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Abstract
With much smaller footprints (approximately a few tens of meters), the data of a laser altimeter are promising for obtaining the sea level near offshore areas, where radar altimeters with larger footprints cannot operate. However, the current ocean surface detection methods for a photon-counting lidar cannot effectively eliminate the noise photons when measuring the sea surface, thereby introducing a ranging bias. In this paper, a new ocean surface detection method is derived based on the JONSWAP (Joint North Sea Wave Project) wave spectrum and LM (Levenberg-Marquardt) nonlinear least-squares fitting. Using the data photons that are captured by the NASA MABEL (Multiple Altimeter Beam Experimental Lidar) photon-counting lidar, the new method is tested and compared to the MABEL standard result. The new method achieved better profile detection of sea surfaces and successfully discarded the noise photons in a sub-layer below the sea surface from the MABEL standard result. By reconstructing the “accumulated waveform”, we found that the noise photons in the sub-layer produce small tails after the main waveform, which introduces an overestimated ranging bias of 9 cm. This difference of 9 cm is similar to the sea level bias of 10 cm that was obtained from the ICESat/GLAS laser altimeter data and the TOPEX/Poseidon radar altimeter data in an earlier study, which limited the use of laser altimeter data. According to the analysis in this paper, we can partially interpret what occurred for the ICESat/GLAS waveform tails when ICESat was measuring sea surfaces. The newly derived method can protect the MABEL and incoming ICESat-2 data photons from noise photon interference and ranging bias when measuring the sea surface.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
ICESat (Ice, Cloud, and land Elevation Satellite) was the first satellite laser altimetry mission for the long-term Earth science and a very successful mission that enabled the estimation of the overall mass changes of the Greenland and Antarctic ice sheets [1]; changes in the sea ice [2], vegetation [3], and ocean [4]; and other environmental changes. The ICESat-2 mission, with advanced photon-counting laser altimetry, will continue to measure the sea ice freeboard and ice sheet elevation and will provide smaller laser footprints with higher spatial density [5]. Prior to launch, an airborne photon-counting lidar, namely, the MABEL (Multiple Altimeter Beam Experimental Lidar), was used as a technology demonstrator for the photon-counting laser altimeter [6]. Compared to traditional altimeters, which record the return waveform, photon-counting lidars measure the distance that is traveled by individual photons. Surface elevations are estimated from the observed photon distributions, which are sometimes referred to as photon clouds [7].
Typically, the return-signal photon clouds suffer from noise photons, such as dark noise and background noise photons. In the daytime, the number of background noise photons typically exceeds the number of signal photons within the range gate. Many algorithms have been developed for optimally extracting the weak laser signal from high background noise during daytime operations. Various universal methods have been derived, e.g., the correlation range receiver (CRR) is gated to accept only those photons that enter the receiver with each laser pulse [8] and adaptive ellipsoid searching (AES) used a spherical noise density estimation model and a morphing ellipsoid that is determined by local principal components to detect the signal photons [9]. Then, for the surface types of sea ice and ice sheets, the surface-finding algorithm has been well used to detect open water and ice surfaces [10–12]. Moreover, for ice sheet surfaces, an adaptive window size with a recursive nearest-neighbor analysis was proposed for the removal of noise photons [13]. For the surface types of buildings and forests, the adaptive ellipsoid searching filter [14], the adaptive density-based model [15], the contour active models [16], and spatial statistical and discrete mathematical concepts [17] were proposed for extracting the surface and the land cover information. In addition, photon-counting lidars can be used to capture the underwater depth images [18] and long distance three-dimensional images [19,20].
A photon-counting lidar can capture the photons of the along-track profile of a surface target. The spatial distribution of the signal photons is related to the shape of the surface target. The current detection algorithms take the profile shapes of various surface types into consideration. A few researchers focused on the ocean surface for both the conventional and the new photon-counting laser altimeters. Jasinski et al. investigated the inland and near-shore water profiles using the MABEL photon-counting lidars [21] and other researchers used the conventional ICESat data to calculate sea level or as an elevation benchmark in inland waters [22–26]. Several methods were derived for extracting the laser signal photons on the ocean surface.
