Crop Growth Monitoring with Drone-Borne DInSAR

: Accurate, high-resolution maps of for crop growth monitoring are strongly needed by precision agriculture. The information source for such maps has been supplied by satellite-borne radars and optical sensors, and airborne and drone-borne optical sensors. This article presents a novel methodology for obtaining growth deﬁcit maps with an accuracy down to 5 cm and a spatial resolution of 1 m, using di ﬀ erential synthetic aperture radar interferometry (DInSAR). Results are presented with measurements of a drone-borne DInSAR operating in three bands—P, L and C. The decorrelation time of L-band for co ﬀ ee, sugar cane and corn, and the feasibility for growth deﬁcit maps generation are discussed. A model is presented for evaluating the growth deﬁcit of a corn crop in L-band, starting with 50 cm height. This work shows that the drone-borne DInSAR has potential as a complementary tool for precision agriculture.


Introduction
Agriculture has a vital role in the economic stability and social development of a country. Effective agricultural management is essential to reduce costs and increase production. The monitoring of crop growth shall be done continuously for accurate support of decision-making [1]. Remote sensing has been an important tool for soil and crop monitoring. Optical remote sensing is widely used; nonetheless, synthetic aperture radar (SAR) remote sensing does not depend on weather conditions or sunlight. Moreover, the radar wavelength is approximately one million times greater than the wavelength of optical systems and provides complementary information about the agriculture parameters.
Several studies have shown the SAR remote sensing capabilities in growth monitoring of various crops [2,3] and crop classification [4][5][6]. Crop growth monitoring has been explored by using techniques based on statistical analysis between radar backscattering and crop height [2,3,7], with the use of techniques like interferometric SAR (InSAR) [8], polarimetric decomposition [9], polarimetric interferometric SAR (Pol-InSAR) [10] and differential SAR Interferometry (DInSAR) [10]. The DInSAR methodology presents high accuracy and spatial resolution, as it takes advantage of the phase difference between images. The most popular application of the DInSAR is the estimation of subsidence maps

Drone-Borne SAR System
The radar prototype, described in [12] and shown in Figure 1, operates in three bands-P, L and C-with five channels: two interferometric C-band antennas, two polarimetric L-band antennas and one P-band antenna. This work only shows results for the DInSAR with the L-band HH-polarization, which corresponds to the transmission and reception of signals with horizontal polarization. The two interferometric C-band antennas are used for calculating the digital surface model, DSM, which is then used in the DInSAR calculation. Nominal height accuracy is better than 1 m RMS with a spatial resolution of 2 m. Table 1 shows the main parameters of radar acquisition for the L-band HH-polarization. The Motion Sensing System (MSS), which includes a single channel GNSS receiver and an inertial measurement unit (IMU), is integrated into the radar, as shown in [12]. There is also a ground station with a single channel GNSS, to provide differential GNSS processing.  The Motion Sensing System (MSS), which includes a single channel GNSS receiver and an inertial measurement unit (IMU), is integrated into the radar, as shown in [12]. There is also a ground station with a single channel GNSS, to provide differential GNSS processing.

SAR Imaging
A circular flight pattern was chosen for generating the growth maps presented in this work. This flight geometry provides images with high resolution, which is given by the following [13]: where c is the speed of light, f c is the signal's central frequency and θ e is the depression angle. Expression (1) is valid for both directions of the ground plane. Additionally, the circular flight pattern helps to reduce shadow effects in the processed image, since it can provide full aspect coverage of the targets of interest [14]. The images were processed by using a time-domain back-projection algorithm, which is a method that is easily applied to nonlinear flight patterns. Let → r l be a radar position along the flight track, and let → p mn be the location of a particular pixel on the image sample grid. The back-projected signal is calculated as follows [15]: where k c is the central wavenumber, and R lmn = | → r l − → p mn |.

