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Theoretical background noise rate over water surface for a photon-counting lidar and its application in land and sea cover classification

Zhiyu Zhang, Yue Ma, Nan Xu, Song Li, Jinyan Sun, and Xiao Hua Wang

Open Access Open Access

Abstract

For photon-counting lidars, the classical theoretical rate of the noise photons reflected by the Earth’s surface is under the assumption that the Earth’s surface is a Lambert reflector, which is obviously not suitable for the water surface. In this paper, the specular reflection theorem is introduced to derive an analytical expression of noise photons arising from the water surface reflection. The verification uses the mean noise rate over water surface, calculated by the raw data photons measured by the Multiple Altimeter Beam Experiment Lidar (MABEL) near the East Coast in the North Carolina, USA. The measured result coincides well with the theoretical noise rate, as both of them equal to 8.4 kHz. In addition, the background noise model also indicates that the background noise rate over the land surface is one order of magnitude larger than that over the water surface, in certain conditions. Hence, a new method, based on the noise rates, is proposed for the Earth’s surface type classification and it performs well in distinguishing all water surfaces from land surfaces in the coastal area. For space-borne or airborne photon-counting lidars, this paper not only fills the gap of theoretical rate of noise photons from the water surface but also provides a fast and effective method to classify the Earth’s surface types. This method is also suitable for distinguishing ice and water in high-latitude sea-ice covered regions, which is the area of most interest of the Ice, Cloud, and Land Elevation Satellite-2 (ICESat-2) mission.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Benefiting from photon-counting detectors with higher detection sensitivity and micro pulse lasers with higher repetition rate, photon-counting lidars can obtain much denser laser photons (points) compared to traditional full-waveform lidars [1,2]. To achieve more precise observation on the Earth’s surface, NASA launched the Ice, Cloud, and Land Elevation Satellite-2 (ICESat-2) in late 2018, which is equipped with a photon-counting lidar [3,4]. However, photon-counting lidars is very sensitive to both laser photons and noise photons (mainly arising from the solar-induced background radiation); therefore, discriminating the signal photons from the raw data is essential to photon-counting lidars since the raw data photons (points) are very noisy. The NASA’s research team proposed a surface-finding algorithm for the ICESat-2 and achieved a good performance on sea ice and ice sheets [5]. However, the surface characteristics vary significantly over different land cover types (e.g., with different reflectivity and profiles) and make the distribution of signal photons quite different. For different land cover types, the adaptive ellipsoid searching method for detecting the land and forest [6], the spatial statistical and discrete mathematical method for detecting the land and forest [7,8], and the Joint North Sea Wave Project (JONSWAP) wave spectrum method for detecting water surface have been proposed to extract the signal photons from the raw noise photons [9].

Accurate information on Earth’s surface types is to select an appropriate method and detect signal photons from raw data, especially in the coastal areas because the distribution of signal photons on the water surface is quite different from that of other surfaces. In previous study, with the help of the national land cover database (NLCD), the along-track Earth’s surface types are obtained from satellite optical images [10]. Although this method is very useful, the ancillary NLCD data is essential and only with a 30 m spatial resolution. Kwok proposed a simpler and more practical method to discriminate water from ice based on the measured noise rate of the photon-counting raw data [11,12]. This method is based on the fact that the noise photons reflected by the ice surface is much stronger than that of the water surface, as the empirical albedo of ice is approximately five times larger than that of water. However, the threshold for classification is determined by an empirical value, rather than an analytical expression. In fact, the noise rate is also related to the solar zenith angle and the atmospheric transmissivity. Degnan derived the analytical expression of the noise photon rates reflected by the land surface and scattered by the atmosphere [13]. Although the theoretical rate of reflected noise photons has been derived, by assuming that the Earth’s surface is a Lambert reflector, it is not suitable for the water surface. Normally, the water surface is considered as randomly distributed specular points.

In this paper, first, we derive a new analytical expression corresponding to solar-induced background noise reflected by the water surface. Second, with given system parameters of a photon-counting lidar and typical environmental parameters (e.g., the solar zenith angle, atmospheric transmissivity, and wind speed above water surface), the theoretical noise rate over land surface (containing noise photons from both the land surface and the atmosphere) and water surface (containing noise photons from both the water surface and the atmosphere) are calculated, respectively. In addition, the theoretical noise rate is verified by the data photons from Multiple Altimeter Beam Experimental Lidar (MABEL), which was used as a technology demonstrator for simulating the ICESat-2 data [14]. Third, based on the ratio of the noise rate over the water surface to the land surface, the along-track Earth’s surface types will be classified as water and land. The validation uses the statistical noise rates from MABEL raw data when the MABEL flew over the east coast of the North Carolina, USA. For space-borne or airborne photon-counting lidars, this paper will not only supplement the missing theoretical rate of noise photons from water surface, but provide a fast and effective method to classify the Earth’s surface types. The classification result is essential to extract signal photons from the noisy raw data precisely.

2. Model and analysis of solar-induced background noise

A photon-counting lidar that is equipped with single photon detectors (SPD) can record only one photon event (PE) within each time bin even though one or more photons are detected. However, the SPD can be triggered by either signal photons (i.e., laser photons reflected by the Earth’s surface) or noise photons. The noise PEs generally contain the dark counts from the dark current of the detector and the noise counts caused by the reflected solar radiation from the target (e.g., the Earth’s surface) and by the atmosphere. Within a single time bin, the probability that a noise photon triggers the SPD is relatively low, but the range gate of a space-borne photon-counting is normally over 1 km in vertical direction and corresponds to thousands of time bins. As a result, the raw data photons of a photon-counting lidar are very noisy, and plenty of noise photons distribute at tens to hundreds of meters above the Earth’s surface in raw data of a photon-counting lidar, which makes it simple to select noise counts and calculate the noise rate. The noise rate in units of Hz usually represents the noise level of a photon-counting lidar. The mean number of noise photons Nn within a short time interval τ (e.g., the range gate) can be calculated as Nn=τ·f, where f is the noise rate.