The JONSWAP wave spectrum is widely used to express the surface profiles [27]. In this paper, for a photon-counting lidar, an ocean surface detection method that is based on the JONSWAP spectrum and LM (Levenberg-Marquardt) nonlinear least-squares fitting is derived to extract the ocean signal photons. Using the MABEL data, the extraction results of this method will be compared to the results of the current surface finding method. The results demonstrate that this method can provide a more precise result when detecting the ocean surface and we determine how to interpret the difference between the mean sea level (MSL) that is calculated via the ICESat data and that calculated via the TOPEX data [22]. In Urban and Schutz’s study, using a reference mean sea surface, a global ICESat ocean elevation bias was found 10 cm lower with respect to TOPEX, thereby limiting the use of laser altimeter data. The MABEL “waveforms” (elevation distributions) are constructed by accumulating along-track signal photons and analyzed. When calculating the centroids of captured waveforms, we find that a laser altimeter will underestimate the elevation (by approximately 9 cm using MABEL data) because of the sub-layer noise photons below the surface profile. If this elevation bias can be precisely calculated and offset, the laser altimeter data can be used in the field for seas and oceans, especially in offshore areas and the Arctic, where radar altimeters cannot operate.
2. Data and method
2.1 MABEL data
A photon-counting lidar system that is equipped with Geiger mode avalanche photodiodes (Gm-APDs) for MABEL or Photomuitpliers (PMTs) for ICESat-2 can only respond to the presence of return signals and record the time tags with the output of 0 or 1; it cannot record the return waveforms [28]. When a photon-counting lidar is carried by an aircraft or a satellite, the return signal consists of along-track data photon clouds. Figure 1 illustrates raw return data (using green filled circles and labeled ‘Raw data’) that are captured by MABEL when the laser is illuminated on sea surfaces. The return data photons with a range gate of 150 m in the vertical direction include both the MABEL laser photons that are reflected by the sea surface and noise photons.
Fig. 1 Typical MABEL data photons, which were captured at 2:36 PM on 09/18/2013 (channel no. 44 at 1064 nm). The raw data photons, which consist of signal photons and noise photons, are illustrated using green filled circles and labeled “Raw data”; the signal photons of the MABEL standard result that correspond to the “high” flags with the best reliabilities are illustrated using blue filled circles and labeled “MABEL result”; and the signal photons that are extracted via the newly derived method are illustrated using red filled circles and labeled “Fitting result”.
MABEL has up to 16 active channels at 532 nm and 8 channels at 1064 nm with changeable viewing angles. For the MABEL detectors, there is a dead time of 2.5 ns. Detector dead time is time required for the detector to recover from a photon event (the triggering of the detector) before the next photon can be detected. Detected photons are time tagged for range determination and have a measured resolution of 83 ps (corresponding to approximately 12 mm in elevation or depth) [7]. At a laser repetition rate of 5 kHz and at an aircraft ground speed of 130 m/s, a laser pulse is emitted every 2.6 cm along the track [21]. Each channel of MABEL separately captures the return signal and generates data photons. Figure 1 illustrates an along-track segment of typical MABEL data photons (channel no. 44 at 1064 nm). Channel no. 44 corresponds to some of the best quality data when illuminating a sea surface because 1) it is the center channel and, thus, corresponds to a smaller nadir angle and can receive more laser photons via the specular reflection of the sea surface and 2) it corresponds to 1064 nm and, thus, has lower penetrance in water compared to 532 nm.
