**2. GB-SAR**

Differential interferometric synthetic aperture radar (DInSAR) by GB-SAR is used to measure the displacement of the target surface [1]. This method has been used for

monitoring landslide slopes [2–4], volcanic lava domes [4, 5], and inspection of large-scale infrastructure facilities such as dams and bridges [6, 7]. However, by conventional GB-SAR, the data for SAR processing is acquired by physically moving a radar unit equipped with a pair of transmitting and receiving antennas on a rail. The size of the rail determines the synthetic aperture length, which is typically about 2 m for 17 GHz GB-SAR. The data acquisition takes several tens of seconds to several minutes for one SAR image. Recently, MIMO radar [8–11], which does not have to move a radar unit, has been proposed to use for GB-SAR applications.

MIMO radar is a multi-static array type radar that has multiple transmitting and receiving antennas. However, MIMO radar transmits electromagnetic wave from one of the transmitting antennas, and the reflected signal is received by all the receiving antennas. Consequently, for a radar system with M transmitting and N receiving antennas, M N independent radar signals can be measured. This is equivalent to acquire radar signal by using M N independent antenna pairs. This concept is called virtual array.

Compared to conventional GB-SAR, MIMO radar can acquire data in a short time by using electronic switches for multiple transmitting and receiving antennas. Since MIMO radar has no mechanical moving parts, it can improve the reliability of longterm operation.

### **3. MIMO GB-SAR**

MIMO radar uses multiple transmitting and receiving antennas independently to form a single SAR image, and a virtual array replaces the physical transmitting and receiving array which is equivalent to an array composed of monostatic radar capable of transmitting and receiving.

**Figure 1** shows the relationship between a physical bistatic radar consisting of a pair of transmitting and receiving antennas and a monostatic radar with a virtual

#### **Figure 1.**

*The relationship between a physical bistatic radar consisting of a pair of transmitting and receiving antennas and a monostatic radar with a virtual array.*

array. Here, *O* is the coordinate origin, *P* is the target position, Tx and Rx are the transmitting and receiving antenna positions, *a* ! *<sup>n</sup>* and *b* ! *<sup>m</sup>* are the position vectors of the transmitting and receiving antennas, and *r* ! is the target position vector. The path length *Rn*,*<sup>m</sup>* is that of the EM wave propagating from the *n*-th Tx antenna to the target and to the *m*-th Rx antenna is given as:

$$R\_{n,m} = \left| \overrightarrow{r} - \overrightarrow{a}\_n \right| + \left| \overrightarrow{r} - \overrightarrow{b}\_m \right| \tag{1}$$

When the target is far enough from the origin compared to the wavelength, it can be approximated by

$$R\_{n,m} = 2\left| \overrightarrow{r} - \frac{\overrightarrow{a}\_n + \overrightarrow{b}\_m}{2} \right| \tag{2}$$

The condition for this approximation [11] is determined by the total length of the transmitting and receiving array *L*Tx, *L*Rx as shown in (3). If (3) is satisfied, the array factor generated by the virtual array will be given by the product of the physical transmitting array factor (4) and the receiving array factor (5), where *λ* is the wavelength, *k* is the wavenumber, and *l* ! is the directional *r* ! vector given by (6).

$$|\overrightarrow{r}| \ge \mathbf{1.24} \sqrt{\frac{L\_{\text{Tx}}^3 + L\_{\text{Rx}}^3}{\lambda}} \tag{3}$$

$$F\_{\rm Tx}(\theta,\phi) = \frac{1}{N} \mathbf{e}^{-jk|\overrightarrow{r}|} \sum\_{n=1}^{N} \mathbf{e}^{-jk\overrightarrow{a}\_{n}\cdot\overrightarrow{l}} \tag{4}$$

$$F\_{\rm Rx}(\theta,\phi) = \frac{1}{M} \mathbf{e}^{-jk\left|\overrightarrow{r}\right|} \sum\_{n=1}^{M} \mathbf{e}^{-jk\overrightarrow{b}\_{m}\cdot\overrightarrow{l}} \tag{5}$$

$$
\overrightarrow{l} = \begin{pmatrix}
\sin\theta\cos\phi \\
\sin\theta\sin\phi \\
\cos\theta
\end{pmatrix} \tag{6}
$$

To prevent the generation of grating lobes in a basic concept for designing array antenna, and the antenna spacing *d* must satisfy the condition *d*<*λ=*2. In MIMO radar, we consider this condition for the virtual array, but not for the physical antenna positions.

Back-projection algorithm is used to reconstruct the SAR image from data acquired by MIMO GB-SAR. The SAR image *I r*!� � is obtained by (7), where, *sn*,*<sup>m</sup>* is the radar waveform (range profile) measured by the combination of the *n*-th transmit antenna and the *m*-th receive antenna.

$$I\left(\overrightarrow{r}\right) = \sum\_{m=1}^{M} \sum\_{n=1}^{N} s\_{n,m}(t) \sigma^{j4\pi R\_{n,m}/c} \tag{7}$$

To estimate the surface displacement of the imaged objects, DInSAR is performed using the phase difference of a pair of SAR images acquired at different times.

Assuming two SAR images acquired at different time as master and slave images, the phase difference Δ*ϕ* between the master image I M and the slave image I S is given by (8).

$$\Delta\phi\left(\overrightarrow{r}\right) = \arctan\left(\frac{\text{Im}\left(I\_M\left(\overrightarrow{r}\right)I\_S^\*\left(\overrightarrow{r}\right)\right)}{\text{Re}\left(I\_M\left(\overrightarrow{r}\right)I\_S^\*\left(\overrightarrow{r}\right)\right)}\right) \tag{8}$$

The actual displacement Δ*d* is obtained by (9), where *λ*<sup>c</sup> is the wavelength at the center frequency.

$$
\Delta d \left( \stackrel{\rightarrow}{r} \right) = \frac{\lambda\_{\text{c}}}{4\pi} \Delta \phi \left( \stackrel{\rightarrow}{r} \right) \tag{9}
$$

### **4. 17 GHz MIMO radar design**

By the recommendation of ITU, 17 GHz is one of the standard frequencies used for GB-SAR all over the world, and it is suitable for the measurement of bare soil ground surface. We use 17 GHz for our system, and the specifications of the MIMO radar that we designed are shown in **Table 1**. The designed antenna arrangement is shown in **Figure 2**. By using these technical specifications, the antenna array factors are simulated and shown in **Figure 3**. **Figure 3** shows the array factors of the transmitting antenna array and the receiving antenna array and the virtual array. The separation of adjacent transmitting antennas is 17.5 mm, which is one wavelength at 17 GHz, and the separation of the adjacent receiving antennas is 131.3 mm, which corresponds to 7.5 wavelengths, and the separation of the adjacent virtual antennas is 4.4 mm, which


#### **Table 1.**

*The technical specification of the 17 GHz MIMO radar.*

#### **Figure 2.**

*The antenna arrangement of the 17 GHz MIMO radar. 15Tx, 16Rx and 240 virtual antennas.*

**Figure 3.** *The antenna factor of the 15* � *16 17 GHz MIMO radar.*

is the 1/4 wavelength. The transmitting and receiving equidistant arrays are separated by 150 mm vertically.

In **Figure 3**, we find that the grating lobes are generated in the physical receiving antenna array. However, since the null points of the transmitting antenna array overlap it and cancel in the virtual array and the radar system has no grating lobes. We should note that the number of physical antennas can drastically be reduced from M � N to M + N by MIMO GB-SAR.
