## 1 Introduction
An overview of their results is provided in Table 1; since all of the studies used thrust control, it is not specified in Table 1.  
Table 1 demonstrates that the obtained effect of VNRS implementation for SST varies significantly for different studies and aircraft concepts. The importance of the aircraft \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Paper** & **Aircraft** & **MTOM, kg** & \
\begin{tabular}{c} **Number of** \\ **Engines** \\ \end{tabular} & **HLD Control** &
\begin{tabular}{c} **Noise Reduction** \\ **due to VNRS** \\ \end{tabular} \\ \hline Grantham and Smith & & & & & \\ (1981) [8] & AST-105-1 & 311,164 & 4 & Yes & 5 EPNdB \\ [PERSON] (1992) [9] & AST-105-1 & 311,164 & 4 & No & 4.3 EPNdB \\ Mirzoyan and Khaletskii & & & & & \\ (2023) [10] & & & & & \\ [PERSON] (2023) [11] & STCA & 55,000 & 3 & Yes & 5.2 EPNdB \\ [PERSON] et al. & & & & & \\ (2024) [12] & & & & & \\ [PERSON] et al. & & & & & \\ (2023) [13] & & & & & \\ \multirow{2}{*}{[PERSON] et al. (2024) [14]} & SBJ & 41,697 & 2 & No & 5.5 EPNdB \\  & SSAL & 144,277 & 4 & No & 14 EPNdB \\ \multirow{2}{*}{[PERSON] et al. (2024) [15]} & TWO & 55,412 & 2 & No & 5.1 EPNdB \\  & THREE & 51,000 & 3 & No & 4.9 EPNdB \\ Akatsuka & & & & & \\ (2024) [16] & S3 & 66,600 & 2 & Yes & 2.1 EPNdB \\ \multirow{2}{*}{[PERSON] and [PERSON] (2024) [17]} & STCA & 55,000 & 3 & Yes & 5.3 EPNdB \\ \hline \hline \end{tabular}
\end{table} 
Table 1: An overview of the results of previous studies of VNRS implementation for SST. concept on VNRS efficiency was also highlighted in [18]. There, a reduced-order model for SST takeoff noise was proposed that showed a difference of 19.2 dB in the noise reduction potential through VNRS implementation between the reference Mach 1.4 aircraft and a Mach 2.2 aircraft. These studies computed aircraft noise with semi-empirical models for different noise sources (jet noise [19], fan noise [20], etc. ), based on engines' parameters at a given point of the takeoff trajectory. A significant amount of attention has been paid to uncertainties [16; 21] and sensitivities to the parameters [22] of these models. Such an approach is excellent for the reference certification procedure [23] or for \"simple\" advanced takeoff procedures (a programmed lapse rate) [6], when a constant thrust reduction is applied soon after clearing the obstacle height. However, optimization for SST takeoff trajectories with this approach has proven to be a formidable and complex task. The complexity of the problem makes it error-prone, which leads some authors to recommend cautiousness when modeling the flights for the novel SST concepts [15]. The studies on SST noise modeling also mention an approach that relies on the so-called noise-power-distance (NPD) data [24]. The approach is used primarily for the computation of noise around airports and scenario assessment, rather than noise at the certification measurement points. In the absence of experimental data for the NPD characteristics of SST, they can be obtained via computation with semi-empirical models [21] for different noise sources and then used for assessing the noise contour areas [15; 25]. In the context of the certification noise, it can be said to start, in a sense, with an answer, because rudimentary noise characteristics for each airplane are given as tabular lookup data [11]. There is, however, a different approach that does not seem to be considered in the previous studies on SST noise reduction with VNRS, but which is commonly used for the optimization of trajectories and high-left-devices configuration for subsonic aircraft. The approach decouples engine noise modeling from trajectory computation. It uses engine noise data obtained for a number of thrust values, typically from static engine tests on open test rigs (Figure 1a) or, less frequently, from tests of engines installed on an aircraft (Figure 1b) as one-third octave noise spectra for a range of azimuthal observation angles.