Ultrasonic Proximal Sensing of Pasture Biomass

: The optimization of pasture food value, known as ‘biomass’, is crucial in the management of the farming of grazing animals and in improving food production for the future. Optical sensing methods, particularly from satellite platforms, provide relatively inexpensive and frequently updated wide-area coverage for monitoring biomass and other forage properties. However, there are also beneﬁts from direct or proximal sensing methods for higher accuracy, more immediate results, and for continuous updates when cloud cover precludes satellite measurements. Direct measurement, by cutting and weighing the pasture, is destructive, and may not give results representative of a larger area of pasture. Proximal sensing methods may also su ﬀ er from sampling small areas, and can be generally inaccurate. A new proximal methodology is described here, in which low-frequency ultrasound is used as a sonar to obtain a measure of the vertical variation of the pasture density between the top of the pasture and the ground and to relate this to biomass. The instrument is designed to operate from a farm vehicle moving at up to 20 km h − 1 , thus allowing a farmer to obtain wide coverage in the normal course of farm operations. This is the only method providing detailed biomass proﬁle information from throughout the entire pasture canopy. An essential feature is the identiﬁcation of features from the ultrasonic reﬂectance, which can be related sensibly to biomass, thereby generating a physically-based regression model. The result is signiﬁcantly improved estimation of pasture biomass, in comparison with other proximal methods. Comparing remotely sensed biomass to the biomass measured via cutting and weighing gives coe ﬃ cients of determination, R 2 , in the range of 0.7 to 0.8 for a range of pastures and when operating the farm vehicle at speeds of up to 20 km h − 1 . capability of through a range of frequencies in an FM chirp, which also provides for high vertical spatial resolution of 11 mm. By through the it is possible to record the reflectance profile, which is related to pasture or biomass density.


Introduction
Biomass describes the food value of pasture for grazing animals. It is the 'dry matter' (DM) weight, per unit area of land, resulting when the pasture is cut to the ground, dried, and weighed [1]. Biomass is therefore an areal density, measured in kg m −2 or kg ha −1 (a hectare is 10 4 m 2 ). Increasing the dry matter of perennial forages remains a crucial factor underpinning the profitability of grazing industries [2].
Pasture biomass may be measured by cutting, drying, and weighing quadrants of pasture. However, this is destructive and time-consuming, as well as unlikely to give information representative of a larger area of pasture. Therefore, this direct evaluation is generally only used for the detailed calibration of other methods. Satellite, aerial, and ground-based platforms equipped with advanced sensors provide the potential for fast, non-destructive, and low-cost monitoring of plant growth, development, and yield in a field environment [3]. The methods currently used include the measurement of parameters such as capacitance, spectral properties, pasture height (known as sward height), and compressed sward height [4][5][6][7][8][9][10][11][12]. A wide range of optical sensors have been used to measure been reported [17,[37][38][39][40][41][42], but these methods suffer from being more complicated in both hardware and analysis. Furthermore, the optical properties of the surface of a 3D object do not in general describe the interior of the object.
The potential for using ultrasound to measure the echo strength from individual grass leaves has been recognized [44], but the influence of factors such as the surface area of each grass blade and their orientation relative to the transducer has been unknown. Previous studies using ultrasound have obtained biomass estimates from sward height, where the ultrasonic sensor is used as a range-finder. The only information used from the ultrasonic signal is the first time of arrival of the echo from the top of the grass. The remainder of the signal has been ignored.
Information about the variation of pasture density throughout the canopy is absent from all the methods outlined above. An analogy can be drawn with human BMI (body mass index) which, like pasture biomass, is a measure of kg m −2 . Human height alone is known to be a poor indicator of BMI, particularly across regions. Much improved estimates are obtained by also weighing, or by profiling in some way, such as via waist measurements.
The current work describes the use of a new ultrasonic instrument which profiles throughout the depth of the pasture from the top of the grass to the ground to obtain both height and density information [45,46]. To the best of the authors' knowledge, this is the first study to investigate the potential of using echoes from throughout the pasture layer to provide enhanced evaluation of pasture properties. The objective is to improve upon height-only methods using a single instrument and using a scattering model approach.
Section 2 develops an ultrasonic pasture meter equation which describes the dependence of received signal strength on instrumental factors and pasture scattering properties. Section 3 provides calibration data for the pasture meter equation and parameters for interpretation of field results. The setup for field evaluations and the nature of received signals, and their connection to pasture properties, are described in Section 4. An acoustic scattering model is developed in Section 5, which provides guidance for multi-parameter regressions to better estimate biomass using vertical pasture density information as well as sward height. Finally, in Section 6, the results of biomass estimation are presented and discussed. Finally, the conclusion is provided in Section 7.

