Author Contributions
Conceptualization, R.W. and L.Y.; methodology, R.W., L.Y., X.Y. and C.Z.; formal analysis, X.Y.; data curation, X.Y.; software, X.Y. and C.Z.; writing— original draft, X.Y., R.W., L.Y., C.Z., T.W. and X.Z.; writing—review and editing, R.W., L.Y., C.Z., T.W. and X.Z.; supervision, R.W. and L.Y.; funding acquisition, R.W. and L.Y. All authors have read and agreed to the published version of the manuscript.
Figure 1.
Schematic diagram of effective sound velocity under wind speed. The X-axis indicates the meridional direction; Y-axis indicates the zonal direction. is the zonal wind; is the meridional wind. , and , are respectively the components of and along the direction of infrasound propagation and the components perpendicular to the direction of infrasound propagation.
Figure 1.
Schematic diagram of effective sound velocity under wind speed. The X-axis indicates the meridional direction; Y-axis indicates the zonal direction. is the zonal wind; is the meridional wind. , and , are respectively the components of and along the direction of infrasound propagation and the components perpendicular to the direction of infrasound propagation.
Figure 2.
(
a) Propagation trajectory of the tau-p model in a phase loop.
R represents the infrasound propagation distance of a phase loop in Equation (
2);
denotes the azimuth angle in
Figure 1,
indicates the angle of elevation;
z represents the altitude of infrasound propagation;
represents the maximum altitude that a phase loop can propagate;
Q represents the lateral offset of infrasound propagation in Equation (
4). (
b) Infrasound propagation trajectories at different elevation angles in a phase loop.
z denotes altitude; the angle values
are from 5 to 25 at intervals of 5.
Figure 2.
(
a) Propagation trajectory of the tau-p model in a phase loop.
R represents the infrasound propagation distance of a phase loop in Equation (
2);
denotes the azimuth angle in
Figure 1,
indicates the angle of elevation;
z represents the altitude of infrasound propagation;
represents the maximum altitude that a phase loop can propagate;
Q represents the lateral offset of infrasound propagation in Equation (
4). (
b) Infrasound propagation trajectories at different elevation angles in a phase loop.
z denotes altitude; the angle values
are from 5 to 25 at intervals of 5.
Figure 3.
Graphs (a–d) represent the sampling numbers of 10, 50, 100, 500 two-dimensional Sobol sampling schematics, respectively. , are two parameters in the range [0–1].
Figure 3.
Graphs (a–d) represent the sampling numbers of 10, 50, 100, 500 two-dimensional Sobol sampling schematics, respectively. , are two parameters in the range [0–1].
Figure 4.
In the first step, the reference data of the atmospheric background parameters with height distribution are obtained from the MSISE00 and HWM93 models. In the second step, the distribution curves of atmospheric background parameters are generated based on the Sobol series sampling of the reference data. In the third step, the sensitivity analysis of atmospheric background parameters is performed using Sobol sensitivity analysis, and the first-order sensitivity index and total Sobol index corresponding to each atmospheric background parameter are calculated.
Figure 4.
In the first step, the reference data of the atmospheric background parameters with height distribution are obtained from the MSISE00 and HWM93 models. In the second step, the distribution curves of atmospheric background parameters are generated based on the Sobol series sampling of the reference data. In the third step, the sensitivity analysis of atmospheric background parameters is performed using Sobol sensitivity analysis, and the first-order sensitivity index and total Sobol index corresponding to each atmospheric background parameter are calculated.
Figure 5.
Distribution curves of each atmospheric parameter generated after Sobol sampling (the number of sampling is 1200). The black line represents the reference atmospheric parameter distribution curve, and the red line represents the atmospheric parameter distribution curve generated after sampling. Graphs (
a–
f) illustrate temperature
T, zonal wind speed
, meridional wind speed
, average atmospheric molar mass
m, specific heat ratio
, and adiabatic sound speed
c, respectively (Data from:
https://kauai.ccmc.gsfc.nasa.gov/instantrun/msis, accessed on 15 May 2021).
Figure 5.
Distribution curves of each atmospheric parameter generated after Sobol sampling (the number of sampling is 1200). The black line represents the reference atmospheric parameter distribution curve, and the red line represents the atmospheric parameter distribution curve generated after sampling. Graphs (
a–
f) illustrate temperature
T, zonal wind speed
, meridional wind speed
, average atmospheric molar mass
m, specific heat ratio
, and adiabatic sound speed
c, respectively (Data from:
https://kauai.ccmc.gsfc.nasa.gov/instantrun/msis, accessed on 15 May 2021).
Figure 6.
(a) Five atmospheric parameters’ first-order Sobol index convergence lines for the infrasound propagation distance R; (b) Five atmospheric parameters’ total Sobol index convergence lines for the infrasound propagation distance R (the blue, dark red, yellow, purple, and light green curves indicate temperature T, zonal wind , meridional wind , mean atmospheric molar mass m, and specific heat ratio , respectively).
Figure 6.
(a) Five atmospheric parameters’ first-order Sobol index convergence lines for the infrasound propagation distance R; (b) Five atmospheric parameters’ total Sobol index convergence lines for the infrasound propagation distance R (the blue, dark red, yellow, purple, and light green curves indicate temperature T, zonal wind , meridional wind , mean atmospheric molar mass m, and specific heat ratio , respectively).
Figure 7.
(a) First-order Sobol index pie charts and (b) total Sobol index pie charts of the five parameters for the infrasound propagation distance R. The five uncertain atmospheric environmental parameters are independently and uniformly distributed. The number of Sobol samples N is 2000.