DInSAR Theory Description
DInSAR is a type of interferometry that provides information on the terrain height displacement between two flights, at different times, following the same flight path. In the DInSAR case, there is no interferometric baseline, so a topographic map is not possible. However, a terrain height displacement over time is perceptible [16]. In the present case, the analyzed terrain is a crop area. Figure 2 presents a circular flight pattern over a crop field. The first survey is carried out at a time t 1 . The second survey follows the same circular flight pattern of the first flight and occurs at a time t 2 . Due to drone instability and weather conditions, the flight patterns are not identical but very similar. Figure 2 presents a circular flight pattern over a crop field. The first survey is carried out at a time . The second survey follows the same circular flight pattern of the first flight and occurs at a time . Due to drone instability and weather conditions, the flight patterns are not identical but very similar. In this example, consider that a first flight is performed on time , during which the crop height is , corresponding to a range from the radar. Then, on time , another flight is executed, when the crop height is now , and is at a range from the radar. Let ∆ = − be the difference in radar range, and let ∆ = − be the growth of the corn crop between these two surveys. It is possible to establish a relationship between these quantities, as shown in Figure 3. The relationship between height and range is given by the following: where is the depression angle. As the drone navigation system performs flight tracks with a position accuracy of about 20 cm at 120 m height, the flight tracks on different dates are assumed to be identical and, therefore, is virtually the same for any flight. The relationship between the radar range, ∆ , and the phase, ∆∅, can be described as follows [17]: In this example, consider that a first flight is performed on time t 1 , during which the crop height is H 1 , corresponding to a range R 1 from the radar. Then, on time t 2 , another flight is executed, when the crop height is now H 2 , and is at a range R 2 from the radar. Let ∆R = R 2 − R 1 be the difference in radar range, and let ∆H = H 2 − H 1 be the growth of the corn crop between these two surveys. It is possible to establish a relationship between these quantities, as shown in Figure 3.
Remote Sens. 2020, 12, 615 4 of 18 Figure 2 presents a circular flight pattern over a crop field. The first survey is carried out at a time . The second survey follows the same circular flight pattern of the first flight and occurs at a time . Due to drone instability and weather conditions, the flight patterns are not identical but very similar. In this example, consider that a first flight is performed on time , during which the crop height is , corresponding to a range from the radar. Then, on time , another flight is executed, when the crop height is now , and is at a range from the radar. Let ∆ = − be the difference in radar range, and let ∆ = − be the growth of the corn crop between these two surveys. It is possible to establish a relationship between these quantities, as shown in Figure 3. The relationship between height and range is given by the following: where is the depression angle. As the drone navigation system performs flight tracks with a position accuracy of about 20 cm at 120 m height, the flight tracks on different dates are assumed to be identical and, therefore, is virtually the same for any flight. The relationship between the radar range, ∆ , and the phase, ∆∅, can be described as follows [17]: The relationship between height and range is given by the following: where θ e is the depression angle. As the drone navigation system performs flight tracks with a position accuracy of about 20 cm at 120 m height, the flight tracks on different dates are assumed to be identical and, therefore, θ e is virtually the same for any flight. The relationship between the radar range, ∆R, and the phase, ∆∅, can be described as follows [17]: Remote Sens. 2020, 12, 615

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For every point of interest in the image sample grid, the mean value of θ e is calculated from the contributions of the depression angles at each drone position along the flight track. If the radar antenna does not illuminate the point of interest at a given drone position, the corresponding contribution is not taken into account. This assessment is based on the antenna aperture angles in azimuth and elevation, considering a circular flight path.
In order to calculate the phase difference between both radar images, it is necessary to determine the phase corresponding to a null vertical displacement or "zero-movement" [16]. That is, it is required to find the phase corresponding to an object within the image for which there was certainly no displacement between the acquisition flights. The zero-movement phase is then subtracted from the phase difference information. Figure 4 shows the scheme used for processing the phase data obtained from two radar images.
Remote Sens. 2020, 12, 615 5 of 18 For every point of interest in the image sample grid, the mean value of is calculated from the contributions of the depression angles at each drone position along the flight track. If the radar antenna does not illuminate the point of interest at a given drone position, the corresponding contribution is not taken into account. This assessment is based on the antenna aperture angles in azimuth and elevation, considering a circular flight path.
In order to calculate the phase difference between both radar images, it is necessary to determine the phase corresponding to a null vertical displacement or "zero-movement" [16]. That is, it is required to find the phase corresponding to an object within the image for which there was certainly no displacement between the acquisition flights. The zero-movement phase is then subtracted from the phase difference information. Figure 4 shows the scheme used for processing the phase data obtained from two radar images. After performing back-projection processing, two images were acquired that have amplitude and phase information. The information of interest is the phase difference between them, calculated as follows: where ( , ) and ( , ) represent the two images, and ( , ) is the image containing the phase difference information. Then, the real and imaginary components of ( , ) are filtered, and so the phase information ∅ is obtained. The next step is to get ∅ from ∅ by using a technique known as phase unwrapping [18]. After that, the zero-movement phase is subtracted. Finally, the resultant value is multiplied by a factor based on Equations (3) and (4), to obtain the vertical displacement or interferometric height difference, ∆ .