Signal Generation and Reception
Assume a tonal sine wave of reference voltage amplitude V ref is fed to a transmitting element. The acoustic pressure output at a reference distance of R ref is: where s tx is the sensitivity of the transmitter in Pa V −1 . The acoustic intensity at a distance R ref from the transmitter is: where z 0 is the acoustic impedance of air [47]. If the range to the pasture is R, and the amplitude of the sine wave driving the transmitter is V tx , the incident intensity at range R is [47]: If scattering occurs from an object having backscattering cross section of σ bs , then the intensity at a microphone which is co-located with the transmitter is I bs , where [48]: Remote Sens. 2019, 11, 2459 4 of 20 At the receiver, the acoustic pressure due to backscattered power is [47]: p rx = (I bs z 0 ) 1/2 .
If the receiver has a sensitivity of s rx in V Pa −1 , and the preamplifier has a gain of G rx , then the voltage output V rx is given by:

Sensor Arrays and Beam-forming
Equation (6) applies to a single speaker and single microphone, and so does not include the beam-forming gain of the arrays used. For small-angle, normal incidence, the output voltage will simple be a factor N tx N rx larger, where N tx is the number of speakers in the transmitter array, and N rx is the number of microphones in the receiver array. If the antenna is approximated by a disk of diameter D, then the angular dependence of both the transmitted and received power is the Airy diffraction pattern [47]. The use of a sensor array changes (6) to: where, and J 1 is the Bessel function of the first kind, k is the wavenumber, and θ is the off-axis angle. The right-hand side of Equation (7) comprises three terms in square brackets. The first is instrumental, the second relates to propagation, and the last term relates to the pasture. With proper calibration, and knowledge of range R, the backscatter cross-section σ bs can be estimated from the received voltage. However, in order to estimate biomass, a relationship between σ bs and biomass needs to be established.

Calibration in the Laboratory
Laboratory measurements were performed to measure the characteristics of the ultrasonic array hardware that would be used for field trials. The measurements presented here were designed to understand how the receiver voltage due to an echo would vary with the effective cross-sectional area of a blade of grass. Measurement were, therefore, made for targets with different cross-sectional areas.
Sonar systems are similar to the new ultrasonic pasture meter in transmitting a short pulse and receiving echoes which are interpreted in terms of known types of targets [49]. A common method for calibrating sonar systems is to use a small solid sphere as a reflecting target [50]. However, the use here of a 20-35 kHz linear FM chirp signal, which has a range resolution of c/(2∆f ) = 11 mm [51], means that echoes from the full sphere diameter were not sensed at once, which is quite unlike a tonal sonar calibration. Therefore, small solid circular disks of radius a were used as calibration targets. Spherical and disk shapes are not close approximations to the shape of a pasture sward, but the purpose of the laboratory calibration was to confirm the veracity of Equation (7). The size parameter range studied was ka = 1.5−10.4, generally within the geometric scattering range. The setup is shown in Figure 1. Note that the green disk is facing the array (in the blue casing), and the apparent misalignment is due to camera perspective distortion.
To confirm that the disks provided results which were representative of grass, measurements were also made with a segment of grass of 24.5 mm length which was trimmed from a 4.5 mm wide blade ( Figure 2). Measurements were conducted of backscatter (0 • incidence). The area of 24.5 × 4.5 = 110 mm 2 is the same as a circular disk of radius a = 6 mm. This small length of blade was chosen so that the blade spanned an angular region over which the beam intensity was nearly constant. The blade segment was supported on a long, very fine, wire of diameter 0.32 mm. Reflection from the wire alone was undetectable.

Sensitivity
For a disk of radius a and sound normally incident: Results of measurements are shown in Figure 3 and relevant parameters in Table 1. The transmitted signal was the linear frequency-modulated (LFM) chirp [51]. The expected response, shown as a solid red line, used the specifications for the MA 40 H1S-R speaker and the SPU1410 LR5H-QB microphone, and a frequency of 24 kHz. Also shown is the response at equivalent disk radius for the blade of pasture. The grass blade segment produced a reflection close to that of a hard disk of equivalent area, indicating that ultrasound did not penetrate through the grass.
Results of measurements are shown in Figure 3 and relevant parameters in Table 1. The transmitted signal was the linear frequency-modulated (LFM) chirp [51]. The expected response, shown as a solid red line, used the specifications for the MA 40 H1S-R speaker and the SPU1410 LR5H-QB microphone, and a frequency of 24 kHz. Also shown is the response at equivalent disk radius for the blade of pasture. The grass blade segment produced a reflection close to that of a hard disk of equivalent area, indicating that ultrasound did not penetrate through the grass.

Beam Pattern
Measurements were also made to determine the beam pattern of the array. This was important, as the beam pattern determined the area of pasture that the ultrasonic sensor would sample in each transmission. In order to obtain the angular beam pattern, the ultrasonic pasture meter was mounted on a Sherline CNC Rotary Indexer. This is a rotary table using a stepping motor and gearbox producing an angular step of 0.0125°. The table and motor/gearbox can be seen beneath the pasture meter in Figure 1. Measurements were made in 0.1° steps on either side of the central direction. The distance between the pasture meter and the closest part of the target was R = 0.78 m, the approximate distance the pasture meter was mounted above the ground on the farm vehicle is described later.
Results of measurements are shown in Figure 3 and relevant parameters in Table 1. The transmitted signal was the linear frequency-modulated (LFM) chirp [51]. The expected response, shown as a solid red line, used the specifications for the MA 40 H1S-R speaker and the SPU1410 LR5H-QB microphone, and a frequency of 24 kHz. Also shown is the response at equivalent disk radius for the blade of pasture. The grass blade segment produced a reflection close to that of a hard disk of equivalent area, indicating that ultrasound did not penetrate through the grass.