2.1 Model of solar-induced background noise

Normally, the noise rate of solar-induced background noise is several orders of magnitude larger than that of the dark current, hence the dark counts of the SPD can be ignored [15]. In the classical model of Degnan, the theoretical rate of noise photons arising from the land surface reflection fL and atmospheric backscatter fA has been derived and can be expressed as Eq. (1) and Eq. (2), respectively [13].

$${f_\textrm{L}} = \frac{{N_\lambda ^0(\Delta \lambda )\pi {{(z{\theta _{\mathop{\textrm r}\nolimits} })}^2}\eta {A_\textrm{r}}}}{{h\nu \pi {z^2}}}{\beta _\textrm{L}}T_\textrm{a}^{1 + \sec {\theta _\textrm{s}}}\cos \psi = \frac{{N_\lambda ^0(\Delta \lambda )\theta _{\mathop{\textrm r}\nolimits} ^2\eta {A_\textrm{r}}}}{{h\nu }}{\beta _\textrm{L}}T_\textrm{a}^{1 + \sec {\theta _\textrm{s}}}\cos \psi$$
$${f_\textrm{A}} = \frac{{N_\lambda ^0(\Delta \lambda )\theta _\textrm{r}^2\eta {A_\textrm{r}}}}{{h\nu }}\frac{{1 - T_a^{1 + \sec {\theta _\textrm{s}}}}}{{4(1 + \sec {\theta _\textrm{s}})}}$$
Nλ0 is the spectral solar irradiance that is measured outside the atmosphere; Δλ is the bandpass of the optical filter; z is the flight altitude of a photon-counting lidar; hv is the energy of a photon at the laser wavelength; η is the system-detector efficiency; θs is the solar zenith angle; θr is the half FOV (field of view) of the receiver; Ar is the area of the receiver aperture; βL is the land surface reflectivity; Ta is the one-way atmospheric transmittance.

Figure 1 illustrates the geometry of the solar irradiance that is reflected by the land surface and finally received by a photon-counting lidar. In Fig. 1, n is the normal of the land surface, and the coordinate system is centered at the FOV center, with the Z-axis pointing to the zenith direction and the plane X-Z containing the line connecting the origin and the center of the Sun. The angle ψ is the angle between the Sun and the normal direction of the surface n and can be calculated by

$$\cos \psi = cos{\sigma _\textrm{L}}cos{\theta _\textrm{s}} + \sin {\sigma _\textrm{L}}\sin {\theta _\textrm{s}}\cos \varphi ,$$
where σL is the surface slope and φ is the azimuth angle between the slope direction and X-axis. The classical Eq. (1) and Eq. (2) well represent the theoretical rate of noise photons arising from the land surface reflection and the atmospheric backscatter; however, in Eq. (1), the Earth’s surface is assumed as a Lambert reflector while the water surface does not satisfy. As a result, in this paper, the specular reflection theorem is introduced to supplement the theoretical rate of noise photons arising from the water surface reflection.

 figure: Fig. 1.

Fig. 1. Geometry of the solar irradiance that is reflected by the land surface and received by a photon-counting lidar.

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The water surface profile is fluctuant and has a certain slope value at each point on water surface. It should be noted that the sea surface is mainly covered with waves and also with a very small fraction of foams or whitecaps when the wind speed is over than 10 m/s. Lancaster has investigated the reflectance from the sea surface and whitecaps based on the data from the NASA’S Geoscience Laser Altimeter System (GLAS) spaceborne laser altimeter. The results indicated that the whitecaps have very small effect on the total sea surface reflectance [16]; therefore, in this paper, only the specular reflection is considered in the derivation over sea surface. Based on the specular reflection theorem and the geometry illustrated in Fig. 2, at each specular point, only when the surface slope σW satisfies the following equation set in Eq. (4), the incident sunlight can be reflected into the lidar receiver.

$$\left\{ {\begin{array}{{l}} {\left( {\frac{\pi }{2} - {\theta_{\textrm{sr}}}} \right) + \textrm{2}\omega + \left( {\frac{\pi }{2} - {\theta_\textrm{s}}} \right) = \pi }\\ {\omega + \left( {\frac{\pi }{2} - {\theta_\textrm{s}}} \right) + {\sigma_\textrm{W}} = \frac{\pi }{2}} \end{array}} \right.$$
In Fig. 2, the optical axis of the FOV points to the nadir direction; ρ is the distance between the FOV center and a given point on water surface; θsr is the angle between the optical axis of the FOV and the reflected sunlight and satisfies θsr≈tanθsr=ρ/z, where z is the flight altitude; n is the normal direction of a given point on water surface; and ω is the reflection angle. Based on Eq. (4), obviously, when the slope σW satisfies σW=(θs–ρ/z)/2, the reflected sunlight can be received by a lidar system.

 figure: Fig. 2.

Fig. 2. Schematic of the geometric constraints for the sunlight to be reflected into the receiver FOV by a sloping water surface. The origin of the coordinate coincides with the center of the receiver FOV, and the Z-axis points to the zenith direction. On the water surface, a polar coordinate is used with the distance ρ between the FOV center and a given point on water surface and the azimuth angle θ.

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Based on the wave spectrum theory, the water surface can be expressed as the superposition of many sine and cosine functions with different wave heights, wavelengths, and phases [17]. These waves are mainly driven by winds above water surface. Many models have been derived to represent the relationship between the wind speed and the slope inclination statistics, e.g., the Cox-Munk model and CALIPSO model [18]. The CALIPSO model is selected and expressed in Eq. (5). The CALIPSO model was based on the data from NASA’s Cloud-Aerosol Lidar with Orthogonal Polarization (CALIOP) spaceborne lidar. In the CALIPSO model, the relationship between the wind speeds and the slope inclination statistics was fitted by the CALIOP sea surface backscatters (at 1064 nm) and the Advanced Microwave Scanning Radiometer (AMSR-E) wind speeds. The CALIOP lidar used the Nd:YAG lasers with two wavelengths of 532 nm and 1064 nm, which are identical with the MABEL photon-counting lidar. Using three piecewise functions, the CALIPSO model has better agreement with the observed data than the classical Cox-Munk model when the wind speeds are very low or very high.