Estimation Model for Corn Crop Growth
Using the DInSAR model described above to retrieve the height difference from the phase information, and taking into account the reflectivity of the corn crop and the soil with weed plants around it, a model for estimating the growth of a corn crop between two dates is proposed, considering the reflectivity contribution at different development stages of the crop. This model is briefly described in Figure 5. The reflectivity data obtained from the input images can be decomposed as follows [2]: After performing back-projection processing, two images were acquired that have amplitude and phase information. The information of interest is the phase difference between them, calculated as follows: where C 1 (i, j) and C 2 (i, j) represent the two images, and V(i, j) is the image containing the phase difference information. Then, the real and imaginary components of V(i, j) are filtered, and so the phase information V∅ is obtained. The next step is to get V∅ uw from V∅ by using a technique known as phase unwrapping [18]. After that, the zero-movement phase is subtracted. Finally, the resultant value is multiplied by a factor based on Equations (3) and (4), to obtain the vertical displacement or interferometric height difference, ∆H.

Estimation Model for Corn Crop Growth
Using the DInSAR model described above to retrieve the height difference from the phase information, and taking into account the reflectivity of the corn crop and the soil with weed plants around it, a model for estimating the growth of a corn crop between two dates is proposed, considering the reflectivity contribution at different development stages of the crop. This model is briefly described in Figure 5.
For every point of interest in the image sample grid, the mean value of is calculated from the contributions of the depression angles at each drone position along the flight track. If the radar antenna does not illuminate the point of interest at a given drone position, the corresponding contribution is not taken into account. This assessment is based on the antenna aperture angles in azimuth and elevation, considering a circular flight path.
In order to calculate the phase difference between both radar images, it is necessary to determine the phase corresponding to a null vertical displacement or "zero-movement" [16]. That is, it is required to find the phase corresponding to an object within the image for which there was certainly no displacement between the acquisition flights. The zero-movement phase is then subtracted from the phase difference information. Figure 4 shows the scheme used for processing the phase data obtained from two radar images. After performing back-projection processing, two images were acquired that have amplitude and phase information. The information of interest is the phase difference between them, calculated as follows: where ( , ) and ( , ) represent the two images, and ( , ) is the image containing the phase difference information. Then, the real and imaginary components of ( , ) are filtered, and so the phase information ∅ is obtained. The next step is to get ∅ from ∅ by using a technique known as phase unwrapping [18]. After that, the zero-movement phase is subtracted. Finally, the resultant value is multiplied by a factor based on Equations (3) and (4), to obtain the vertical displacement or interferometric height difference, ∆ .