Beam Pattern
Measurements were also made to determine the beam pattern of the array. This was important, as the beam pattern determined the area of pasture that the ultrasonic sensor would sample in each transmission. In order to obtain the angular beam pattern, the ultrasonic pasture meter was mounted on a Sherline CNC Rotary Indexer. This is a rotary table using a stepping motor and gearbox producing an angular step of 0.0125°. The table and motor/gearbox can be seen beneath the pasture meter in Figure 1. Measurements were made in 0.1° steps on either side of the central direction. The distance between the pasture meter and the closest part of the target was R = 0.78 m, the approximate distance the pasture meter was mounted above the ground on the farm vehicle is described later.

Beam Pattern
Measurements were also made to determine the beam pattern of the array. This was important, as the beam pattern determined the area of pasture that the ultrasonic sensor would sample in each transmission. In order to obtain the angular beam pattern, the ultrasonic pasture meter was mounted on a Sherline CNC Rotary Indexer. This is a rotary table using a stepping motor and gearbox producing an angular step of 0.0125 • . The table and motor/gearbox can be seen beneath the pasture meter in Figure 1. Measurements were made in 0.1 • steps on either side of the central direction. The distance between the pasture meter and the closest part of the target was R = 0.78 m, the approximate distance the pasture meter was mounted above the ground on the farm vehicle is described later.  The measured beam pattern for intensity is shown in Figure 4. The pattern very closely matched the Airy disk diffraction pattern over the central lobe. However, the side lobes, due to the finite number of sensors dominating from a radius of 0.2 m, projected onto the ground at a distance of 0.78 m (a beam angle of 14°). These side lobes were -20 dB, down from the -3 dB width at 68 mm radius, and are acceptable. Also shown, in green, are the measurement standard deviation around the measured pattern. The measurement error is very small. The asymmetry in the side lobe intensities is to be expected because the locations of the sensors were not symmetric. The measured beam pattern for intensity is shown in Figure 4. The pattern very closely matched the Airy disk diffraction pattern over the central lobe. However, the side lobes, due to the finite number of sensors dominating from a radius of 0.2 m, projected onto the ground at a distance of 0.78 m (a beam angle of 14 • ). These side lobes were -20 dB, down from the -3 dB width at 68 mm radius, and are acceptable. Also shown, in green, are the measurement standard deviation around the measured pattern. The measurement error is very small. The asymmetry in the side lobe intensities is to be expected because the locations of the sensors were not symmetric. The measured beam pattern for intensity is shown in Figure 4. The pattern very closely matched the Airy disk diffraction pattern over the central lobe. However, the side lobes, due to the finite number of sensors dominating from a radius of 0.2 m, projected onto the ground at a distance of 0.78 m (a beam angle of 14°). These side lobes were -20 dB, down from the -3 dB width at 68 mm radius, and are acceptable. Also shown, in green, are the measurement standard deviation around the measured pattern. The measurement error is very small. The asymmetry in the side lobe intensities is to be expected because the locations of the sensors were not symmetric.

Theoretical Considerations
Based on the laboratory calibration, Equation (7) is a good representation of the instrument operating principles. However, a disk shape is not an adequate model for a blade of grass. A closer approximation is a finite cylinder of radius a and length L, for which [50]: Here, θ is the angle between the propagation direction and the normal to the cylinder axis. For 24 kHz and L = 0.14 m (i.e., a blade lying across the full beam width at half-power), σ bs drops Remote Sens. 2019, 11, 2459 7 of 20 to half its maximum at θ = 1.5 • . This means that the scattering is very directional. An even better scattering model would be a thin flat strip [52]. For that case, the backscatter is very sensitive to the orientation to the horizontal of the flat surface of the strip. Furthermore, blades may lie partially across the active beam area, or be partially obscured by other blades.
The net result is that modelling the detailed scattering geometry was not possible, and a statistical approach was necessary.

Field Experiment Setup
Field trials were performed to evaluate the operational performance using the hardware described in the previous section. The design goal was an instrument which will operate in real-time from a moving platform. A series of experiments were conducted with the instrument mounted on a farm vehicle, with the vehicle moving at constant speeds of 5, 10, 15, and 20 km h −1 (1.4 to 5.6 m s −1 ). Figure 5 shows the farm vehicle with the instrument, which is at the front of the aluminium frame. Also mounted on this frame was a much larger array, not discussed here.  The measured beam pattern (solid black line) and ±one standard deviation (green). Also shown is the Airy diffraction pattern at 24 kHz (blue line) and the half-power range limits (red lines).