$$\left\{ {\begin{array}{{c}} {{s^2} = 0.146\sqrt w ,(w < 7\,{\mathop{\textrm m}\nolimits} \textrm{/}{\mathop{\textrm s}\nolimits} )}\\ {{s^2} = 0.003 + 0.00512w,(7 \le w < 13.3\,{\mathop{\textrm m}\nolimits} \textrm{/}{\mathop{\textrm s}\nolimits} )}\\ {{s^2} = 0.138{{\log }_{10}}w - 0.084,(w \ge 13.3\,{\mathop{\textrm m}\nolimits} \textrm{/}{\mathop{\textrm s}\nolimits} )} \end{array}} \right.$$
Each illuminated specular point on water surface scatters like a tangent sphere with a radius equals to the mean of major axis ra and minor axis rb, and its optical scattering cross section can be expressed as SRCS =πδ <rarb>, where < > is the mean function and δ is the reflection coefficient between the water and air surface (approximately equals to 2% at 532 nm and 1064 nm), which can be calculated by Fresnel’s formula [19]. By substituting the SRCS into Barrick’s theory, the reflectivity βW at a given point with respect to ρ can be expressed in the middle expression of Eq. (6) [20]. As analyzed in Eq. (4), when the slope σW satisfies σW=(θs–ρ/z)/2, the reflected sunlight can be received by a lidar system. As a result, the middle expression of Eq. (6) is then transformed into the right expression that is respect to ρ.
$${\beta _\textrm{W}}(\rho ) = \frac{{\delta {{\left[ {1 + {{\arctan }^2}\left( {\frac{\rho }{z}} \right)} \right]}^2}}}{{4\pi {s^2}}}\exp ( - \frac{{{{\tan }^2}{\sigma _\textrm{W}}}}{{{s^2}}}) = \frac{{\delta {{\left[ {1 + {{\arctan }^2}\left( {\frac{\rho }{z}} \right)} \right]}^2}}}{{4\pi {s^2}}}\exp \left[ { - \frac{{{{\left( {{\theta_\textrm{s}} - \frac{\rho }{z}} \right)}^2}}}{{4{s^2}}}} \right]$$
In Eq. (1), the solar irradiance on land surface is expressed as Nλ0(Δλ)·π(θr­·z)2 and the land reflectivity is βL·cosψ. Similar with Eq. (1), the theoretical rate of noise photons reflected by the water surface yields
$$\begin{array}{l} {f_\textrm{W}} = \frac{{N_\lambda ^0(\Delta \lambda )\eta {A_\textrm{r}}}}{{{z^2}h\nu }}T_a^{1 + \sec {\theta _s}}\int_0^{2\pi } {\int_0^{z{\theta _\textrm{r}}} {{\beta _\textrm{W}}(\rho )} } \rho \textrm{d}\rho \textrm{d}\theta \\ = \frac{{N_\lambda ^0(\Delta \lambda )\eta {A_\textrm{r}}\delta }}{{4\pi {s^2}{z^2}h\nu }}T_a^{1 + \sec {\theta _s}}{\int_0^{2\pi } {\int_0^{z{\theta _\textrm{r}}} {\left( {1 + {{\arctan }^2}\left( {\frac{\rho }{z}} \right)} \right)} } ^2}\exp \left[ { - \frac{{{{\left( {{\theta_\textrm{s}} - \frac{\rho }{z}} \right)}^2}}}{{4{s^2}}}} \right]\rho \textrm{d}\rho \textrm{d}\theta \\ \approx \frac{{N_\lambda ^0(\Delta \lambda )\eta {A_\textrm{r}}\delta }}{{2sh\nu }}T_a^{1 + \sec {\theta _\textrm{s}}}\left\{ {2s\{ \exp ( - \frac{{{\theta_\textrm{s}}^2}}{{4{s^2}}}) - \exp [ - \frac{{{{({\theta_\textrm{r}} - {\theta_s})}^2}}}{{4{s^2}}}]\} + \sqrt \pi {\theta_\textrm{s}}[erf(\frac{{{\theta_\textrm{s}}}}{{2s}}) + erf(\frac{{{\theta_\textrm{r}} - {\theta_\textrm{s}}}}{{2s}})]} \right\}, \end{array}$$
where θ is the integration variable of the azimuth angle on the water surface. In the first line of Eq. (7), The reflectivity on water surface βW(ρ) is in place of the βLcosψ in Eq. (1). In Eq. (7), the upper limit of the first and second integration (i.e., zθr) indicates that only the solar irradiance within the FOV of a photon-counting lidar is considered, which corresponds to the π(θr­z)2 in Eq. (1). It should be noted that, in the derivation of Eq. (7), because the radius ρ=z·tanθsr within the FOV is much less than the altitude z (ρ/z<<1), the following approximation is adopted, i.e., arctan(ρ/z) ≈ρ/z and (1+ρ/z)2≈1.

2.2 Analysis of solar-induced background noise

In Eqs. (1)–(2) and Eq. (7), the theoretical rates of noise photons arising from the land reflection, atmospheric backscatter, and water reflection are proposed, respectively. In addition to the system parameters of a photon-counting lidar, all three noise sources are influenced by the solar zenith angle and atmospheric transmittance, and in particular, the background noise from the land surface varies with the land surface reflectivity while the background noise from the water surface is influenced by the wind speed. In this section, for a typical photon-counting lidar, the noise rates of three sources are analyzed in detail under different environmental parameters (i.e., the solar zenith angle, atmospheric transmittance, land reflectivity, and wind speed above water surface). It should be noted that the solar irradiance Nλ0 is also determined by the solar zenith angle. The system parameters of the MABEL photon-counting lidar used in the model are shown in Table 1. The MABEL was borne on a high-altitude aircraft (at 20 km), which includes 95% of the total atmosphere; therefore, the MABEL was used as a technology demonstrator and produced data similar to the space-borne ICESat-2 [14].

Tables Icon

Table 1. Instrument parameters of the MABEL.

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The theoretical relationships between the noise rates (in units of kHz) from different noise sources and the solar zenith angles are illustrated in Fig. 3. Figure 3(a) corresponds to the background noise from the atmosphere, and the solid, dashed, and dash-dotted blue curves correspond to the atmospheric transmittance of 0.7, 0.8, and 0.9, respectively. The background noise rates from the land surface and water surface are separately analyzed in Figs. 3(b) and 3(c) with the atmospheric transmittance of Ta=0.8. The background noise rates from the land surface are simulated in Fig. 3(b), when the land surface reflectivity equals to 0.3 (using solid curve), 0.5 (using dashed curve), and 0.7 (using dash-doted curve). Meanwhile, the noise rates from the water surface correspond to different wind speeds (5 m/s, 10 m/s, and 15 m/s) are plotted using different curve shapes in Fig. 3(c).

 figure: Fig. 3.