Estimation Model for Corn Crop Growth
Using the DInSAR model described above to retrieve the height difference from the phase information, and taking into account the reflectivity of the corn crop and the soil with weed plants around it, a model for estimating the growth of a corn crop between two dates is proposed, considering the reflectivity contribution at different development stages of the crop. This model is briefly described in Figure 5. The reflectivity data obtained from the input images can be decomposed as follows [2]: The reflectivity data obtained from the input images can be decomposed as follows [2]: Remote Sens. 2020, 12, 615 where σ T is the total reflectivity measured by the radar, σ corn represents the corn reflectivity, σ wp is the reflectivity of the soil with weed plants that can be measured in an adjacent area to the corn crop and σ db is the reflectivity of the double-bounce between the crop stem and the soil, as illustrated in Figure 6. At different stages of the crop growth, the proportions of these reflectivities vary. In early stage growth, shown in Figure 6a, the portion due to double bounce is the most significant, while in late-stage growth, pictured in Figure 6b, the contribution of the corn crop is dominant.
Remote Sens. 2020, 12, 615 6 of 18 where is the total reflectivity measured by the radar, represents the corn reflectivity, is the reflectivity of the soil with weed plants that can be measured in an adjacent area to the corn crop and is the reflectivity of the double-bounce between the crop stem and the soil, as illustrated in Figure 6. At different stages of the crop growth, the proportions of these reflectivities vary. In early stage growth, shown in Figure 6a, the portion due to double bounce is the most significant, while in late-stage growth, pictured in Figure 6b, the contribution of the corn crop is dominant. Consider that can be estimated from the reflectivity of areas close to the crop so that it can be subtracted from Equation (6); thus, the resulting reflectivity, , is as follows: Assuming that is proportional to by a contribution factor ϵ [0,1], then: Also, the double-bounce contribution can be estimated from and as follows: As the presence of stems and branches is dominant during the early stages of crop growth, the contribution factor, , is negligible and, from Equation (9), ≈ . On the other hand, in late stages of crop growth, the double-bounce contribution decreases drastically due to the presence of leaves; thus, the contribution factor ≈ 1 and, from Equation (8), ≈ . Consider that σ wp can be estimated from the reflectivity of areas close to the crop so that it can be subtracted from Equation (6); thus, the resulting reflectivity, σ R , is as follows: Assuming that σ corn is proportional to σ R by a contribution factor K R ∈ [0, 1], then: Also, the double-bounce contribution can be estimated from K R and σ R as follows: Remote Sens. 2020, 12, 615 7 of 18 As the presence of stems and branches is dominant during the early stages of crop growth, the contribution factor, K R , is negligible and, from Equation (9), σ db ≈ σ R . On the other hand, in late stages of crop growth, the double-bounce contribution decreases drastically due to the presence of leaves; thus, the contribution factor K R ≈ 1 and, from Equation (8) To calculate the height difference measured between two different days, consider the two different stages of crop growth shown in Figure 7.
Remote Sens. 2020, 12, 615 7 of 18 To calculate the height difference measured between two different days, consider the two different stages of crop growth shown in Figure 7. Assume that, on Day 1, the height of the crop is ( ) , the height of the weed plants is ( ) and the difference between them is ( ) = ( ) − ( ) . Similarly, for Day 2, consider that ( ) = ( ) − ( ) . Then, the total estimated height variation, ∆ = ( ) − ( ) , can be calculated as: where ∆ = ( ) − ( ) is the height variation corresponding to the crop and ∆ = ( ) − ( ) is the height variation due to the growth of weed plants. Equation (10) is only valid when contributions can be measured separately.
The radar measures a sum of complex contributions. The amplitude of the return signal is proportional to √ , and the interferometric phase is proportional to ∆ , as seen in Equations (3) and (4). As the ratio ⁄ approaches unity, ∆ becomes the major contribution in Equation (10). The same reasoning can be made for and ∆ . In that way, a simple solution is to correct Equation (10) by weighting each contribution by its respective reflectivity. First, let Equation (7) can be rewritten as follows: are the corn, weed and double-bounce contributing factors, respectively. Now, assuming that each contribution can be weighted by its respective contributing factor, the difference in height measure by the radar between two consecutive days can be expressed by the following: Then, from Expressions (12), (14) and (15) comes the following: Assume that, on Day 1, the height of the crop is H corn (1) , the height of the weed plants is H wp (1) and the difference between them is H (1) = H corn(1) − H wp (1) . Similarly, for Day 2, consider that H (2) = H corn(2) − H wp (2) . Then, the total estimated height variation, ∆H = H (2) − H (1) , can be calculated as: where ∆H corn = H corn(2) − H corn(1) is the height variation corresponding to the crop and ∆H wp = H wp(2) − H wp(1) is the height variation due to the growth of weed plants. Equation (10) is only valid when contributions can be measured separately. The radar measures a sum of complex contributions. The amplitude of the return signal is proportional to √ σ T , and the interferometric phase is proportional to ∆H, as seen in Equations (3) and (4). As the ratio σ corn /σ T approaches unity, ∆H corn becomes the major contribution in Equation (10). The same reasoning can be made for σ wp and ∆H wp . In that way, a simple solution is to correct Equation (10) by weighting each contribution by its respective reflectivity. First, let Equation (7) can be rewritten as follows: where are the corn, weed and double-bounce contributing factors, respectively. Now, assuming that each contribution can be weighted by its respective contributing factor, the difference in height measure by the radar between two consecutive days can be expressed by the following: Then, from Expressions (12), (14) and (15) comes the following: Next, a new correction factor K = 1/ √ K R is defined based on field measurement data as follows: where ∆H corn( f m) represents the height difference data measured on the crop. This new correction factor (K) is only valid for the corn model. Finally, the estimated growth between two dates in a corn crop can be described as follows:

