**2. Methods**

#### *2.1. Velocity Analysis of CMP Data*

The common mid-point (CMP) measurement in a GPR survey is a multi-o ffset measurement that is primarily used to obtain an estimated velocity of the EM wave in the ground. By increasing the antenna preparation symmetrically, the path of the EM wave varies while keeping the point of reflection fixed. Thus, the di fference in two-way travel time enables wave velocity to be estimated [2].

The velocity analysis by CMP gather assumes that the subsurface media is horizontally layered. The two-way travel time of reflected waves can be obtained by

$$t(x\_i) = \sqrt{t\_0^2 + \frac{x\_i^2}{v\_{rms}^2}}\tag{1}$$

where *xi* is the antenna o ffset of the *i*-th channel, *t*0 is the two-way travel time of zero o ffset, and *vrms* is the root mean square velocity.

The velocity spectrum can be obtained by transforming the data from the o ffset versus two-way time domain to the stacking velocity versus two-way zero-o ffset time domain [23]. The velocity spectrum shows the picks that correspond to the best coherency of the signal along a hyperbolic trajectory over the entire spread length of the CMP gather. The coherency can be computed in several ways [24]. However, in the velocity analysis of GPR data, the simple and e ffective measures are the average stacked amplitude ( *A*) and the energy of the average stacked amplitude (*E*), which can suppress interference signals with large amplitude. The *A* and *E* can be defined by

$$A = \frac{1}{N} \sum\_{j=1}^{M} \left| \sum\_{i=1}^{N} f\_{i, j + r\_i} \right| \tag{2}$$

$$E = \sum\_{j=1}^{M} \left(\frac{1}{N} \sum\_{i=1}^{N} f\_{i, j + r\_i} \right)^2 \tag{3}$$

where *fi*,*j*+*ri* is the amplitude value of the *j*<sup>+</sup>*r*-th point on the *i*-th trace in which *r* is a delay associated with normal moveout (NMO). *N* is the number of traces in the CMP gather. *M* is the number of sampling points in a stack time window. Each stacking velocity (*vrms*) is scanned according to Equation (1). When the scanning velocity is equal to the *vrms*, *E* reaches the maximum value, which shows a focused energy cluster in the velocity spectrum. After the stacking velocity is picked up, the interval velocity can be calculated by Dix formula [25], and the expression is as follows:

$$w\_{\rm int,n} = \sqrt{\frac{t\_{0,n}v\_{rms,n}^2 - t\_{0,n-1}v\_{rms,n-1}^2}{t\_{0,n} - t\_{0,n-1}}} \tag{4}$$

where *vint*,*<sup>n</sup>* is the interval velocity of the *n*th layer; *vrms*,*<sup>n</sup>* is the stacking velocity to the bottom of the *n*th layer; *t*0,*n* is the two-way travel time with zero o ffset to the bottom of the *n*th layer; *vrms*,*n*−<sup>1</sup> is the stacking velocity to the bottom of the *n* − 1th layer; and *<sup>t</sup>*0,*<sup>n</sup>*−<sup>1</sup> is the two-way travel time with zero offset to the bottom of the *n* − 1th layer.

After obtaining the interval velocities (*vint*) of radar wave propagating in the subsurface, the relative dielectric constant ε*r* in low loss medium can be obtained by

$$v\_{\rm int} = \frac{c}{\sqrt{\varepsilon\_r}}\tag{5}$$

where *c* = 0.3 m/ns is the velocity of EM wave in air.

#### *2.2. Optimal Gather in Velocity Analysis*

As the o ffset increases, the primary reflections from the interface of each layer will be interfered by multiple reflections and refractions in CMP gathers. In the media composed of thin multi-layers, this interference is so serious that it makes the velocity spectra unable to give correct velocities of EM waves. In addition, the CMP gathers obtained in small o ffsets can be interfered by direct wave. Since the layers are shallow and thin, the reflected wave is close to the direct wave, whose strong energy makes the reflected wave difficult to be identified. Therefore, it is crucial to use high-quality primary reflections from layer interfaces for velocity analysis to obtain accurate velocities.