Fig. 3. Solar-induced background noise rates from different sources. (a) Curves of background noise from the atmosphere fA versus the solar zenith angle θs when the atmospheric transmittance Ta equals to 0.7, 0.8, and 0.9. (b) Curves of background noise from the land surface fL versus the solar zenith angle θs when the land surface reflectivity βL equals to 0.3, 0.5, and 0.7. The slope of the land surface is set as 5 degrees and the azimuth angle is 0 degree. (c) Curves of background noise from the water surface fw versus the solar zenith angle θs when the wind speed above water surface w equals to 5, 10, and 15 m/s.

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As shown in Fig. 3, the background noise of all three sources will drop when the solar zenith angle increases, especially in Fig. 3(c) the noise rates of water surface drastically drop when the solar zenith angle exceeds 20 degrees. In Fig. 3(a), it can be found that the background noise from the atmosphere will drop when the atmospheric transmittance improves (e.g., in a clear sky). When the atmospheric transmittance is fixed, the background noise from the land surface rises with the land surface reflectivity. However, the background noise from the water surface is a little complex, as it rises with the increasing wind speed, when the solar zenith angle is larger than 20 degrees and this trend will opposite, when the solar zenith angle is between 0 and 20 degrees. This phenomenon is in accordance with what Bufton’s study found in the laser reflection over water surface [19], as the physical process of the laser and the sunlight reflected over the water surface are similar.

In addition, the noise rates of three sources are quite different. The reflectivity from 0.3 to 0.7 is typical value for land surface or its covering (e.g., rocks, soils, sands, forests, etc.), and it will be as much as 0.9 for ice and snow surface. The noise rate from the land surface is normally several times larger than those from the atmosphere and water surface, even if the land reflectivity only ranges from 0.3 to 0.7. When the solar zenith angle is beyond 20 degrees, the noise rate from land surface will be approximately an order of magnitude larger than that from water surface. In most regions and time, the solar zenith angle will satisfy this condition (>20 degrees), especially for the snowy regions at high latitudes.

3. Land cover classification using noise rates

3.1 Ratios of different noise rates

As mentioned above, the noise rates of different sources are quite different, and the specific ratio between different sources will be derived and analyzed in this section. The ratio of background noises from the land surface to that from the water surface can be expressed as

$${P_1} = \frac{{{f_\textrm{L}}}}{{{f_\textrm{W}}}} = \frac{{2s\theta _\textrm{r}^2{\beta _\textrm{L}}\cos \psi }}{{\delta \left\{ {2s\{ \exp ( - \frac{{{\theta_\textrm{s}}^2}}{{4{s^2}}}) - \exp [ - \frac{{{{({\theta_\textrm{r}} - {\theta_\textrm{s}}z)}^2}}}{{4{s^2}}}]\} + \sqrt \pi {\theta_s}[{\textrm{erf}}(\frac{{{\theta_\textrm{s}}}}{{2s}}) + {\textrm{erf}}(\frac{{{\theta_\textrm{r}} - {\theta_\textrm{s}}}}{{2s}})]} \right\}}}.$$
Similarly, the ratio of background noise from the atmosphere to that from the water surface is
$$\begin{aligned}{P_\textrm{2}} &= \frac{{{f_\textrm{A}}}}{{{f_\textrm{W}}}} \\ & = \frac{{s\theta _\textrm{r}^2({1 - T_\textrm{a}^{1\textrm{ + }\sec {\theta_\textrm{s}}}} )}}{{2\delta T_\textrm{a}^{1\textrm{ + }\sec {\theta _\textrm{s}}}({1\textrm{ + }\sec {\theta_\textrm{s}}} )\left\{ {2s\{ \exp ( - \frac{{{\theta_\textrm{s}}^2}}{{4{s^2}}}) - \exp [ - \frac{{{{({\theta_\textrm{r}} - {\theta_\textrm{s}}z)}^2}}}{{4{s^2}}}]\} + \sqrt \pi {\theta_s}[{\textrm{erf}}(\frac{{{\theta_\textrm{s}}}}{{2s}}) + {\textrm{erf}}(\frac{{{\theta_\textrm{r}} - {\theta_\textrm{s}}}}{{2s}})]} \right\}}}.\end{aligned}$$
As illustrated in Fig. 4, the ratio P1 (using solid blue curve) and P2 (using dashed red curve) are calculated when the atmospheric transmittance Ta is 0.8, the slope of land surface σL is 5 degrees, the azimuth angle φ is 0 degree and the land surface reflectivity βL is 0.5. The results indicate that, when the solar zenith angle is smaller than 20 degrees, the background noises from the water surface and the atmosphere are at the same level, whereas the background noise from the land surface is five to ten times larger than that from the water surface or the atmosphere. Moreover, when the solar zenith angle exceeds 40 degrees, the background noise contributed by the water surface is negligible.

 figure: Fig. 4.

Fig. 4. Ratio of noise rate P1 (using solid blue curve) and the ratio of noise rate P2 (using dashed red curve) vary with solar zenith angles with the atmospheric transmittance Ta of 0.8, the land surface reflectivity βL of 0.5, the wind speed of w = 8m/s, the land slope σL of 5 degrees, and the azimuth angle φ of 0 degree.

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It should be noted that, the noise photons captured by a photon-counting lidar will include both the atmospheric backscatter and Earth’s surface reflection, and we cannot distinguish the noise photons that are reflected by the Earth’s surface from the noise photons that are scattered by the atmosphere. As a result, when a photon-counting lidar flies over land surfaces, the total noise rate fnL reflected by the land and scattered by the atmosphere is fnL=fL+fA (the sum of Eq. (1) and Eq. (2)). Similarly, the noise rate over the water surfaces includes both the noise photons reflected by the water surface and scattered by the atmosphere, i.e., fnW=fW+fA (the sum of Eq. (2) and Eq. (7)). If the atmospheric condition above a certain area is identical, the ratio of the noise rates fnL to fnW can be expressed as