Experimental Site
The experimental site covers an area of 300 m × 300 m, located at the School of Agricultural Engineering of the University of Campinas (UNICAMP), as shown in Figure 8. The survey flights occurred on the following dates: 11 December 2018; and 17 April, 2 July, 17 July and 22 August 2019.
Remote Sens. 2020, 12, 615 8 of 18 Next, a new correction factor = 1 ⁄ is defined based on field measurement data as follows: where ∆ ( ) represents the height difference data measured on the crop. This new correction factor ( ) is only valid for the corn model.
Finally, the estimated growth between two dates in a corn crop can be described as follows:

Experimental Site
The experimental site covers an area of 300 m × 300 m, located at the School of Agricultural Engineering of the University of Campinas (UNICAMP), as shown in Figure 8. The survey flights occurred on the following dates: 11 December 2018; and 17 April, 2 July, 17 July and 22 August 2019.

Field Measurements
Height measurements of coffee and corn crops, depicted in Figure 8b,c, respectively, were carried out over nine months. In this period, the corn reached the maturity stage in December 2018, and a new crop was set in July 2019. Coffee reached the maturity stage in March 2019. The corn crop is structured with 14 rows, where 20 height measurements were made per row. The heights were

Field Measurements
Height measurements of coffee and corn crops, depicted in Figure 8b,c, respectively, were carried out over nine months. In this period, the corn reached the maturity stage in December 2018, and a new crop was set in July 2019. Coffee reached the maturity stage in March 2019. The corn crop is structured with 14 rows, where 20 height measurements were made per row. The heights were measured from the soil to the top of the corn crop. The following measurements were taken on the same days of the flight surveys: 2 July, 17 July and 22 August 2019. The field measurements of the corn crop height are subjected to the natural variability of visual measurement errors.
During the survey period, corn was sown in two areas: field 01 and 02, as shown in Figure 8a. Hereafter, fields 01 and 02 will be labeled as C1 and C2, respectively, where "C" denotes the corn culture.

Drone-Borne SAR Data Acquisition
Three trihedral corner reflectors with square sides and an edge length of 0.6 m were used as a ground and radiometric calibration reference. The resulting radar cross-section is 20 dBsm.
On each acquisition date, the experiment with the drone-borne system was executed as follows: • Mount three corner reflectors on the test site, for planimetric and radiometric calibration purposes; • Place the GNSS ground station close to the initial position of the drone and start the GNSS recording; • Perform each flight over the experimental site, following the subsequent procedure: turn on the drone and the radar, wait 15 min for simultaneous and stationary recording of ground station and radar GNSS data, take-off, perform the same circular flight track, land, wait 15 min for simultaneous and stationary recording of ground station and radar GNSS data, and turn-off the radar and the drone; • Dismount the GNSS ground station and the drone. Download the acquired data for processing.

Drone-Borne SAR Data Processing
The collected data were processed as follows: • Differential GNSS processing of the ground station and the radar GNSS receivers; • IMU and differential GNSS data fusion for generating position and antenna orientation history; • Radar data processing at each acquisition date, according to Section 2.2: range compression and back-projection for the azimuth compression. The output is a geocoded single-look-complex (SLC) image; • Verification of the absolute position of the corner reflectors in the geocoded SLC images; • Differential interferometric processing with data from previous acquisitions, as defined in Section 2.3; • Production of the crop growth map, as described in Section 2.4; • Generation of the corresponding multi-look images with 30 cm × 30 cm sampling.

Drone-Borne SAR Images
The circular flights over the experimental site were carried out by following the path shown in Figure 9. Figure 10 displays the SAR images acquired with circular flights on different acquisition dates. Each image depicts an area of 300 m × 300 m, with 30 cm sampling in both directions. Figure 10b shows the area C2 with the corn crop in a mature stage. The new seeding of area C2 took place in July 2019, shown in Figure 10d, and this area presents increasing reflectivity through Figure 10e,f, until reaching its maturity.