Theoretical Considerations
Based on the laboratory calibration, Equation (7) is a good representation of the instrument operating principles. However, a disk shape is not an adequate model for a blade of grass. A closer approximation is a finite cylinder of radius a and length L, for which [50]: Here, θ is the angle between the propagation direction and the normal to the cylinder axis. For 24 kHz and L = 0.14 m (i.e., a blade lying across the full beam width at half-power), σbs drops to half its maximum at θ = 1.5°. This means that the scattering is very directional. An even better scattering model would be a thin flat strip [52]. For that case, the backscatter is very sensitive to the orientation to the horizontal of the flat surface of the strip. Furthermore, blades may lie partially across the active beam area, or be partially obscured by other blades.
The net result is that modelling the detailed scattering geometry was not possible, and a statistical approach was necessary.

Field Experiment Setup
Field trials were performed to evaluate the operational performance using the hardware described in the previous section. The design goal was an instrument which will operate in real-time from a moving platform. A series of experiments were conducted with the instrument mounted on a farm vehicle, with the vehicle moving at constant speeds of 5, 10, 15, and 20 km h -1 (1.4 to 5.6 m s -1 ). Figure 5 shows the farm vehicle with the instrument, which is at the front of the aluminium frame. Also mounted on this frame was a much larger array, not discussed here.  The vehicle was operated at four speeds, 5, 10, 15, and 20 km h −1 (1.4, 2.8, 4.2, and 5.6 m s −1 ), as nearly as the driver could manage by watching the vehicle speedometer. This means that the actual platform speed varied somewhat. Three strips of pasture, or 'transects', were repeatedly passed over with the pasture meter. Each transect was traversed at each of the four speeds, and generally there were several 'passes' at each speed. Care was taken to follow the same wheel tracks at each pass over a transect, and to avoid any disturbance between the wheel tracks of the pasture being sampled. Following the vehicle operations, the pasture was cut, dried, and weighed from 0.5 × 0.5 m 'quadrats' along the transects.
In order to compare ultrasonic measurements with the ground truth biomass obtained by cutting, a vehicle position registration scheme was required. The ultrasound profiles needed to be located within the 0.5 m cut areas, so registration to better than 0.1 m was ideally required. Smartphone and mapping grade GPS do not achieve this accuracy [53]. Using real-time differential corrections allows navigation to within one to two meters of any location depending on the service and the GPS receiver, and only under the very best conditions can 0.1 m accuracy be obtained. Therefore, an alternative registration method was devised. This involved setting posts at intervals along the edge of a transect and using an infrared sensor to detect each post as the vehicle passed by. The ultrasonic and infrared signals were simultaneously sampled. This allowed the farm bike speed to be measured and the location of each ultrasonic transmission to be estimated. The method was under review during the field experiments, because some of the posts were not registered by the infrared sensor in bright sunlight. This meant that a different scheme was used for each transect. Transect 1 had 21 posts at 1 m intervals, with the overall transect length being 20 m. Detection of a post caused the transmission of three ultrasonic pulses, at 50 ms intervals, except for the first post, for which there were six ultrasonic pulses transmitted at 50 ms intervals. Post number 9 was omitted, so that the gap could be an extra reference point. Biomass quadrats were centered on each post position, as shown in Figure 6. a vehicle position registration scheme was required. The ultrasound profiles needed to be located within the 0.5 m cut areas, so registration to better than 0.1 m was ideally required. Smartphone and mapping grade GPS do not achieve this accuracy [53]. Using real-time differential corrections allows navigation to within one to two meters of any location depending on the service and the GPS receiver, and only under the very best conditions can 0.1 m accuracy be obtained. Therefore, an alternative registration method was devised. This involved setting posts at intervals along the edge of a transect and using an infrared sensor to detect each post as the vehicle passed by. The ultrasonic and infrared signals were simultaneously sampled. This allowed the farm bike speed to be measured and the location of each ultrasonic transmission to be estimated. The method was under review during the field experiments, because some of the posts were not registered by the infrared sensor in bright sunlight. This meant that a different scheme was used for each transect. Transect 1 had 21 posts at 1 m intervals, with the overall transect length being 20 m. Detection of a post caused the transmission of three ultrasonic pulses, at 50 ms intervals, except for the first post, for which there were six ultrasonic pulses transmitted at 50 ms intervals. Post number 9 was omitted, so that the gap could be an extra reference point. Biomass quadrats were centered on each post position, as shown in Figure  6.       Figure 8 shows a waterfall display of all the reflectance profiles from Pass 4 at 10 km h −1 from Transect 3. There are features of this plot common to all speeds and transects. Firstly, the top of the pasture is relatively easily seen as the region above which the reflectivity does not vary. A simple thresholding scheme allows this point to be estimated. Secondly, there are frequently reflectance peaks near the expected ground (vertical distance 0 m in the figure), but not always so. In practice, reflectivity is windowed in a region around the expected range to the ground and the position of the maximum within that window taken to be the ground position. Thirdly, all profiles exhibit a relatively small number of well-defined peaks, each of which has similar properties to the expected chirp response to a point reflector [51]. This suggests that a few individual blades of grass are contributing to the reflectance for each profile. The correlation between peak positions in adjacent profiles is not high, so the dominant grass blades do not clearly extend across profiles. Fourthly, there are reflections at ranges greater than the instrument-to-ground range, labelled on this plot as being 'below ground'. These arise from multiple reflections.