$$\begin{array}{l} P = \frac{{{f_{\textrm{nL}}}}}{{{f_{\textrm{nW}}}}} = \frac{{{f_\textrm{L}} + {f_\textrm{A}}}}{{{f_\textrm{W}}\textrm{ + }{f_\textrm{A}}}}\\ = \frac{{2s\theta _\textrm{r}^2\left[ {{\beta_\textrm{L}}T_\textrm{a}^{1\textrm{ + }\sec {\theta_\textrm{s}}}\cos \psi + \frac{{1 - T_\textrm{a}^{1\textrm{ + }\sec {\theta_\textrm{s}}}}}{{4({T_\textrm{a}^{1\textrm{ + }\sec {\theta_\textrm{s}}}} )}}} \right]}}{{\delta T_\textrm{a}^{1\textrm{ + }\sec {\theta _\textrm{s}}}\left\{ {2s\{ \exp ( - \frac{{{\theta_\textrm{s}}^2}}{{4{s^2}}}) - \exp [ - \frac{{{{({\theta_\textrm{r}} - {\theta_\textrm{s}}z)}^2}}}{{4{s^2}}}]\} + \sqrt \pi {\theta_s}[{\textrm{erf}}(\frac{{{\theta_\textrm{s}}}}{{2s}}) + {\textrm{erf}}(\frac{{{\theta_\textrm{r}} - {\theta_\textrm{s}}}}{{2s}})]} \right\} + \frac{{1 - T_\textrm{a}^{1\textrm{ + }\sec {\theta _\textrm{s}}}}}{{4({T_\textrm{a}^{1\textrm{ + }\sec {\theta_\textrm{s}}}} )}}}}. \end{array}$$
The contour maps of the ratio P under different conditions are illustrated in Fig. 5. In Fig. 5(a), the ratio P varies with the combination of solar zenith angle and atmospheric transmittance, when the wind speed and the land surface reflectivity equal to 8 m/s and 0.5, respectively. It can be found that with a land surface reflectivity of 0.5, the ratio P always remains over three when the solar zenith angle exceeds 20 degrees and the atmospheric transmittance is above 0.7. In Fig. 5(b), the atmospheric transmittance is a constant 0.8 and the wind speed equals to 8 m/s, and the ratio P will remain larger than three, when the land surface reflectivity and the solar zenith angle exceed 0.3 and 20 degrees, respectively.

 figure: Fig. 5.

Fig. 5. Contour maps of the ratio P and the line colors represent different values of P. (a) The ratio P versus the combination of the solar zenith angle θs and the atmospheric transmittance Ta. The abscissa and vertical ordinate value correspond to the solar zenith angle θs and the atmospheric transmittance Ta, respectively. (b) The ratio P versus with the combination of the solar zenith angle θs and the land surface reflectivity βL. The vertical ordinate value corresponds to the land surface reflectivity βL.

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From Fig. 5, with system parameters of a given photon-counting lidar and certain environmental parameters (e.g., the solar zenith angle θs>20 degrees, the atmospheric transmittance Ta>0.8, and land reflectivity βL>0.3), the ratio P will be always larger than three, which indicates that the density of noise photons over the land surface will be significantly larger than that over water surface. Given the raw data photons measured by a photon-counting lidar, the statistical noise rate of each along-track segment can be estimated. Then, a threshold (the ratio P) can be used to distinguish whether the photon-counting lidar was over the land surface or the water surface based on the statistical noise rate.

3.2 Land-water classification method

When a photon-counting lidar works, the parameters of the location, date, and time are known, by which the solar zenith angle θs and solar irradiance Nλ0 can be estimated [21]. In addition, the system parameters of the lidar are definite, and the wind speed above water surface can be obtained from the global National Centers for Environmental Prediction (NCEP) dataset [22]. In these settings, if the atmospheric transmittance exceeds 0.8 (in clear sky), the land reflectivity is larger than 0.3 (the land and its covering normally satisfy), and the solar zenith angle θs is beyond 20 degrees, the ratio P will be larger than three, which is sufficient to distinguish water from land.

The procedure of land-water classification is illustrated in Fig. 6. First, when a photon-counting lidar works, if the solar zenith angle is larger than 20 degrees and the atmospheric transmittance is larger than 0.8, the land-water classification method can be implemented. Second, substituting the atmospheric transmittance and wind speed from the NCEP dataset, the parameters of the experimental location, date, and time, and the system parameters of the used lidar into Eq. (2) and Eq. (7), the theoretical noise rate over water surface (fnW=fW+fA) can be estimated. Third, set the theoretical ratio P (normally using P = 3) as the classification threshold, i.e., the surface type will be classified as ‘water surface’ when the measured noise rate fnS is less than three times of theoretical noise rates fnW over water surface; otherwise, the surface type corresponds to ‘land surface’. In other words, if the ratio of the mean statistical noise rates fnS of the current segment to the theoretical noise rate fnW is smaller than the threshold (i.e., 3 times), this segment is classified as ‘water surface’; otherwise, this segment is classified as ‘land surface’.

 figure: Fig. 6.

Fig. 6. Flow chat of the land-water classification method based on the theoretical background noise models.

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The statistical noise rates fnS within a 10 m along-track distance can be calculated segment by segment in along-track direction, and the statistical noise rates in the ten adjacent segments are then averaged. Because the range gate of a space-borne photon-counting is normally over 1 km in vertical direction, in coastal areas, nearly all photons that locate in elevation from tens to hundreds meters above the Earth’s surface are noise photons, and these noise photons can be used to calculate the statistical along-track noise rates fnS.

4. Verification of derived model and classification

4.1 Study lidar data and experimental area

To verify the derived theory, the raw lidar data from the MABEL photon-counting lidar is used. The MABEL has up to 24 channels with different viewing angles (i.e., eight channels correspond to laser wavelength of 1064 nm and sixteen channels correspond to laser wavelength of 532 nm), and each channel can work independently and separately capturing data photons [23]. However, the optical fiber was damaged in the MABEL and the actual transmitted laser energy was only approximately 1/7 compared to its designed laser energy [11]. As a result, the raw photons captured by the channel 44 in 1064 nm with a relatively higher quality was used because 1) the channel 44 has a nearly nadir incidence, and 2) the 1064 nm wavelength locates in the atmospheric window and has a lower energy loss. With a 5 kHz laser pulse repetition rate and an approximately 200 m/s flight speed, the along-track interval is approximately 4 cm between contiguous laser pulses [10]. The raw data photons in the MABEL L2A standard dataset are based on the WGS84 coordinate frame and each photon corresponds to its unique latitude, longitude, and elevation coordinates.