Qualitative Validation of Growth Deficit Maps
DInSAR technique was applied to the campaigns of 5 December and 11 December 2018, and 17 April 2019, on coffee, sugar cane and corn crops. According to the field data, the corn crop was present only on the first two dates mentioned, and the other crops had an insignificant height variation between the three campaign dates. DInSAR provided a compatible result for these crops, showing a negligible height variation during those dates.
The first trial for a crop growth map was performed with the coffee crop, presented in Figure 8. No field measurement of the crop height variation was determined. However, a rough growth estimate of about 10 cm was expected, as the coffee crop was quite mature in December. Because of a small morphology alteration in the crop due to its maturity, the interferogram had a low but acceptable coherence of 0.2.
A growth map corresponding to a coffee crop of approximately 1700 m 2 is shown in Figure 11. The mean height growth between those dates with DInSAR is 11 cm, with a standard deviation of 6 cm. The south area of the coffee crop presented less growth than the north area. A mask was used to discard all targets that do not correspond to the coffee crop. Moreover, a 15 × 15 moving average filter was used over the interferogram, with a pixel spacing of 30 cm × 30 cm. Only data with a coherence greater than 0.1 were considered valid data. This result has motivated a more in-depth study of DInSAR for growth maps.

Qualitative Validation of Growth Deficit Maps
DInSAR technique was applied to the campaigns of 5 December and 11 December 2018, and 17 April 2019, on coffee, sugar cane and corn crops. According to the field data, the corn crop was present only on the first two dates mentioned, and the other crops had an insignificant height variation between the three campaign dates. DInSAR provided a compatible result for these crops, showing a negligible height variation during those dates.
The first trial for a crop growth map was performed with the coffee crop, presented in Figure 8. No field measurement of the crop height variation was determined. However, a rough growth estimate of about 10 cm was expected, as the coffee crop was quite mature in December. Because of a small morphology alteration in the crop due to its maturity, the interferogram had a low but acceptable coherence of 0.2.
A growth map corresponding to a coffee crop of approximately 1700 m 2 is shown in Figure 11. The mean height growth between those dates with DInSAR is 11 cm, with a standard deviation of 6 cm. The south area of the coffee crop presented less growth than the north area. A mask was used to discard all targets that do not correspond to the coffee crop. Moreover, a 15 × 15 moving average filter was used over the interferogram, with a pixel spacing of 30 cm × 30 cm. Only data with a coherence greater than 0.1 were considered valid data. This result has motivated a more in-depth study of DInSAR for growth maps.      Figure 12 shows the corresponding SAR images for the acquisition dates of 11 December 2018, and 17 April 2019, where the yellow squares represent the coffee crop for which the growth map of Figure 11 was produced. (a)

Quantitative Validation of Growth Deficit Maps
SLC images of circular flight tracks were used to validate the growth estimation model considering the reflectivity contribution. The high resolution allows a much easier analysis, as the cornfield C2 is rather small.
To obtain the phase reference, known as the zero-movement phase, an object that would remain static during all flights was searched. Initially, it was thought to use the dihedral corner reflectors. However, it would not be possible to leave the dihedral corner reflectors at all times in the study area, because other daily activities are carried out there. Besides, a dihedral corner reflector can only be observed during a small portion of the circular track. Therefore, a metal fence near the crop area was chosen as a reference, as it is a fixed object and can be observed during the entire flight (see Figure 13). (a)

Quantitative Validation of Growth Deficit Maps
SLC images of circular flight tracks were used to validate the growth estimation model considering the reflectivity contribution. The high resolution allows a much easier analysis, as the cornfield C2 is rather small.
To obtain the phase reference, known as the zero-movement phase, an object that would remain static during all flights was searched. Initially, it was thought to use the dihedral corner reflectors. However, it would not be possible to leave the dihedral corner reflectors at all times in the study area, because other daily activities are carried out there. Besides, a dihedral corner reflector can only be observed during a small portion of the circular track. Therefore, a metal fence near the crop area was chosen as a reference, as it is a fixed object and can be observed during the entire flight (see Figure 13).