Reflecting Objects
The peaks above a given threshold of 0.25 V 2 were identified using the CLEAN algorithm [54]. In this algorithm, the peak was found and then removed, assuming the shape predicted from the chirp response. The peak in the modified, 'cleaned' profile was then found and the process repeated. The CLEAN algorithm enables an enhanced vertical resolution. An example is given in Figure 9. Overlapping peaks are also evident.
A sward height of around 0.1 m corresponds to a biomass of around 0.2 kg m −2 (see Figure 13 below), or a bulk density of around ρ = 2 kg m −3 . Taking the 3 dB width of the ultrasonic beam as 0.14 m, the volume sampled was around V s = 1.5 x 10 −3 m 3 , so the dry mass within the beam was around ρV s = 3 x 10 −3 kg. Figure 10 shows the number distribution per profile of peaks exceeding the 0.25 V 2 threshold, with an average of 11 blades sensed in each profile. The dry mass per sensed blade would therefore be around 3 x 10 −4 kg. DM is typically 35%, giving a blade density of 350 kg m −3 , so an average sensed blade volume would be 8 x 10 −7 m 3 . Assuming each grass blade is of width 5 mm and lies across this volume, the thickness of a grass blade would be around 1 mm. While this calculation is very crude, this estimated blade thickness is far too high. The conclusion is that not all blades were sensed, because of the variation in blade orientation. m, the volume sampled was around Vs = 1.5 x 10 m , so the dry mass within the beam was around ρVs = 3 x 10 -3 kg. Figure 10 shows the number distribution per profile of peaks exceeding the 0.25 V 2 threshold, with an average of 11 blades sensed in each profile. The dry mass per sensed blade would therefore be around 3 x 10 -4 kg. DM is typically 35%, giving a blade density of 350 kg m -3 , so an average sensed blade volume would be 8 x 10 -7 m 3 . Assuming each grass blade is of width 5 mm and lies across this volume, the thickness of a grass blade would be around 1 mm. While this calculation is very crude, this estimated blade thickness is far too high. The conclusion is that not all blades were sensed, because of the variation in blade orientation.   The probability distribution of the magnitude of the peaks, shown in Figure 11, reflects the probability distribution of the blade orientations and the variation in blade size. It is beyond the scope of the present work to attempt to separate these two effects. The mean magnitude of a peak is 1.5 V 2 . Most of the variability in numbers of peaks is in the smaller magnitude peaks, meaning that the  The probability distribution of the magnitude of the peaks, shown in Figure 11, reflects the probability distribution of the blade orientations and the variation in blade size. It is beyond the scope of the present work to attempt to separate these two effects. The mean magnitude of a peak is 1.5 V 2 . Most of the variability in numbers of peaks is in the smaller magnitude peaks, meaning that the The probability distribution of the magnitude of the peaks, shown in Figure 11, reflects the probability distribution of the blade orientations and the variation in blade size. It is beyond the scope of the present work to attempt to separate these two effects. The mean magnitude of a peak is 1.5 V 2 . Most of the variability in numbers of peaks is in the smaller magnitude peaks, meaning that the discrete random nature of the number of blades of grass reflecting from within the sensitive volume of this instrument is only a few percent.

Multiple Scattering
The profiles shown in Figures 8 and 9 show 'distance above ground' as being negative for some reflectance peaks. These peaks occurred from ranges further than the distance from the instrument to the ground and arose from secondary scattering. Figure 12 shows the geometry. The path could either be a ground reflection followed by a pasture reflection (as shown) or a pasture reflection followed by a ground reflection. The extra path length beyond the instrument-ground distance was (x 2 + h 2 ) 1/2 , but since the distance to the reflector was half the path length, the apparent 'below ground' distance was (x 2 + h 2 ) 1/2 /2. For a sward height of 0.2 m and a 3 dB beam half-width of 0.07 m, the maximum extra distance for secondary scattering would be 0.11 m. This is consistent with what is shown in Figure 9.
The fact that secondary scattering is always seen as discrete scattering events means that there is always penetration of the ultrasound to the ground, although the ground may not be the dominant scatterer. The small number of physically small scattering elements from the pasture means that secondary scattering from a grass blade onto another grass blade and then onto the receivers was insignificant.
Remote Sens. 2019, 11, x FOR PEER REVIEW 12 of 22 discrete random nature of the number of blades of grass reflecting from within the sensitive volume of this instrument is only a few percent.

