},A_{2}, A_{3}, A_{4}, andA_{5} being the adaptation parameters to be obtained based on the field signal propagation measurements in a nonlinear square sense. It is given by eq (11):
$$ H=\sum \limits_{m=1}^n{\left({Y}_m{\hat{Y}}_m\right)}^2=\sum \limits_{m=1}^n{\left({Y}_m{A}_1{A}_2{\log}_{10}\left({f}_{t(MHz)}\right)+{A}_3{\log}_{10}\left({H}_{tr}\right){A}_4{\log}_{10}\left({H}_r\right)+{A}_5{\log}_{10}\left({d}_c\right)\right)}^2 $$
(11)
where n indicates measured signal propagation data number. From (11), the challenge of determining the values of A_{1},A_{2},A_{3}, A_{4}, and A_{5} based on field measurement can be transformed into an optimisation problem as given by (12). The evolving partial derivatives are given in eq.(13) to (17). Details of the mathematical procedures are available [35,36,37]:
$$ \underset{A_1,{A}_2,..,{A}_n}{\min }H\left({A}_1,{A}_2,..,{A}_n\right)=\underset{A_1,{A}_2,..,{A}_n}{\min}\sum \limits_{m=1}^n{\left({Y}_m{\hat{Y}}_m\right)}^2 $$
(12)
$$ \frac{\partial H}{\partial {A}_1}=2\sum \limits_{i=1}^n\;\left({Y}_m{\hat{Y}}_m\;\right)\;\frac{\partial {\hat{Y}}_m}{\partial {A}_1}=0 $$
(13)
$$ \frac{\partial H}{\partial {A}_2}=2\sum \limits_{i=1}^n\;\left({Y}_m{\hat{Y}}_m\;\right)\;\frac{\partial {\hat{Y}}_m}{\partial {A}_2}=0 $$
(14)
$$ \frac{\partial H}{\partial {A}_3}=2\sum \limits_{i=1}^n\;\left({Y}_m{\hat{Y}}_m\;\right)\;\frac{\partial {\hat{Y}}_m}{\partial {A}_3}=0 $$
(15)
$$ \frac{\partial H}{\partial {A}_4}=2\sum \limits_{i=1}^n\;\left({Y}_m{\hat{Y}}_m\;\right)\;\frac{\partial {\hat{Y}}_m}{\partial {A}_4}=0 $$
(16)
$$ \frac{\partial H}{\partial {A}_5}=2\sum \limits_{i=1}^n\;\left({Y}_m{\hat{Y}}_m\;\right)\;\frac{\partial {\hat{Y}}_m}{\partial {A}_5}=0 $$
(17)
By setting \( \frac{\partial H}{\partial {A}_1}={f}_1\left({A}_1,{A}_2,{A}_3\right)\frac{\partial H}{\partial {A}_2}={f}_2\left({A}_1,{A}_2,{A}_3\right) \), and \( \frac{\partial H}{\partial {A}_3}={f}_3\left({A}_1,{A}_2,{A}_3\right) \)
The expressions in (13) to (17) are converted as given in (18):
$$ \left\{\begin{array}{l}{f}_1\left({A}_1,{A}_2,{A}_3,{A}_4,{A}_5\right)=0\\ {}{f}_2\left({A}_1,{A}_2,{A}_3,{A}_4,{A}_5\right)=0\\ {}{f}_3\left({A}_1,{A}_2,{A}_3,{A}_4,{A}_5\right)=0\\ {}{f}_3\left({A}_1,{A}_2,{A}_3,{A}_4,{A}_5\right)=0\\ {}{f}_3\left({A}_1,{A}_2,{A}_3,{A}_4,{A}_5\right)=0\end{array}\right. $$
(18)
Equation (18) expresses the nonlinear equation; solving for the values of A_{1}, A_{2}, A_{3}, A_{4}, and A_{5} by employing analytical techniques is generally a complex task. A commonly used method of solving the above complex equation is the GaussNewton method [38]. Still, its application during the iterative implementation process requires a full rank matrix, thus becoming a significant limitation of the algorithm.
In this work, we engaged the LevenbergMarquart method [39,40,41] to upturn the limitation of the Gaussian Newton algorithm in resolving the nonlinear equation in (18). Thus, by employing the LevenbergMarquart algorithm, eq (18) is transmuted as given in (19):
$$ \underset{A\in {\Re}^3}{\min }f(A)=\frac{1}{2}{\left\Vert f(A)\right\Vert}^2=\frac{1}{2}\sum \limits_{i=1}^5{f}_i^2(A),A=\left({A}_1,{A}_2,{A}_3,{A}_4,{A}_5\right) $$
(19)
The LevenbergMarquart method can be resolved with the following eq (20)–(21):
$$ \left(J{\left({x}_q\right)}^TJ\left({x}_q\right)+{\mu}_qI.\right)\Delta {x}_q=J{\left({x}_q\right)}^Tf\left({x}_q\right),\mu \ge 0 $$
(20)
$$ \Rightarrow \Delta {x}_q={\left(J{\left({x}_q\right)}^TJ\left({x}_q\right)+{\mu}_qI.\right)}^{1}\left(J{\left({x}_q\right)}^Tf\left({x}_q\right)\right)\mu \ge 0 $$
(21)
J(x_{q}) = f^{I}(A)= Jacobian Matrix
where
I ∈ ℜ^{m × m} and μ express the damping term and the identity matrix introduced by LeverbergMarquart into the classical GaussNewton algorithm [38] to improve its performance.
Accordingly, utilizing the LevenbergMarquart method, the parameters A = (A_{1}, A_{2}, A_{3}, A_{4}, A_{5}) can be obtained iteratively using eq (22):
$$ {x}_{q+1}={x}_q+{\left(J{\left({x}_q\right)}^TJ\left({x}_q\right)+{\mu}_qI.\right)}^{1}\left(J{\left({x}_q\right)}^Tf\left({x}_q\right)\right),\mu \ge 0 $$
(22)
Thus, for μ = 0, the expression in eq (20) becomes the classical GaussNewton algorithm given by eq (23):
$$ \left(J{\left({x}_q\right)}^TJ\left({x}_q\right)\right)\Delta {x}_q=J{\left({x}_q\right)}^Tf\left({x}_q\right) $$
(23)
Similarly, eq (22) is simplified as given in (24):
$$ {x}_{q+1}={x}_q+{\left(J{\left({x}_q\right)}^TJ\left({x}_q\right)\right)}^{1}\left(J{\left({x}_q\right)}^Tf\left({x}_q\right)\right) $$
(24)
To implement the LM algorithm, the following steps are engaged intuitively:
LM algorithm implementation steps

I.
Initialise guess parameters,x_{o} for x at iteration q = 0, 1, 2, …

II.
Select the Lagrange multiplierλ_{q}for each step q

III.
Calculate theΔx_{q}with its expression in eq. (21)

IV.
Calculate x_{q + 1} = x_{q} + Δx_{q}

V.
Evaluate Δx_{q} = x_{q + 1} − x_{q}at the initial parameter,x_{o}

VI.
For smaller Δx_{q}values, check the rate of convergence

VII.
If convergence rate is acceptable, stop the calculation or else go a step (IV).