Quantitative Validation of Growth Deficit Maps
SLC images of circular flight tracks were used to validate the growth estimation model considering the reflectivity contribution. The high resolution allows a much easier analysis, as the cornfield C2 is rather small.
To obtain the phase reference, known as the zero-movement phase, an object that would remain static during all flights was searched. Initially, it was thought to use the dihedral corner reflectors. However, it would not be possible to leave the dihedral corner reflectors at all times in the study area, because other daily activities are carried out there. Besides, a dihedral corner reflector can only be observed during a small portion of the circular track. Therefore, a metal fence near the crop area was chosen as a reference, as it is a fixed object and can be observed during the entire flight (see Figure 13).
(a) To obtain the zero-movement phase, profiles were drawn for the interferogram over the area corresponding to the metal fence, Figure 13c. The profiles are shown in Figure 14a,b. The zeromovement phase was calculated by using the average of the profiles, which, in this example, resulted in 0.76 rad. Then, the height growth of the cornfield C2 was monitored, as shown in Table 2. These data, together with the reflectivity data for the corn crop and a neighboring area (see Figure 6), were used to estimate the correction factor, , of Equation (17), which varies along with the phenological life of the corn crop. To obtain the zero-movement phase, profiles were drawn for the interferogram over the area corresponding to the metal fence, Figure 13c. The profiles are shown in Figure 14a,b. The zero-movement phase was calculated by using the average of the profiles, which, in this example, resulted in 0.76 rad. To obtain the zero-movement phase, profiles were drawn for the interferogram over the area corresponding to the metal fence, Figure 13c. The profiles are shown in Figure 14a,b. The zeromovement phase was calculated by using the average of the profiles, which, in this example, resulted in 0.76 rad. Then, the height growth of the cornfield C2 was monitored, as shown in Table 2. These data, together with the reflectivity data for the corn crop and a neighboring area (see Figure 6), were used to estimate the correction factor, , of Equation (17), which varies along with the phenological life of the corn crop. Then, the height growth of the cornfield C2 was monitored, as shown in Table 2. These data, together with the reflectivity data for the corn crop and a neighboring area (see Figure 6), were used to estimate the correction factor, K, of Equation (17), which varies along with the phenological life of the corn crop.  Figure 15 displays three values for the correction factor, K, estimated by using the available radar data and field measurement data from the three different periods presented in Table 2. Figure 15 also shows the fitted curve (blue line) for the correction factor, K, as a function of the corn phenology stage.   Figure 15 displays three values for the correction factor, , estimated by using the available radar data and field measurement data from the three different periods presented in Table 2. Figure  15 also shows the fitted curve (blue line) for the correction factor, , as a function of the corn phenology stage. Due to the small amount of data, there is some flexibility when choosing the type of equation that best fits the experimental data. An exponential fitting curve was chosen, and the resultant expression is as follows: where represents the average number of days after planting, calculated over each data collection period, as displayed in Table 3. Equation (19) is a first attempt to find an expression for , using the few available data, and it is calculated with MATLAB Curve Fitting Toolbox [19]. Table 3 also shows the height-difference data between two dates obtained by field measurements in the corn crop and estimates from the proposed model, using the radar data together with the correction factor, .  Figure 16 presents the crop growth maps derived from the same acquisition intervals displayed in Figure 15. Rapid growth can be seen from 2 July to 17 July, though growth is noticeably slower between 17 July and 22 August. Due to the small amount of data, there is some flexibility when choosing the type of equation that best fits the experimental data. An exponential fitting curve was chosen, and the resultant expression is as follows: where d represents the average number of days after planting, calculated over each data collection period, as displayed in Table 3. Equation (19) is a first attempt to find an expression for K, using the few available data, and it is calculated with MATLAB Curve Fitting Toolbox [19]. Table 3 also shows the height-difference data between two dates obtained by field measurements in the corn crop and estimates from the proposed model, using the radar data together with the correction factor, K.  Figure 16 presents the crop growth maps derived from the same acquisition intervals displayed in Figure 15. Rapid growth can be seen from 2 July to 17 July, though growth is noticeably slower between 17 July and 22 August.