Multiple Scattering
The profiles shown in Figures 8 and 9 show 'distance above ground' as being negative for some reflectance peaks. These peaks occurred from ranges further than the distance from the instrument to the ground and arose from secondary scattering. Figure 12 shows the geometry. The path could either be a ground reflection followed by a pasture reflection (as shown) or a pasture reflection followed by a ground reflection. The extra path length beyond the instrument-ground distance was (x 2 + h 2 ) 1/2 , but since the distance to the reflector was half the path length, the apparent 'below ground' distance was (x 2 + h 2 ) 1/2 /2. For a sward height of 0.2 m and a 3 dB beam half-width of 0.07 m, the maximum extra distance for secondary scattering would be 0.11 m. This is consistent with what is shown in Figure 9.
The fact that secondary scattering is always seen as discrete scattering events means that there is always penetration of the ultrasound to the ground, although the ground may not be the dominant scatterer. The small number of physically small scattering elements from the pasture means that secondary scattering from a grass blade onto another grass blade and then onto the receivers was insignificant.  discrete random nature of the number of blades of grass reflecting from within the sensitive volume of this instrument is only a few percent.

Multiple Scattering
The profiles shown in Figures 8 and 9 show 'distance above ground' as being negative for some reflectance peaks. These peaks occurred from ranges further than the distance from the instrument to the ground and arose from secondary scattering. Figure 12 shows the geometry. The path could either be a ground reflection followed by a pasture reflection (as shown) or a pasture reflection followed by a ground reflection. The extra path length beyond the instrument-ground distance was (x 2 + h 2 ) 1/2 , but since the distance to the reflector was half the path length, the apparent 'below ground' distance was (x 2 + h 2 ) 1/2 /2. For a sward height of 0.2 m and a 3 dB beam half-width of 0.07 m, the maximum extra distance for secondary scattering would be 0.11 m. This is consistent with what is shown in Figure 9.
The fact that secondary scattering is always seen as discrete scattering events means that there is always penetration of the ultrasound to the ground, although the ground may not be the dominant scatterer. The small number of physically small scattering elements from the pasture means that secondary scattering from a grass blade onto another grass blade and then onto the receivers was insignificant.

Biomass and Reflectance
The biomass is: where ρ is the bulk density of B, H is the depth of the pasture, and height h is measured upward from the ground. The bulk density includes the empty space between grass blades. Assume that at height h there are n v (h) identical grass blades per unit volume, each having mass m(h) and back-scattering cross section σ bs (h). In the height interval h to h + dh, the biomass is: The backward scattered acoustic power, dP, from the same volume is: where P i is the power incident on the area at depth h. Combining Equations (12) and (13), where β will be called the "blade areal density". Like B, β is a mass per unit area.

Height Variation within the Pasture Layer
Both the blade areal density and the back-scattered acoustic power may be expected to vary with depth within the pasture layer. Expanding the blade areal density as a polynomial in h, giving: where c n = b n /P i and: The R n are measures of the shape of the reflectivity profile. For example, and: are the power-weighted mean height and height variance within the pasture layer. R 3 is a measure of skewness and R 4 a measure of kurtosis. Equation (16) is a calibration equation which estimates biomass B for each ultrasonic profile using the R n derived from the profile reflectivity and using the constant calibration coefficients c n . It predicts that B = 0 when H = 0, in accordance with expectations, but in contrast to calibration equations for other pasture biomass instruments, such as the Rising Plate meter or C-Dax. Underpinning this approximation is the major assumption that β(h) has a constant shape for all sward heights and varieties. Essentially, this is assuming that if the mass m(h) of a blade in a layer increases, then the back-scattering cross section σ bs (h) increases proportionally. While not physically unreasonable, this assumption can only be tested through field investigations, in which the biomass is measured by cutting, drying, and weighing. Note that this assumption does not restrict the shape of the overall biomass profile.

Field Calibration Methodology
Equation (16) provides a basis for field calibration, in which biomass B q is measured via cutting, drying, and weighing at a number of quadrats q = 1, 2, . . . , Q together with ultrasonic pasture meter profiles providing backscattered power dP j,q at height intervals j in each quadrat. The regressors R n,q are obtained from these measurements and a multiple linear regression performed to estimate the coefficients c n .
If these calibrations are performed over a number of pasture sites, seasons, and varieties, then the assumption that the c n is constant can be checked, and also how well the overall model explains variance in measured B.

Relationship to Other Methods
From equations (13) and (17): providing the term in brackets is constant throughout the pasture layer. This is equivalent to the approximation made by the ultrasonic sward and the C-Dax, for which it is assumed that the biomass is proportional to sward height H. From our field data (see below) we can estimate the residuals between our profile approximation (16) and the depth-only approximation (20).

Field Results
The field data are extensive and only a subset of results will be discussed here. A following publication will contain an exhaustive presentation and discussion of all transect data, as well as data from a non-moving platform.

Biomass Versus Sward Height
The sward height H can be estimated solely from the ultrasonic profile based on the position of a reflectance peak in the vicinity of the expected ground location. This allows for vertical movement of the farm vehicle due to its suspension. Biomass B was measured by cutting the pasture, drying the sample, and weighing it, and taking into account the area cut.
From Table 2, Transect 3 provided six passes over the same pasture at 10 km h −1 . The results of linear regressions of the following form: are shown in Figure 13. The six colors represent the six different passes. The coefficient of determination, R 2 , varies from 0.61 to 0.75, with a mean value of 0.66. Over all passes for this transect, at all speeds, the slope is µ ρ = 0.97 ± 0.05 kg m −3 and the intercept is B 0 = 0.088 ± 0.005 kg m −2 . In common with the methods used by the C-Dax, ultrasonic sward stick, and other biomass estimations based on sward height, this regression model is non-physical because it predicts a non-zero biomass for zero sward height.

