## 4 Joint Inversion
### Multi-Order Joint Inversion of S-Wave Velocity Structure  
Real strata are often not the ideal models of increasing velocity. When there is a low velocity or high velocity layer sandwiched between a layer in the middle, the energy distribution of each model of the surface wave will change, and the velocity curves of each model will also exhibit the phenomenon of mode crossover. Researchers [42] compared the dispersion curves extracted from the numerical simulation of the Rayleigh wave field with those calculated from the dispersion equation, and analyzed the energy distribution characteristics of the multi-order mode in the Rayleigh wave field. When dispersion of higher-mode energy dominates, the development of the fundamental surface wave cannot be observed in the dispersion energy spectrum. It is very necessary to introduce the joint inversion of fundamental and higher-mode surface wave dispersion. At present, the joint inversion of fundamental and higher-mode dispersion curves is mainly applied in the direction of the active source of the Rayleigh surface wave, and the inversion effect has been theoretically and practically verified. However, in the past, background noise dispersion curves were mostly inverted using the fundamental dispersion curve, and the joint inversion effect of measured data needs to be verified by more application cases.  
In the paper, we take the dispersion curves of measured noise data in Figure 4 as an example. An initial number of 10 layers was adopted for the inversion model of the fundamental and higher-mode dispersion curves, and the values of S-wave velocity, P-wave velocity, density, layer thickness, and depth of the model layers were given. S-wave velocity and P-wave velocity were converted according to a certain Poisson's ratio, and the layer thickness increment factor was 1.1. The minimum S-wave velocity value was 100 m/s, the maximum S-wave velocity value was 3000 m/s, the maximum number of iterations was 20, and the root mean square error limit was 0.10 m/s.  
Figure 6a shows a monotonically increasing trend of shear wave velocity in the stratum of the fundamental mode dispersion inversion, which does not match the stratum structure of the test area. With the addition of the first higher-mode dispersion information, Figure 7a shows the S-wave velocity inversion in both shallow 12.4 m and deep 67.4 m, which is closer to the real stratum situation. Figure 6b shows the dispersion curve of measured noise data picked up in the fundamental mode and the theoretical dispersion curve calculated after the initial model iteration, and Figure 7b shows the dispersion curve of the inversion in the fundamental mode, and the first higher-mode fitting the measured data dispersion curve. It can be seen that the joint inversion of the multi-order dispersion with the addition of higher-mode dispersion has a higher resolution of the stratum and can distinguish more detailed layers. In the dispersion curve below the 3 Hz band, the multi-order mode fits the measured dispersion curve better than the fundamental mode does. In the dispersion curve above 15 Hz, the fundamental and higher-mode dispersion shows a trend of mode crossover, which is close to the occurrence of a missing stratum, or inversion in real strata.  
The joint inversion of the fundamental and higher-mode dispersion of measured noise data through the frequency-Bessel transform results in a S-wave velocity structure that is more consistent with the sedimentary characteristics of real strata, laying a reliable theoretical basis for the calculation and interpretation of two-dimensional profiles, and at the same time providing a new method for the detection of karst development under the Quaternary overburden.  
Figure 6: Inversion results of fundamental mode. (**a**) S-wave velocity structure of fundamental mode inversion. (**b**) Dispersion curve of measured noise data picked up in fundamental mode and the theoretical dispersion curve calculated after the initial model iteration.