Discussion
Determining the zero-movement phase is an essential step for estimating crop growth. The zeromovement phase is not null due to external factors such as the system's thermal noise, the difference between flight paths from distinct dates and internal changes of the radar system between different flights [16]. The chosen reference object was a metal fence because it is always fixed, thus providing more robust results than moving objects, for which precise positioning between separate flights is not guaranteed. Moreover, it can be seen from all directions in a circular flight track, and it is typically found near crop fields.
Corner reflectors are not practical reference points for DInSAR, as they should be well fixed and kept clean during the DInSAR observation period. The authors intend to continue using corner reflectors, only for purposes of planimetric accuracy and height calibration. Luneburg lenses could also be used to better calibrate the back-projection processor [20].
No other publications were found that use the DInSAR technique on a corn crop to estimate growth, but some similar works are noteworthy. Erten et al. [10] used the DInSAR technique, with the TANDEM-X satellite, over a rice cultivation area, and were able to estimate growth with an RMSE

Discussion
Determining the zero-movement phase is an essential step for estimating crop growth. The zero-movement phase is not null due to external factors such as the system's thermal noise, the difference between flight paths from distinct dates and internal changes of the radar system between different flights [16]. The chosen reference object was a metal fence because it is always fixed, thus providing more robust results than moving objects, for which precise positioning between separate flights is not guaranteed. Moreover, it can be seen from all directions in a circular flight track, and it is typically found near crop fields.
Corner reflectors are not practical reference points for DInSAR, as they should be well fixed and kept clean during the DInSAR observation period. The authors intend to continue using corner reflectors, only for purposes of planimetric accuracy and height calibration. Luneburg lenses could also be used to better calibrate the back-projection processor [20].
No other publications were found that use the DInSAR technique on a corn crop to estimate growth, but some similar works are noteworthy. Erten et al. [10] used the DInSAR technique, with the TANDEM-X satellite, over a rice cultivation area, and were able to estimate growth with an RMSE of 18 cm. Cao et al. [21] used an airborne L-band radar system to measure landslide with a precision of a few centimeters.
In this work, a first attempt to monitor growth was carried out with a coffee crop, as described in Section 3.2. The coffee had a barely noticeable height variation between 11 December 2018, and 17 April 2019. However, using DInSAR, it was possible to measure an 11 cm growth, with a standard deviation of 6 cm. No model will estimate the growth of a crop that hardly grows.
As for the corn crop growth using correction factor K, the errors obtained from 2 July to 17 July 2019, and from 17 July to 22 August 2019, were 5 cm and 2 cm, respectively, while from 2 July to 22 August 2019, the error was 15 cm. Therefore, the longest period between data acquisitions produces the largest error, whereas shorter acquisition periods provide more accurate results. Therefore, to achieve a more accurate growth model oriented to the corn crop, it is necessary to retrieve a better estimation of the variable K, and thus more flights are required. Another way to develop this model would be to use the two interferometric C-band antennas to estimate crop height [22].
The corn crop was chosen due to its rapid growth. Nevertheless, this work could be extended to other crops, such as sugar cane or coffee. The case of sugar cane, which grows slower than corn, is of particular interest to the authors; therefore, more campaigns will be needed over time.

Conclusions
This work proposes a novel method for estimating the growth of different crops by executing circular flight paths with a drone-borne DInSAR operating in the L-band with HH polarization. First tests on late-stage coffee, corn and sugar cane crops have shown that it is possible to reliably estimate small height variations. In the case of coffee, the growth within approximately four months was estimated at 11 cm, with a standard deviation of 6 cm. Such a growth rate is difficult to perceive visually or to measure with conventional tools.
Furthermore, a method is proposed to estimate the growth of a corn crop, taking into account its phenological development. Although more campaigns on the area of interest are still necessary, these first-stage results show a strong agreement with the measured field data. The largest error was 15 cm, corresponding to the longest period between data acquisitions, for which the crop growth was approximately 55 cm. In contrast, errors of up to 5 cm were obtained for shorter acquisition periods.
The images obtained from the circular flights were processed with the back-projection algorithm using a 30 cm × 30 cm sampling. In future work, a more accurate growth estimation is expected since back-projection images shall be processed with a 5 cm × 5 cm sampling.
The authors are motivated to continue with this line of research, taking as a starting point the methods and first-stage results obtained so far. More consistent results are expected in the near future, when more data will be available.