The optimal gathers include mainly the primary reflections who suffer little interference from multi-waves, refracted waves, and direct waves. These gathers are determined and used to calculate the velocity spectrum. The reflection angle can be used to explain the operation of optimal gather.

The reflection angle θ can be calculated using the sketch shown in Figure 1. *x*1 represents a small offset, and the CMP gathers acquired in *x*1 are supposed to be contaminated with direct waves. *x*2 represents a large offset, and the CMP gathers acquired in *x*2 are supposed to be contaminated with multi-waves and refracted waves. The blue arrows in Figure 1 show the corresponding antenna locations when effective reflection from the interface can be obtained, and the reflection angle θ ranges from θ1 to θ2. The optimal gather for the velocity analysis should be in this angle range.

**Figure 1.** A description for reflection angle and optimal gather. The locations marked by arrows represent the optimal gather range.

In this paper, we can analyze the influence range of interference waves in CMP gathers through numerical simulation because the railway subgrade structure is known. It can help to determine the optimal gather in field CMP data.

## *2.3. Topp Formulae*

A well-known empirical relationship between the relative dielectric constant and volumetric moisture content at frequencies between 1 MHz and 1 GHz was developed by Topp et al. [13]. This empirical relationship is independent of soil type, soil density, soil temperature, and soluble salt content, and it is determined by compiling data for many soils under varying moisture conditions:

$$
\varepsilon\_7 = 3.03 + 9.3\theta + 146.0\theta^2 - 76.7\theta^3 \tag{6}
$$

where θ is the volumetric soil moisture content, which is equal to the product of the water saturation and porosity. A polynomial expression for moisture content as a function of the relative dielectric constant, ε*r*, was reported as well:

$$\theta = -5.3 \times 10^{-2} + 2.92 \times 10^{-2} \varepsilon\_r - 5.5 \times 10^{-4} \varepsilon\_r^2 + 4.3 \times 10^{-6} \varepsilon\_r^3. \tag{7}$$

## *2.4. Induced Polarization*

Induced polarization (IP) is a geophysical method that is mainly used in metal ore exploration and groundwater search. In recent years, with the development of sensitive instruments and methods to improve the signal-to-noise ratio, the IP method has been widely used in the field of environment and engineering [26–28]. IP may reflect the moisture content through the membrane effect, which is related to variations in the mobility of ions in fluids at grain scales in a porous medium under the influence of the electrical field. The polarizability generated by the water-bearing sand gravel layer is higher than that of non-water bearing structure.

In time domain IP, a primary current is injected in the ground for a period *T* using a pair of electrodes A and B. The difference of electrical potential is measured using several bipoles of electrodes M and N. The secondary voltage on a bipole MN decays over time after the shut-down of the primary current. The rate of this decay reflects the strength of the polarization effect. The simplest way to measure the IP effect with time-domain equipment is to compare the residual voltage *V*(*t*) existing at a time t after the current is cut off with the steady voltage *Vc* during the current flow interval. It is not possible to measure potential at the instant of cutoff because of large transients caused by breaking the current circuit. On the other hand, *V*(*t*) must be measured before the residual has decayed to the noise level. Therefore, the polarizability, η(*t*) can be obtained by

$$
\eta(t) = \frac{V(t)}{V\_{\mathfrak{L}}} \times 100\%.\tag{8}
$$

#### **3. Velocity Analysis of CMP Models**

#### *3.1. Railway Subgrade*

Figure 2 shows a typical railway subgrade structure, which can be roughly divided into 4 layers. These are constituted with ballast, graded gravel, medium to coarse-grained sand, and A, B group filling from top to bottom. There is a sheet of geotextile in the middle of the 0.2 m sand layer (marked by a red line in Figure 2).

**Figure 2.** The profile of a typical railway subgrade.

The horizontal layered structure of the railway subgrade satisfies the assumption of velocity analysis through CMP gather. However, due to the shallow depth and the thin multi-layers, the reflected waves from the interfaces of horizontal layers will be interfered by multiple waves and refracted waves. Once the hyperbolic trajectory of the reflected wave in CMP gathers is damaged, the velocity spectrum will not obtain the picks representing the best coherency and failure to pick up the accurate velocity.