MABEL data (Level 2A, Release 9) for the ocean surface survey that is considered in this paper were captured on 09/18/2013 and are available from the NASA ICESat-2 website (https://icesat.gsfc.nasa.gov/icesat2/data/mabel/data/browse/index.html); we refer to them as Dataset 1 [29]. The flight routes of the MABEL system are illustrated in Fig. 2 and all the data were captured when the MABEL laser was illuminated on the sea surface near the east coast of Portsmouth, USA. In each data set, the raw data photons were recorded for 1 min of along-track flight for every available channel. The data sets contain photon arrival times of flight from reflected laser photons (i.e., signal photons), solar background and backscatter in the atmosphere (i.e., background noise photons) and, to a lesser degree, detector noise (i.e., dark noise photons) [12]. The data sets also contain the flight velocity in the east and north directions and the MABEL position that corresponds to each laser pulse in latitude and longitude coordinates (captured by a Novatel GNSS/INS device on the aircraft). Using the flight velocity and MABEL position data, the return signal can be calculated and plotted as the data photons in Fig. 1 (with the abscissa of the along-track distance and the vertical coordinate of the elevation).
Fig. 2 Flight routes of the MABEL system (using red curves) on 09/18/2013 when the MABEL laser was illuminated on the sea surface near the east coast of Portsmouth, USA. These data photons were captured when intermittent clouds covered the ocean; hence, some of the data photons are contaminated by the cloud scattering effect.
In addition, each data set contains measurements of the laser signal photons, which are the standard result. The standard result was extracted via the surface-finding algorithm and labeled “ph_class” in the data sets. The algorithm is based on histograms of photon arrival times in 25 m along-track segments and 10 m vertical bins and assumes a random distribution of background photons and a symmetric return pulse [12]. In the standard result, the flags of all photons are classified as noise, buffer, low, medium, and high. In Fig. 1, the data photons, which are represented by blue filled circles, are the photons with “high” flags, which correspond to the highest reliabilities.
2.2 Extraction method
2.2.1 Sea surface profile based on the wave spectrum
The JONSWAP (Joint North Sea Wave Project) wave spectrum is widely used to express the surface profiles. The profile of the sea surface, which is denoted as z(x, y), can be expressed as the superposition of cosine and sine waves with various amplitudes, phases, angular frequencies, and direction angles [30]:
where x and y represent the planimetric coordinates in the east and north directions, respectively; ζi, ωi, αi and εi represent the amplitude, angular frequency, direction angle and wave phase, respectively; g is the gravitational acceleration of 9.8 m/s2; and m is the number of superposed cosine and sine waves. These parameters in Eq. (1) are mainly determined by the wind speed above the sea surface and are also related to the peak enhancement factor and the wind fetch [27]. The number of superposed cosine and sine waves also depends on the wind speed.Based on the JONSWAP spectrum, when the wind speed is less than 5.5 m/s, the PSD (Power Spectral Density) is mainly concentrated in the angular frequency range [1.1, 4.0] and to finely illustrate details of the sea surface, the angular frequency interval should be less than 0.2. Hence, the angular frequency ωi is varied from 1.1 to 4.0 with an interval of 0.1 and m is set to 30. Thus, if the wind speed above the sea surface, the peak enhancement factor and the wind fetch are known, all the parameters in Eq. (1) can be calculated. Then, the sea surface profiles are generated via the method of IFFT (Inverse Fast Fourier Transform) to simulate a profile of the sea surface.
2.2.2 Fitting sea surface profiles
Based on Eq. (1), the captured data photons of the MABEL system can be used as input data to fit the instantaneous profile of the sea surface and extract the signal photons. The return signal of each MABEL channel consists of along-track data photons. Thus, Eq. (1) should be converted into Eq. (2), in which the along-track distance is used in place of the planimetric coordinates and the direction angle.
