## 3 Simulated Measurements
### Single Sound Source  
The aim of this simulation was to compare the reconstruction performances of the four methods for a single sound source. The source was placed at (0, 0, 0) m. The distance between the measurement plane and the origin of coordinates was 0.1 m, while the distance between the reconstruction plane and the origin of coordinates was 0.02 m. The equivalent source plane was located at 0.001 m from the real sources. Random noise with a signal-to-noise ratio (SNR) of 20 dB was added to the pressures to simulate the actual measurement. The reconstruction plane and equivalent source plane were meshed by 21 \(\times\) 21 grids with a uniform interspacing of 0.025 m. There were 64 measurement points which were randomly selected from 441 points located on 21 \(\times\) 21 grids (with 0.025 m intervals) to simulate a random array, as shown in Figure 2.  
Figure 2: The array with 64 randomly distributed microphones.  
To quantitatively evaluate the performance of ERDCESM, the reconstruction error was defined as Equation (21):  
\[L_{error}=\frac{\mathbf{P}_{r}-\mathbf{P}_{t2}}{\mathbf{P}_{t2}}\times 100\% \tag{21}\]  
where \(\mathbf{P}_{t}\) and \(\mathbf{P}_{r}\) are the theoretical and reconstructed pressure, respectively.  
According to Equation (21), the reconstruction error for a single source in a frequency range from 50 Hz to 3000 Hz is shown in Figure 3. The results at each frequency were obtained 10 times to produce an average reconstruction error. It was observed that the reconstruction accuracy of ERDCESM and CESM were much higher than TRESM and CMESM. The results of CMESM were not satisfactory because, for the source that distributed sparsely in the space domain, its spatial acoustic modes were spatially continuous rather than sparse. In this condition, it was not appropriate to use the \(l_{1}\) norm as a constraint term. The unsatisfactory reconstruction effect of TRESM was mainly due to insufficient measurement points. When the frequency range was higher than 500 Hz, the accuracy of ERDCESM was almost as good as that of CESM. Viewed as a whole, ERDCESM obtained a satisfactory reconstruction effect in a relatively wide frequency range.  
Figure 4 illustrates the theoretical sound source map on the reconstruction plane and the corresponding maps calculated by TRESM, CESM, CMESM, and ERDCESM. For the source at 500 Hz, it is easy to see that ERDCESM and CESM have good consistency with the theoretical source map. The reconstructed source maps based on TRESM and CMESM present a hot-spot at the given position, which is much larger than that on the corresponding maps using other two methods. When the source was set at 2000 Hz, a similar conclusion could be derived that CESM and ERDCESM produced a better resolution than TRESM and CMESM.  
To better and vividly illustrate the performance of ERDCESM-\(\
u\), Figure 5 shows the pressure distribution on the reconstruction plane at 500 and 2000 Hz. As can be seen from Figure 5a,e, with the order \(\
u\) increasing, the identified peak sound pressure almost kept constant and approximated the theoretical value. For the source at 500 Hz, it was observed that although ERDCESM with order \(\
u=1\) (Figure 5c) and \(\
u=3\) (Figure 5d) could not fully reconstruct the sound pressure distribution, they could provide a more accurate source position than that of ERDCESM with \(\
u=0\) (Figure 5b). When the source was at 2000 Hz, a similar conclusion was derived.  
Figure 3: Reconstructed error for a single sound source. Frequency interval is 50 Hz.  
Figure 4: The calculated sound maps of (**a**) Theoretical map at 500 Hz, (**b**) TRESH map at 500 Hz, (**c**) CESM map at 500 Hz, (**d**) CMESM map at 500 Hz, (**e**) ERDCESM map at 500 Hz, (**f**) Theoretical map at 2000 Hz, (**g**) TRESH map at 2000 Hz, (**h**) CESM map at 2000 Hz, (**i**) CMESM map at 2000 Hz and (**j**) ERDCESM map at 2000 Hz for a single source.