As shown in Section 3, we use the objective function *F*_{HNGPS-CA}(*x*_{ms}, *y*_{ms}) in (5) rather than the LS obcetive function (*F*_{LS}(*x*_{ms}, *y*_{ms})) shown in (6). To verify this choice, let us look at the objective functions *F*_{HNGPS-CA}(*x*_{ms}, *y*_{ms}) and *F*_{LS}(*x*_{ms}, *y*_{ms}).

For sake of simplicity, from now on we will drop the argument (*x*_{ms}*, y*_{ms}) in *F*_{LS}(*x*_{ms}*, y*_{ms}) and *F*_{HNGPS-CA}(*x*_{ms}, *y*_{ms}). Thus, we can write (6) as follows

{F}_{\mathsf{\text{LS}}}=\sum _{i=1}^{3}{\alpha}_{i}{f}_{i}

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where

{f}_{i}={\left({l}_{i}-{R}_{i}\right)}^{2}.

From the definition of *l*_{
i
} in (3), the function *f*_{
i
} can be simplified to

{f}_{i}={\left({\eta}_{i}+{\mu}_{i}\right)}^{2}.

(13)

Let us recall that we intend to minimize *η*_{
i
} which always has a nonnegative value, i.e., *η*_{
i
} ≥ 0. Also, *η*_{
i
} is actually a distance and function of the estimated MS coordinates \left({\widehat{x}}_{\mathsf{\text{ms}}},\phantom{\rule{0.3em}{0ex}}{\u0177}_{\mathsf{\text{ms}}}\right), where {^} represents the estimated value of the variable, i.e., for example \widehat{v} is the estimated value of *v*. Since *η*_{
i
} is a positive error which increases as the estimated MS coordinates \left({\widehat{x}}_{\mathsf{\text{ms}}},\phantom{\rule{0.3em}{0ex}}{\u0177}_{\mathsf{\text{ms}}}\right) get further from the actual MS coordinates (*x*_{ms}, *y*_{ms}), then *η*_{
i
} is a convex surface which has a minima (zero value) at (*x*_{ms}, *y*_{ms}). Now, because *μ*_{
i
} is a noise signal it can have both negative and positive values. If it has a positive value then the surface *η*_{
i
} + *μ*_{
i
} has a value which is greater than zero, i.e., *η*_{
i
} + *μ*_{
i
} *>* 0 and has a minima at (*x*_{ms}, *y*_{ms}). Thus, *f*_{
i
} = (*η*_{
i
} + *μ*_{
i
})^{2} will be convex in this case and *F*_{LS} which is the summation of positively weighted convex functions (*f*_{
i
}) for *i* = 1, 2, 3 will be convex in this case as well [9]. Whereas, if *μ*_{
i
} is negative, i.e., *μ*_{
i
} *<* 0, then parts of the surface *η*_{
i
} + *μ*_{
i
} will be below zero and other parts will be above zero with the minimum located at (*x*_{ms}, *y*_{ms}) to be below zero. When taking the square of *η*_{
i
} + *μ*_{
i
}, i.e., (*η*_{
i
} + *μ*_{
i
})^{2}, then, the point (*x*_{ms}, *y*_{ms}) will have a local maxima and the surface surrounding it will be decreasing until it hits the zero level. Then, the surface (*η*_{
i
} + *μ*_{
i
})^{2} is reflected back to the increasing mode as we get further from the point (*x*_{ms}, *y*_{ms}), causing (*η*_{
i
} + *μ*_{
i
})^{2} to be a non-convex surface, i.e., *f*_{
i
} will be non-convex in this case, causing *F*_{LS} which is the summation of positively weighted non-convex functions (*f*_{
i
}) for *i* = 1, 2, 3 will be non-convex in this case [9]. This is illustrated in Figure 2. Thus, when minimizing *F*_{LS} we will reach an unwanted local minimum which is not located at (*x*_{ms}, *y*_{ms}). So, there will be an error in locating the MS.

Similarly, *F*_{HNGPS-CA} in (5) can be written as follows

{F}_{\mathsf{\text{HNGPS}}-\mathsf{\text{CA}}}=\sum _{i=1}^{3}{\alpha}_{i}{g}_{i}

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where

{g}_{i}={\left(\stackrel{\u0304}{{l}_{i}}-{R}_{i}\right)}^{2}.

From the definition of \left({\stackrel{\u0304}{l}}_{i}\right) in (4), the function *g*_{
i
} can be simplified to

{g}_{i}={\left({\eta}_{i}+{\mu}_{i}+{R}_{s}+{\mu}_{s}\right)}^{2}.

(15)

Let us introduce the variable *h*_{
i
} which is defined as

{h}_{i}={\mu}_{i}+{R}_{s}+{\mu}_{s}\phantom{\rule{0.3em}{0ex}}.

Thus,

{g}_{i}={\left({\eta}_{i}+{h}_{i}\right)}^{2}\phantom{\rule{0.3em}{0ex}}.

Because *R*_{
s
} is the distance between the MS and the satellite which is much greater than zero, i.e., *R*_{
s
} *>>* 0, and because *R*_{
s
} *>> μ*_{
i
} and *R*_{
s
} *>> μ*_{
s
}, then *h*_{
i
} *>>* 0 for all realistic values of *μ*_{
i
} and *μ*_{
s
}. Thus, the function *g*_{
i
} is the square of the summation of a positive surface (*η*_{
i
}) which has a minimum value of zero at (*x*_{ms}, *y*_{ms}) and a positive variable (*h*_{
i
}). So, *g*_{
i
} is always a convex function and has a minimum value at (*x*_{ms}, *y*_{ms}). This is illustrated in Figure 3. Thus, *F*_{HNGPS-CA} is convex as well because the positively weighted sum of convex functions is also convex [9] (recall that {F}_{\mathsf{\text{HNGPS}}-\mathsf{\text{CA}}}={\Sigma}_{i=1}^{3}\phantom{\rule{0.3em}{0ex}}{\alpha}_{i}{g}_{i}). This will assure that when minimizing *F*_{HNGPS-CA} we will not reach an unwanted local minimum as the case for minimizing *F*_{LS}, but rather we will reach the true MS location (*x*_{ms}, *y*_{ms}).