where d is the along-track distance. Typically, the sea level is a negative value in the WGS84 elevation benchmark; hence, a constant offset, which is denoted as off, is added into Eq. (2) to represent this difference between benchmarks. To effectively extract the signal photons, all the data sets in this experiment on the sea surface are tested and the best scheme is as follows:- (A) As illustrated in Fig. 1, there are a substantial number of noise photons in the returned raw data, which will interfere with the fitting results. To eliminate the gross error, a preliminary data processing procedure is needed before the fitting process. In each data set, the raw data photons are uniformly divided into 20 segments in the vertical direction. The elevation length of each vertical segment is equal to the range gate (approximately 150 m) that divides the vertical segments (20 segments). The histogram of the number of photons in each vertical segment is calculated. The photon densities of segments that contain signal and noise photons are higher compared to those that contain only noise photons; hence, we only reserve the segments for which the photon density is 1.2 times larger than the average photon density of all segments. The remaining data photons are used as the input data to fit the sea surface profile via Eq. (2).
- (B) In the fitting process, the initial parameters (the amplitude ζi of the corresponding angular frequency ωi) can be calculated based on the JONSWAP wave spectrum, which is denoted as S(ω) and expressed as Eq. (3), at a specified wind speed U10 (10 m above the sea surface), a wind fetch X, and a peak enhancement factor γ.
where ; the peak of the spectrum, which is denoted as ωp, is ; and σ is the width of the spectrum peak. The data that are used in this paper were captured at a wind condition of approximately 5 m/s (i.e., U10 that is read from the MABEL wind data is 5 m/s) and, typically, the peak enhancement factor is γ = 3.3 and the wind fetch is assumed as X = 30 km. The width σ is equal to 0.09 when ω is larger than ωp; otherwise, σ is equal to 0.07. The amplitudes ζ can be expressed as . The angular frequency ωi is set from 1.1 to 4.0 with a frequency interval dω of 0.1 and i is set as i = 1, 2, 3,…, 30; then, the 30 corresponding amplitudes ζi are calculated; all 30 phases εi are set to 0; and the elevation offset off is set to −20 m. More accurately estimated initial parameters will decrease the number of iterations and prevent becoming trapped in a local minimum [31].
- (C) All 91 fitted parameters are estimated iteratively by substituting the initial parameters and the coordinates of the remaining photons (the along-track distances d and their corresponding elevations z) via the LM algorithm based on least-squares criteria. Using the fitted parameters, the surface profile can be calculated. The fitting error (FE) of each remaining data photon z(d) is calculated by subtracting the fitting curve and the root mean square error (RMSE) is calculated between the fitting curve and all remaining data photons.
- (D) When the absolute FE of a remaining photon is larger than twice the RMSE, this remaining photon is categorized as a noise photon and discarded. Three additional fitting processes are run to further discard the noise photons, as described in Procedure (C). If the current RMSE exceeds 1 m, the remaining photons will be discarded when their corresponding absolute FEs are larger than twice the RMSE; otherwise, the remaining photons will be discarded when their corresponding absolute FEs are larger than 3 times the RMSE.
- (E) Because the along-track distance of each MABEL data set is several kilometers, too many data photons cannot achieve a small fitting RMSE. In every data set, the remaining data photons are divided into specific along-track segments, typically, with an along-track distance of 500 m. In each along-track segment, the selected data photons are used as the input data to fit the sea surface profile, as described in Procedure (C). If the current RMSE exceeds 1.5 m, the remaining photons will be discarded when their corresponding absolute FEs exceed the RMSE; otherwise, the remaining photons will be discarded when their corresponding absolute FEs exceed twice the RMSE. Finally, 2 additional fitting processes are run in each along-track segment. If the current RMSE exceeds 0.5 m, the remaining photons will be discarded when their corresponding absolute FEs exceed the RMSE; else, the remaining photons will be discarded when their corresponding absolute FEs exceed 3 times the RMSE.
3. Result comparison and discussions
3.1 Result comparison
Using procedures (A) to (E), the signal photons are extracted via the newly derived method. In Fig. 1, a typical data set (channel no. 44 at 1064 nm) was selected for comparison among the data photons of the raw data, the signal photons of the MABEL result, and the signal photons of the newly derived method. In Figs. 3(a) and 3(b), two along-track segments of the data in Fig. 1 are selected and magnified. In Figs. 3(a) and 3(b), the signal photons that are extracted from the MABEL result and the newly derived method are illustrated using blue filled circles and red circles, respectively, and the final fitting curve is plotted as a sky-blue curve.