Biomass Estimation Using the Reflectivity Profile
Regression models were evaluated with regressors H, R0, R1, R2, and R3 in various combinations. These models did not have a constant intercept, except for the model in equation (21), so they predicted that B → 0 as H → 0, which is physically reasonable. Results from the four models listed in Table 4 are shown here.

Model Regression equation Number N of regressors
3 Model 1 was simply dependence on sward height expressed by Equation (21). Model 2 was the N = 2 model from Equation (16). Models 3 and 4 used regressor H instead of R0 for the reason that it was found that much higher values of R 2 are obtained. Figure 14 shows the adjusted R 2 values obtained for the four models and for each of the 14 passes from Transect 3. The adjusted R 2 is related to R 2 via the following: Adjusted R 2 allows for the inflation of R 2 , which occurs as more regressors are added. As can be seen, the use of R0 gives a poor estimation of B. The two models, 3 and 4, which included profile information performed better than the depth-only model 1 across all passes at all speeds. Pass 1 exhibited some problems because of difficulty in registration with the reflectors. There was an indication that estimation of biomass at 20 km h -1 was slightly worse than at other speeds.
More important than R 2 is the estimation error, σB, for biomass based on these regressions. Figure  15 shows this estimation error for the four models and 14 passes. Excluding Pass 1, the mean estimation error for the four models was 340, 610, 460, and 400 kg m -2 . The best performance was by Figure 13. Linear regressions of biomass versus sward height for six passes at 10 km h −1 from Transect 3.

Biomass Estimation Using the Reflectivity Profile
Regression models were evaluated with regressors H, R 0 , R 1 , R 2 , and R 3 in various combinations. These models did not have a constant intercept, except for the model in equation (21), so they predicted that B → 0 as H → 0, which is physically reasonable. Results from the four models listed in Table 4 are shown here.
Model 1 was simply dependence on sward height expressed by Equation (21). Model 2 was the N = 2 model from Equation (16). Models 3 and 4 used regressor H instead of R 0 for the reason that it was found that much higher values of R 2 are obtained. Figure 14 shows the adjusted R 2 values obtained for the four models and for each of the 14 passes from Transect 3. The adjusted R 2 is related to R 2 via the following: Adjusted R 2 allows for the inflation of R 2 , which occurs as more regressors are added. As can be seen, the use of R 0 gives a poor estimation of B. The two models, 3 and 4, which included profile information performed better than the depth-only model 1 across all passes at all speeds. Pass 1 exhibited some problems because of difficulty in registration with the reflectors. There was an indication that estimation of biomass at 20 km h −1 was slightly worse than at other speeds.

Model Regression Equation Number N of Regressors
More important than R 2 is the estimation error, σ B , for biomass based on these regressions. Figure 15 shows this estimation error for the four models and 14 passes. Excluding Pass 1, the mean estimation error for the four models was 340, 610, 460, and 400 kg m −2 . The best performance was by the sward height model, B = B 0 + µ ρ H, and then the three-regressor model B = c 0 H + c 1 R 1 + c 2 R 2 . Figure 16 shows residuals at one of the best passes, Pass 3, for the model B = c 0 H + c 1 R 1 . Residuals did not show a well-defined increase with biomass B or sward height H, so are probably more closely related to model error rather than measurement error.

Model Resilience
From the data shown in Figure 15, the estimation error varied around 12% over Passes 2 to 14 for models B = B 0 + µ ρ H and B = c 0 H + c 1 R 1 , and around 21% for model B = c 0 H + c 1 R 1 + c 2 R 2 . The variation of the two fitted coefficients, c 0 and c 1 , for B = c 0 H+c 1 R 1 are shown in Figure 17. The first coefficient varied little over Passes 2 to 14. The second coefficient appeared to increase slightly with vehicle speed, but this is as likely to be a registration outcome as a genuine speed dependence.
Remote Sens. 2019, 11, x FOR PEER REVIEW 16 of 22 the sward height model, B = B0 + μρH, and then the three-regressor model B = c0H + c1R1 + c2R2. Figure  16 shows residuals at one of the best passes, Pass 3, for the model B = c0H + c1R1. Residuals did not show a well-defined increase with biomass B or sward height H, so are probably more closely related to model error rather than measurement error.

Model Resilience
From the data shown in Figure 15, the estimation error varied around 12% over Passes 2 to 14 for models B = B0 + μρH and B = c0H + c1R1, and around 21% for model B = c0H + c1R1 + c2R2. The variation of the two fitted coefficients, c0 and c1, for B = c0H+c1R1 are shown in Figure 17. The first coefficient varied little over Passes 2 to 14. The second coefficient appeared to increase slightly with vehicle speed, but this is as likely to be a registration outcome as a genuine speed dependence.