On 21/09/2012, from east to west, the MABEL flew over the east coast of the North Carolina, United States with a very clear sky [24]. In Fig. 7(a) (using red curves), the MABEL flight trajectory was first over the Atlantic Ocean; then flew over the banks that separate the Croatan Sound from the Atlantic Ocean; entered into the Croatan Sound that has many islands and shoals; crossed the vegetation covered land (was once over a slim lake in the middle of this route); entered into the East Lake; and was finally over another vegetation covered land. The used data in Fig. 7(b) last approximately 5 minutes (from 21:37 AM to 21:41 AM of UTC time) and covered approximately 60 km along-track data photons. The local wind speed above water surface was 16 m/s, which is obtained from the bilinear interpolation result of the NCEP dataset. The wind speed from NCEP is identical with that measured by the Oregon Inlet Marina Station from the National Oceanic and Atmospheric Administration (NOAA). The station is located at [35° 47.7’ N, 75° 32.9’ W], inside the Croatan Sound, nearby the MABEL trajectory and its position is marked by an orange filled circle in Fig. 7(a). As a result, the wind speed required in classification can be well estimated by the NCEP dataset.

 figure: Fig. 7.

Fig. 7. (a) MABEL trajectory (using a red solid curve) on high-resolution satellite image on 21/09/2012 near the east coast of North Carolina, USA. The location of the Oregon Inlet Marina Station at [35° 47.7’ N, 75° 32.9’ W] is marked by an orange filled circle. The MABEL flight trajectory was first over the Atlantic Ocean; then flew over the banks that separate the Croatan Sound from the Atlantic Ocean; entered into the Croatan Sound that has many islands and shoals; crossed the vegetation covered land (was once over a slim lake in the middle of this route); entered into the East Lake; and finally was over another vegetation covered land. (b) Green points correspond to raw data photons captured by the MABEL and the vertical red dashed curves correspond to the classified boundary between the water and land surface. It should be noted that the latitude and longitude coordinates of all data photons are transformed into the along-track distance. The origin of the along-track distance is set as the beginning of the MABEL trajectory (the east end of the red curve in Fig. 7(a) in the map).

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4.2 Verification of the noise rate model

To verify the derived background noise model, the theoretical background noise rates reflected by the land and water surface as well as atmospheric backscatter are calculated and compared to the statistical noise rates from the MABEL raw data photons in Fig. 7(b). It should be noted that when a photon-counting lidar flies over land areas in the daytime, we cannot distinguish the noise photons that are reflected by the Earth’s surface from that are scattered by the atmosphere. It will be similar when a photon-counting lidar is over water surface. The noise rate over the Earth’s surface extracted from the MABEL raw photons is denoted by fnS.

Given that the range gate of the MABEL is 1500 m in vertical direction and because the land elevation is normally lower than 400 m in coastal areas, the statistical noise rates fnS can be easily obtained using the raw photons in the range from 400 m to 900 m in ellipsoidal height. With known parameters of the location, date, and time, the solar zenith angle θs and solar irradiance Nλ0 are calculated by Reda’s algorithm [25] and Frouin’s algorithm [21], respectively. The results are shown in Table 2. It should be noted that the local time was at 16:37 AM because this area belongs to the time zone of GMT −5. Then, the given MABEL system parameters, the in-situ wind speed above water surface (16 m/s), and an assumed atmospheric transmittance Ta of 0.9 (because the weather condition is a very clear sky) are substituted to calculate the theoretical noise rates fnL and fnW.

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Table 2. Parameters of the location, date, and time when MABEL flew over the experimental area.

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The comparison between the theoretical calculation and the statistical result is shown in Table 3. The theoretical noise rate over the water surface is identical with that is extracted from MABEL raw photons. The mean statistical noise rate from the MABEL above lands is within the range of the theoretical calculation because the reflectivity of the soil surface and the covered vegetation normally varies from 0.3 to 0.5. Therefore, the statistical noise rates from the MABEL raw photons agree well with the theoretical noise rates that are derived above.

Tables Icon

Table 3. Comparison between the theoretical and statistical noise rate.

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4.3 Classification performance based on noise rates

The solar zenith angle exceeded 70 degrees when the MABEL flew over this area, and with a very clear sky, the atmospheric transmittance is definitely larger than 0.8. As a result, the classification method based on the noise rates can be implemented in this experiment. Using the parameters in last section, in Fig. 8(a), the blue solid and red dashed curves correspond to theoretical noise rates fnW and fnL, respectively. The contour map of ratio P is illustrated in Fig. 8(b), which indicates that the water surface can be easily distinguished from the land because the ratio P is always larger than the classification threshold (the threshold is set to three). In fact, as shown in Fig. 4, when the solar zenith angle exceeds 40 degrees, the noise rate reflected by the water surface fW is at least one order of magnitude lower than both the noise rate reflected by the lands fL and the noise rate scattered by the atmosphere fA, i.e., the fW is negligible; the ratio P is nearly equal to P = fnL/fnW= (fL+fA)/fA. As a result, in this experiment, the ratio P is only related to the atmospheric transmittance and the reflectivity of the land and its covering.

 figure: Fig. 8.

Fig. 8. (a) The theoretical noise rate fnW over the water surface with different wind speeds w (using blue solid curve) versus the theoretical background noise above the lands fnL with different reflectivity βL (using dashed red curve). The atmospheric transmittance Ta equals to 0.9. (b) The contour map of ratio P varies with different land and vegetation reflectivity βL and atmospheric transmittances Ta.

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According to the classification procedure in Fig. 6, the theoretical noise rate over water surface is fnW=8.4 kHz (in Table 2), and the classification threshold is three times larger than the theoretical noise rate over water surface, i.e., 25.2 kHz in this experiment. Then, the statistical noise rate of each along-track segment fnS is calculated. As illustrated in Fig. 7(a), the classified results agree well with the high-resolution satellite image in Fig. 7(a). In Fig. 7(b), a very slim lake near the along-track distance of 38 km is successfully discriminated by the proposed method and marked by an orange box in Fig. 7(b). In addition, the along-track segment between 3 km and 11 km in Fig. 7 (marked by a blue box in Fig. 7(b)) are enlarged and illustrated in Fig. 9. In Fig. 9(a), the MABEL had flown over the land surface for six times (marked by the white numbers), and the proposed classifier has successfully discriminated the six land surfaces (marked by the blue numbers in Fig. 9(b)). The classification result is identical with that of the high-resolution image.

 figure: Fig. 9.