Fig. 3 Two typical along-track segments of the data set that was obtained at 2:36 PM on 09/18/2013 (channel no. 44 at 1064 nm). (a) The left figure corresponds to the relative along-track distance in the range from 1500 m to 2000 m; and (b) the right figure corresponds to the relative along-track distance in the range from 2000 m to 2500 m. The signal photons that are extracted via the MABEL method are labeled “MABEL results” and the signal photons extracted via the newly derived method are labeled “Fitting results”.
In Figs. 1, 3(a), and 3(b), the new method outperforms the MABEL surface-finding algorithm and the signal photons on the profile of the sea surface are precisely detected. The fitting curve of the JONSWAP wave spectrum is well in accordance with the photons on the sea surface profile. Some of the photons of the MABEL result are located in a sub-layer that is below the profile of the sea surface; the distribution of these photons is similar to that of the photons on the sea surface profile. The photons in the sub-layer may arise from the very slight laser penetration in the water column at 1064 nm, the forward-scattering effect in the atmosphere, and the after-pulsing effect of the hardware detector [32–34].
To demonstrate this phenomenon more clearly, the accumulated elevation distributions, which are also referred to as “accumulated waveforms”, of the signal photons in Figs. 3(a) and 3(b) are generated and illustrated in Figs. 4(a) and 4(b), where the vertical axis represents the accumulated probability at the corresponding elevation. A photon-counting lidar transmits a laser pulse with much lower energy (normally 2 or 3 orders of magnitude) compared to a traditional lidar system. Lower transmitted laser energy enables a higher repetitive frequency to be achieved, thereby recording every photon event. Therefore, the “accumulated waveforms” that are generated by the signal photons (photon events) are similar to the waveforms that are captured by a traditional laser altimeter.
Fig. 4 Accumulated elevation distributions or “accumulated waveforms” correspond to the signal photons in Figs. 3(a) and 3(b), respectively. (a) The left figure corresponds to the accumulated waveform in the range from 1500 m to 2000 m; and (b) the right figure corresponds to the accumulated waveform in the range from 2000 m to 2500 m. The accumulated waveforms of the new method (labeled as “Fitting waveform”) and the MABEL result (labeled as “MABEL waveform”) are shown as red and blue solid curves, respectively, and their corresponding centroids are shown as red and blue dashed lines, respectively.
The tails of return waveforms cannot be easily discarded by setting simple threshold values, because the sea level anomaly and mesoscale variability features (approximately several hundred kilometers) may make the local sea level quite different from another area. In addition, in small area with a diameter of a few kilometers, the mean sea surface also varies with several meters arising from the swell and other effects. The data in Fig. 1 prove this effect, i.e., the along-track mean sea surface (in the WGS84 elevation benchmark) varies from approximately −44 m to −46 m. Figure 5 illustrates the signal photons (in Fig. 1) that are extracted by this new method and are zoomed in from 3000 m to 5000 m in the along-track direction.
Fig. 5 Signal photons of the sea surface that is extracted by the new derived method (Zooming in from 3000 m to 5000 m in the along-track direction).
3.2 Discussions
In Figs. 4(a) and 4(b), the accumulated waveforms of the new method (red curves) are nearly Gaussian in shape and in accordance with the theoretical waveform model when a traditional laser altimeter is measuring above the ocean surface [35], while the accumulated waveforms of the MABEL result (blue curves) have small tails after the main peak of the waveforms. The waveforms that correspond to sea surfaces of the traditional laser altimeter, namely, the ICESat/GLAS (Geoscience Laser Altimeter System), also have small tails after the main peak of the waveform. Using the official software (IDL_visualizer), a typical GLAS received waveform and its ranging centroid is read and illustrated in Fig. 6. This waveform was captured on 03/04/2004 in the east Pacific, and its unique index (UINDEX) and shot number are also shown in Fig. 6. The amplitude of the received waveform is defined by the quantization counts of the GLAS analog/digital converter [36].
