## 4 Synthetic Example Tests  
In this study, one example was used to demonstrate the performances of the GCV and the UPRE methods using the RGSVD algorithm. For comparison, the CG method and the GSVD method were also used in the GCV and UPRE methods. The following tests were run on a computer with a 3.00 GHz processor and 32 GB RAM.  
The model consists of two cuboids, and Figure 1a shows its 3D perspective view. The dimensions of the two cuboids are both 300 m \(\times\) 300 m \(\times\) 200 m, and the depths of their tops are both 50 m. Both of them have a density contrast of 1 g/cm\({}^{3}\). The gravity data were generated on a grid with 31 \(\times\) 21 = 651 points and 50 m \(\times\) 50 m spacings. Meanwhile, random Gaussian noise, whose standard deviation is 0.02 mGal plus 2% of each datum, were incorporated in the gravity data (Figure 1b). The subsurface was divided into 30 \(\times\) 20 \(\times\) 10 = 6000 cells, and dimension of each cell is 50 m \(\times\) 50 m \(\times\) 50 m. In the following inversions, the density range was 0-1 g/cm\({}^{3}\) and \(K_{\text{max}}\) was set as 50.  
First, we used the GCV method to determine the regularization parameter in inversion. In the GCV method, the CG method, the GSVD, and the RGSVD algorithm were used, respectively. Figure 2a-c show the inversion results from the GCV methods using the CG method, the GSVD, and the RGSVD algorithm, respectively. These three inversion results are very similar: two source bodies with cuboid shape are in the actual position of the true model. Their corresponding GCV function curves are shown in Figure 3a-c, and these three curves have similar trends. Meanwhile, Table 1 records the corresponding optimal regularization parameters \(\mu_{opt}\), and the corresponding elapsed times for the process of choosing the regularization parameter. The GCV methods using the GSVD and the RGSVD algorithm have very similar \(\mu_{opt}\), and the \(\mu_{opt}\) of the GCV method using the CG method is slightly large. The GCV method using the RGSVD algorithm has the shortest elapsed time, at only 1.3 s.  
Then, the UPRE method was implemented in inversion. Similarly, the UPRE method used the CG method, GSVD, and the RGSVD algorithm, respectively. Their inversion results are shown in Figure 2d-f, respectively. These inversion results are similar to those obtained using the GCV method. The curves for the UPRE function are demonstrated in Figure 3d-f, and they also have similar trends. Table 1 also lists the value of optimal regularization parameters and the elapsed times for the UPRE method. Their optimal regularization parameters are the same as them from the GCV method, and the elapsed times are close to them from the GCV method. The UPRE method using the RGSVD algorithm also has the shortest elapsed time.  
Through the above comparative test, it was concluded that the GCV and the UPRE methods using the RGSVD algorithm can quickly provide an optimal regularization parameter that can generate a satisfying inversion result.  
Figure 1: (**a**) 3D perspective view of the model consisting of two cuboids; (**b**) gravity data produced by this model.  
Figure 3: The GCV functions using (**a**) the CG method, (**b**) the GSVD, and (**c**) the RGSVD algorithm; the UPRE functions using (**d**) the CG method, (**e**) the GSVD, and (**f**) the RGSVD algorithm. The blue curve is GCV function or UPRE function, and the red \(\odot\) is the optimal regularization parameter.  
Figure 2: Inversion results from different regularization methods. The generalized cross-validation (GCV) methods using (**a**) the conjugate gradient (CG) method, (**b**) the generalized singular value decomposition (GSVD), and (**c**) the randomized generalized singular value decomposition (RGSVD algorithm; the unbiased predictive risk estimator (UPRE) methods using (**d**) the CG method, (**e**) the GSVD, and (**f**) the RGSVD algorithm. In all these panels, the black lines indicate the real position of the model.