Fig. 9. (a) High-resolution image of the enlarged along-track segment between 3 km and 11 km in Fig. 7(b). In this enlarged segment, the MABEL had flown over land surface for six times (that are marked by the white numbers). (b) Distribution of the MABEL raw data photon of the enlarged along-track segment between 3 km and 11 km in Fig. 7(b). The proposed classifier has successfully detected the land surfaces for six times (that are marked by the blue numbers) and agrees well with the land cover captured by the high-resolution image in Fig. 9(a).

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It should be noted that the waves near the shore are different from that in oceans. Fortunately, when the proposed noise rate model is used to classify the water surface from the land near the shore, the noise rate from the land surface is five to ten times larger than that from the water surface or the atmosphere. Moreover, when the solar zenith angle exceeds 40 degrees, the background noise contributed by the water surface is negligible. The experiment using MABEL raw photons shown in Fig. 7 and Fig. 9 verified the performance of the proposed classifier.

5. Conclusion

In this paper, the theoretical model of the solar-induced background noise arising from the water surface reflection is proposed based on the specular reflection theorem and fills the gap of the classical model, which assumes the Earth’s surface to be a Lambert reflector. According to the proposed model, the noise level from water surface is influenced by the solar zenith angle, the atmospheric transmittance and the wind speed above water surface. The comparison among the solar-induced noise from the land surface, water surface, and the atmosphere indicates that, when the solar zenith angle is smaller than 20 degrees, the noise rate from the water surface and the atmosphere are at the same level. However, the noise rate from the land surface is five to ten times larger than that from the water surface or the atmosphere. Moreover, when the solar zenith angle exceeds 40 degrees, the background noise contributed by the water surface is negligible. The data photons and system parameters of the MABEL photon-counting lidar were used to verify the theoretical noise model near the east coast of the North Carolina, USA. The theoretical noise rate over the water surface is 8.4 kHz, which is identical with the statistical noise rates calculated from the MABEL’s raw data photons.

The analysis of theoretical noise rates reveals an obvious difference of noise level between the water surface and the land surface, therefore, a new classification method was proposed to classify the Earth’s surface types, only using the captured noise photons. Despite the signal photons are difficult to extract, the noise rate of the background noise is easy to be estimated precisely, because the range gate is one-kilometers in vertical while the elevation of the Earth’s surface mostly takes up from −100 to 400 m in coastal areas, i.e., there are sufficient time bins containing only noise photons. Using the MABEL raw data photons that were captured in the experimental area (at the mid-latitude area), the classification method performed well and successfully distinguished all water surfaces from land surfaces. Compared to the current along-track classification method based on the ancillary NLCD data and with a 30 m classification resolution, the new method can not only directly and precisely obtain the land and water types without other ancillary data, but also achieve a better along-track classification resolution of 10 m.

The along-track land cover classification is essential for data of photon-counting lidar because the signal photon detecting algorithm is directly related to the surface types. This method utilizes the ‘useless’ noise photons and only consumes little processing capacity while achieves an excellent classification accuracy in coastal areas. In addition, the classification method has great potential to detect the open water from the ice surface in high-latitude sea ice region (e.g., the Arctic Ocean), which is the most interested area of the ICESat-2 mission. As the solar zenith angle and the reflectivity of the ice surface is even larger, the ratio of the noise over land surface to the water surface is even higher (i.e., the differences between the noise levels over the water surface and the ice surface will be more significant).

Funding

National Natural Science Foundation of China (41801261); China Postdoctoral Science Foundation (2016M600612, 20170034); Youth Science and Technology Innovation Fund Project of Anhui Province Key Laboratory of Water Conservancy and Water Resources (KY201703); National Major Science and Technology Projects of China (11-Y20A12-9001-17/18, 42-Y20A11-9001-17/18); State Key Laboratory of Geo-Information Engineering (SKLGIE2018-Z-3-1); Fundamental Research Funds for the Central Universities (2042019kf1001).

Acknowledgments

We thank the Goddard Space Flight Center (GSFC) for distributing the MABEL data, the National Oceanic and Atmospheric Administration (NOAA) for distributing the in-situ wind speed data, and the University Corporation for Atmospheric Research for distributing the NCEP data. This is publication number 74 of the Sino-Australian Research Centre for Coastal Management.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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Figures (9)
Fig. 1.
Fig. 1. Geometry of the solar irradiance that is reflected by the land surface and received by a photon-counting lidar.

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Fig. 2.
Fig. 2. Schematic of the geometric constraints for the sunlight to be reflected into the receiver FOV by a sloping water surface. The origin of the coordinate coincides with the center of the receiver FOV, and the Z-axis points to the zenith direction. On the water surface, a polar coordinate is used with the distance ρ between the FOV center and a given point on water surface and the azimuth angle θ.

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Fig. 3.
Fig. 3. Solar-induced background noise rates from different sources. (a) Curves of background noise from the atmosphere fA versus the solar zenith angle θs when the atmospheric transmittance Ta equals to 0.7, 0.8, and 0.9. (b) Curves of background noise from the land surface fL versus the solar zenith angle θs when the land surface reflectivity βL equals to 0.3, 0.5, and 0.7. The slope of the land surface is set as 5 degrees and the azimuth angle is 0 degree. (c) Curves of background noise from the water surface fw versus the solar zenith angle θs when the wind speed above water surface w equals to 5, 10, and 15 m/s.

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Fig. 4.
Fig. 4. Ratio of noise rate P1 (using solid blue curve) and the ratio of noise rate P2 (using dashed red curve) vary with solar zenith angles with the atmospheric transmittance Ta of 0.8, the land surface reflectivity βL of 0.5, the wind speed of w = 8m/s, the land slope σL of 5 degrees, and the azimuth angle φ of 0 degree.

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Fig. 5.
Fig. 5. Contour maps of the ratio P and the line colors represent different values of P. (a) The ratio P versus the combination of the solar zenith angle θs and the atmospheric transmittance Ta. The abscissa and vertical ordinate value correspond to the solar zenith angle θs and the atmospheric transmittance Ta, respectively. (b) The ratio P versus with the combination of the solar zenith angle θs and the land surface reflectivity βL. The vertical ordinate value corresponds to the land surface reflectivity βL.