Discussion and Conclusions
Biomass is a crucial parameter in optimally managing the farming of grazing animals. Direct biomass measurement, via cutting and weighing, is destructive and time-consuming, as well as potentially giving results not representative of the larger pasture area. Consequently, a number of indirect methods have arisen. Sensors on satellite platforms have the advantages of providing data with a large areal coverage, at modest cost, and reasonable repetition rates. Furthermore, the use of wavelengths over a wide spectral range can provide not just biomass information, but also forage quality information. Nevertheless, farmers also may prefer a more immediate hands-on estimation method, which they can deploy during the normal course of their farm operations under any weather conditions. Of these proximal methods, the most convenient provide biomass estimation from a moving farm vehicle. However, such methods generally correlate an estimate of the pasture height with measured biomass, and the correlations are not robust across pasture species and seasons.
A new low-frequency ultrasound instrument was described, which has the capability of both sensing the depth of the pasture and penetrating through the pasture to the ground. The instrument achieves this by using a range of frequencies in an FM chirp, which also provides for high vertical spatial resolution of 11 mm. By penetrating through the pasture, it is possible to record the reflectance profile, which is related to pasture or biomass density.
The calibration of the new instrument was found to agree closely with theoretical expectations based on sensor specifications and the arrangement of sensors in spiral arrays. The ultrasonic beam pattern provided a -3 dB footprint diameter of 0.14 m, which is sufficiently small to avoid gross variations in biomass across the diameter, while being sufficiently large that statistics of small numbers of blades of grass being sampled is not a problem. Estimates of scattering strength agree with observations, lending confidence to the understanding of the principles upon which the methodology is based.

Discussion and Conclusions
Biomass is a crucial parameter in optimally managing the farming of grazing animals. Direct biomass measurement, via cutting and weighing, is destructive and time-consuming, as well as potentially giving results not representative of the larger pasture area. Consequently, a number of indirect methods have arisen. Sensors on satellite platforms have the advantages of providing data with a large areal coverage, at modest cost, and reasonable repetition rates. Furthermore, the use of wavelengths over a wide spectral range can provide not just biomass information, but also forage quality information. Nevertheless, farmers also may prefer a more immediate hands-on estimation method, which they can deploy during the normal course of their farm operations under any weather conditions. Of these proximal methods, the most convenient provide biomass estimation from a moving farm vehicle. However, such methods generally correlate an estimate of the pasture height with measured biomass, and the correlations are not robust across pasture species and seasons.
A new low-frequency ultrasound instrument was described, which has the capability of both sensing the depth of the pasture and penetrating through the pasture to the ground. The instrument achieves this by using a range of frequencies in an FM chirp, which also provides for high vertical spatial resolution of 11 mm. By penetrating through the pasture, it is possible to record the reflectance profile, which is related to pasture or biomass density.
The calibration of the new instrument was found to agree closely with theoretical expectations based on sensor specifications and the arrangement of sensors in spiral arrays. The ultrasonic beam pattern provided a -3 dB footprint diameter of 0.14 m, which is sufficiently small to avoid gross variations in biomass across the diameter, while being sufficiently large that statistics of small numbers of blades of grass being sampled is not a problem. Estimates of scattering strength agree with observations, lending confidence to the understanding of the principles upon which the methodology is based.
A model was developed for the scattering of ultrasound by pasture. This model makes a connection between observations of reflectance and the biomass in the field of view. From this connection, a number of suitable regressors were suggested between ultrasonic observables and biomass. Field studies conducted at vehicle speeds of 5 to 20 km h −1 , together with direct biomass measurements, allowed a number of regression models to be evaluated. It was found that the best models included both pasture height and at least one measure of reflectance variation within the pasture layer. By including variation within the pasture layer, biomass estimation was significantly improved. The coefficient of variation, R 2 , if pasture height alone was used, was in the range 0.6 to 0.7 in these studies, compared with a range of 0.7 to 0.8 if profile information was included. The model also appeared to be resilient in that the regression coefficients were stable with vehicle speed and with a range of pasture covers.
The integrated ultrasonic reflectance, which provides extra information, depends on the amplitude of reflected ultrasound from each pasture blade within the sampled volume of pasture. A crude calculation, based on numbers of signal peaks from within the pasture canopy, suggested variations in blade orientation cause a substantial number of blades to return low amplitudes. This means that the regressors based on reflectance are in practice averages over the probability distribution of blade orientation. Nevertheless, this averaging appears to be robust, since multiple passes over the same pasture gave similar results in spite of the disturbance caused by the farm vehicle having passed above the pasture.
Some of the unexplained variance (or the fact that R 2 is less than 1) was undoubtedly due to remaining difficulties in registering the location of ultrasonic profiles with the location of cut quadrats used for reference biomass measurements. A further contribution to unexplained variance will be from pasture moisture variations, although this is thought to be a small influence since multiple passes over pasture on different days and times of day showed undetectable regression variations.
The instrument is small, low-power, and easy to use, and has the potential to be a readily accessible and useful addition to precision agriculture.