Figure 7.
(a) First-order Sobol index pie charts and (b) total Sobol index pie charts of the five parameters for the infrasound propagation distance R. The five uncertain atmospheric environmental parameters are independently and uniformly distributed. The number of Sobol samples N is 2000.
Figure 8.
(a) Five atmospheric parameters’ first-order Sobol index convergence lines for the maximum height of infrasound propagation ; (b) Five atmospheric parameters’ total Sobol index convergence lines for the maximum height of infrasound propagation (the blue, dark red, yellow, purple, and light green curves indicate temperature T, zonal wind , meridional wind , mean atmospheric molar mass m, and specific heat ratio , respectively).
Figure 8.
(a) Five atmospheric parameters’ first-order Sobol index convergence lines for the maximum height of infrasound propagation ; (b) Five atmospheric parameters’ total Sobol index convergence lines for the maximum height of infrasound propagation (the blue, dark red, yellow, purple, and light green curves indicate temperature T, zonal wind , meridional wind , mean atmospheric molar mass m, and specific heat ratio , respectively).
Figure 9.
(a) First-order Sobol index pie charts and (b) total Sobol index pie charts of the five parameters for the infrasound propagation maximum height. The five uncertain atmospheric environmental parameters are independently and uniformly distributed. The number of Sobol samples N is 5000.
Figure 9.
(a) First-order Sobol index pie charts and (b) total Sobol index pie charts of the five parameters for the infrasound propagation maximum height. The five uncertain atmospheric environmental parameters are independently and uniformly distributed. The number of Sobol samples N is 5000.
Figure 10.
(a) Five atmospheric parameters’ first-order Sobol index convergence lines for the infrasound propagation travel time t in a phase loop; (b) Five atmospheric parameters’ total Sobol index convergence lines for the infrasound propagation travel time t in a phase loop (the blue, dark red, yellow, purple, and light green curves indicate temperature T, zonal wind , meridional wind , mean atmospheric molar mass m, and specific heat ratio , respectively).
Figure 10.
(a) Five atmospheric parameters’ first-order Sobol index convergence lines for the infrasound propagation travel time t in a phase loop; (b) Five atmospheric parameters’ total Sobol index convergence lines for the infrasound propagation travel time t in a phase loop (the blue, dark red, yellow, purple, and light green curves indicate temperature T, zonal wind , meridional wind , mean atmospheric molar mass m, and specific heat ratio , respectively).
Figure 11.
(a) First-order Sobol index pie charts and (b) total Sobol index pie charts representing the five parameters for travel time t. The five uncertain atmospheric environmental parameters are independently and uniformly distributed. The number of Sobol samples N is 2400.
Figure 11.
(a) First-order Sobol index pie charts and (b) total Sobol index pie charts representing the five parameters for travel time t. The five uncertain atmospheric environmental parameters are independently and uniformly distributed. The number of Sobol samples N is 2400.
Table 1.
The range of values of the polynomial curve fitting coefficients for the five atmospheric parameters.
Table 1.
The range of values of the polynomial curve fitting coefficients for the five atmospheric parameters.
| | | | | | | |
---|
T | min | −1.8359 | 5.49439 | −0.00181 | 0.20020 | −8.048 | 299.09 |
max | −1.8363 | 5.49549 | −0.00180 | 0.20060 | −7.889 | 330.57 |
| min | 9.880 | −4.377 | 0.000655 | −0.03674 | 0.511 | −18.52 |
max | 9.882 | −4.376 | 0.000657 | −0.03666 | 0.521 | 6.17 |
| min | 1.0201 | −4.410 | 0.000639 | −0.0355 | 0.665 | −18.40 |
max | 1.0203 | −4.409 | 0.000640 | −0.0353 | 0.679 | 11.04 |
m | min | −0.000308 | 0.01608 | 27.51 | | | |
max | −0.000307 | 0.01612 | 30.41 | | | |
| min | 6.3877 | −0.000378 | 1.37 | | | |
max | 6.4005 | −0.000377 | 1.43 | | | |
Table 2.
First Sobol index S and Total Sobol index ST of each atmospheric parameter for the infrasound propagation distance R.
Table 2.
First Sobol index S and Total Sobol index ST of each atmospheric parameter for the infrasound propagation distance R.
| T | | | m | |
---|
S | 0.3768 | 0.2684 | 0.3528 | 0.0016 | 0.0004 |
| 0.3533 | 0.2649 | 0.3814 | 0.0003 | 0.0001 |
Table 3.
First Sobol index S and Total Sobol index ST of each atmospheric parameter for the infrasound propagation maximum height.
Table 3.
First Sobol index S and Total Sobol index ST of each atmospheric parameter for the infrasound propagation maximum height.
| T | | | m | |
---|
S | 0.8496 | 0.0095 | 0.0727 | 0.0659 | 0.0033 |
| 0.8563 | 0.0093 | 0.0847 | 0.0490 | 0.0007 |
Table 4.
First-order Sobol index S and total Sobol index of each atmospheric parameter for infrasound propagation travel time in a phase loop.
Table 4.
First-order Sobol index S and total Sobol index of each atmospheric parameter for infrasound propagation travel time in a phase loop.
| T | | | m | |
---|
S | 0.2685 | 9.7098 | 0.0045 | 0.6301 | 0.0960 |
| 0.3396 | 2.6454 | 0.0036 | 0.5771 | 0.0795 |