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Fig. 6.
Fig. 6. Flow chat of the land-water classification method based on the theoretical background noise models.

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Fig. 7.
Fig. 7. (a) MABEL trajectory (using a red solid curve) on high-resolution satellite image on 21/09/2012 near the east coast of North Carolina, USA. The location of the Oregon Inlet Marina Station at [35° 47.7’ N, 75° 32.9’ W] is marked by an orange filled circle. The MABEL flight trajectory was first over the Atlantic Ocean; then flew over the banks that separate the Croatan Sound from the Atlantic Ocean; entered into the Croatan Sound that has many islands and shoals; crossed the vegetation covered land (was once over a slim lake in the middle of this route); entered into the East Lake; and finally was over another vegetation covered land. (b) Green points correspond to raw data photons captured by the MABEL and the vertical red dashed curves correspond to the classified boundary between the water and land surface. It should be noted that the latitude and longitude coordinates of all data photons are transformed into the along-track distance. The origin of the along-track distance is set as the beginning of the MABEL trajectory (the east end of the red curve in Fig. 7(a) in the map).

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Fig. 8.
Fig. 8. (a) The theoretical noise rate fnW over the water surface with different wind speeds w (using blue solid curve) versus the theoretical background noise above the lands fnL with different reflectivity βL (using dashed red curve). The atmospheric transmittance Ta equals to 0.9. (b) The contour map of ratio P varies with different land and vegetation reflectivity βL and atmospheric transmittances Ta.

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Fig. 9.
Fig. 9. (a) High-resolution image of the enlarged along-track segment between 3 km and 11 km in Fig. 7(b). In this enlarged segment, the MABEL had flown over land surface for six times (that are marked by the white numbers). (b) Distribution of the MABEL raw data photon of the enlarged along-track segment between 3 km and 11 km in Fig. 7(b). The proposed classifier has successfully detected the land surfaces for six times (that are marked by the blue numbers) and agrees well with the land cover captured by the high-resolution image in Fig. 9(a).

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Tables (3)

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Table 1. Instrument parameters of the MABEL.

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Table 2. Parameters of the location, date, and time when MABEL flew over the experimental area.

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Table 3. Comparison between the theoretical and statistical noise rate.

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Equations (10)

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f L = N λ 0 ( Δ λ ) π ( z θ r ) 2 η A r h ν π z 2 β L T a 1 + sec ⁡ θ s cos ⁡ ψ = N λ 0 ( Δ λ ) θ r 2 η A r h ν β L T a 1 + sec ⁡ θ s cos ⁡ ψ
f A = N λ 0 ( Δ λ ) θ r 2 η A r h ν 1 − T a 1 + sec ⁡ θ s 4 ( 1 + sec ⁡ θ s )
cos ⁡ ψ = c o s σ L c o s θ s + sin ⁡ σ L sin ⁡ θ s cos ⁡ φ ,
{ ( π 2 − θ sr ) + 2 ω + ( π 2 − θ s ) = π ω + ( π 2 − θ s ) + σ W = π 2
{ s 2 = 0.146 w , ( w < 7 m / s ) s 2 = 0.003 + 0.00512 w , ( 7 ≤ w < 13.3 m / s ) s 2 = 0.138 log 10 w − 0.084 , ( w ≥ 13.3 m / s )
β W ( ρ ) = δ [ 1 + arctan 2 ( ρ z ) ] 2 4 π s 2 exp ⁡ ( − tan 2 σ W s 2 ) = δ [ 1 + arctan 2 ( ρ z ) ] 2 4 π s 2 exp ⁡ [ − ( θ s − ρ z ) 2 4 s 2 ]
f W = N λ 0 ( Δ λ ) η A r z 2 h ν T a 1 + sec ⁡ θ s ∫ 0 2 π ∫ 0 z θ r β W ( ρ ) ρ d ρ d θ = N λ 0 ( Δ λ ) η A r δ 4 π s 2 z 2 h ν T a 1 + sec ⁡ θ s ∫ 0 2 π ∫ 0 z θ r ( 1 + arctan 2 ( ρ z ) ) 2 exp ⁡ [ − ( θ s − ρ z ) 2 4 s 2 ] ρ d ρ d θ ≈ N λ 0 ( Δ λ ) η A r δ 2 s h ν T a 1 + sec ⁡ θ s { 2 s { exp ⁡ ( − θ s 2 4 s 2 ) − exp ⁡ [ − ( θ r − θ s ) 2 4 s 2 ] } + π θ s [ e r f ( θ s 2 s ) + e r f ( θ r − θ s 2 s ) ] } ,
P 1 = f L f W = 2 s θ r 2 β L cos ⁡ ψ δ { 2 s { exp ⁡ ( − θ s 2 4 s 2 ) − exp ⁡ [ − ( θ r − θ s z ) 2 4 s 2 ] } + π θ s [ erf ( θ s 2 s ) + erf ( θ r − θ s 2 s ) ] } .
P 2 = f A f W = s θ r 2 ( 1 − T a 1  +  sec ⁡ θ s ) 2 δ T a 1  +  sec ⁡ θ s ( 1  +  sec ⁡ θ s ) { 2 s { exp ⁡ ( − θ s 2 4 s 2 ) − exp ⁡ [ − ( θ r − θ s z ) 2 4 s 2 ] } + π θ s [ erf ( θ s 2 s ) + erf ( θ r − θ s 2 s ) ] } .
P = f nL f nW = f L + f A f W  +  f A = 2 s θ r 2 [ β L T a 1  +  sec ⁡ θ s cos ⁡ ψ + 1 − T a 1  +  sec ⁡ θ s 4 ( T a 1  +  sec ⁡ θ s ) ] δ T a 1  +  sec ⁡ θ s { 2 s { exp ⁡ ( − θ s 2 4 s 2 ) − exp ⁡ [ − ( θ r − θ s z ) 2 4 s 2 ] } + π θ s [ erf ( θ s 2 s ) + erf ( θ r − θ s 2 s ) ] } + 1 − T a 1  +  sec ⁡ θ s 4 ( T a 1  +  sec ⁡ θ s ) .
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