Fig. 6 Typical ICESat/GLAS waveform of the sea surface. This waveform, which is shown as a blue solid curve, was captured on 03/04/2004 in the east Pacific and the corresponding centroid is shown as a blue dashed line.
These small tails will introduce a ranging bias (overestimate the range) for a laser altimeter. The average elevation of the laser footprint on the sea surface will be obtained from the centroid of the best-fit Gaussian distribution to the returned waveform [32]. In Fig. 6, the centroid of this waveform is read by the IDL_visualizer software and illustrated as a blue dashed line. Due to the tails, there is a small difference between the centroid line and the waveform peak and the relative time of the centroid line exceeds that of the waveform peak, thereby overestimating the ranging result. The centroid of each waveform is calculated and illustrated as a dashed line in Figs. 4(a) and 4(b). The MABEL result also has an overestimated ranging bias that is similar to the GLAS result in Fig. 6, while the new method prevents this ranging bias.
The GLAS system and channel no. 44 of the MABEL system were both nearly normally incident to the sea surface; hence, the overestimated ranging bias is nearly equal to the underestimated elevation bias of the sea surface. For all the valid data sets in this experiment (using channel no. 44 at 1064 nm), we calculated the centroid difference between the MABEL result (with the ranging bias) and the new method (without the ranging bias). The average centroid difference over all data sets is 0.6 ns (which corresponds to a 9.0 cm ranging bias) and the MABEL result yields a lower sea level. The difference between the mean sea level that is calculated from the ICESat data and that from the TOPEX data was investigated by Urban and Schutz and a global ICESat ocean elevation bias (within the latitude range from 65°S to 65°N) of approximately −10 cm with respect to TOPEX was identified, although the ICESat laser altimetry results match the TOPEX detection results for major sea level anomaly and mesoscale variability features [22].
Few researchers have performed investigations to interpret the lower calculated elevation of the mean sea level. MABEL has the same laser wavelength at 1064 nm as GLAS. MABEL was flown aboard either the ER-2 or Proteus Aircraft at 20 km or above 95% Earth’s atmosphere [21], while GLAS is a satellite laser altimeter above Earth’s entire atmosphere. These two lidar systems were subjected to similar measurement conditions. Benefiting from a more sensitive detector and a laser with a higher repetition rate, MABEL can capture signal photons with more detail, reconstruct the echo waveform, and reproduce what occurred for the small tails that GLAS cannot record, namely, the noise photons in a sub-layer below the sea surface introduce the tails into returning waveforms and a ranging bias of nearly 10 cm.
4. Conclusions
In this paper, a new ocean surface detection method that is based on the JONSWAP wave spectrum and LM nonlinear least-squares fitting is derived and used to extract the signal photons of MABEL data sets that are captured on the sea surface near Portsmouth, USA. Compared to the MABEL standard result, the new method achieved a better profile detection result of sea surfaces and successfully discarded the noise photons in a sub-layer below the sea surface from the MABEL standard result. By reconstructing the “accumulated waveform”, we found that the noise photons in the sub-layer produced small tails after the main waveform, thereby introducing an overestimated ranging bias when measuring the sea surface. This ranging bias is calculated in this experiment and is approximately 9 cm; hence, an underestimated sea surface elevation bias of 9 cm is introduced.
Compared to the classical TOPEX data, the ICESat/GLAS data yield a lower sea level. This difference of 9 cm is similar to the sea level bias of 10 cm that was obtained from the ICESat data and the TOPEX data. Based on the analysis in this paper, we can partially interpret what occurred with the ICESat/GLAS waveform tails when ICESat was measuring sea surfaces. The newly derived method can protect the MABEL and incoming ICESat-2 data photons from noise photon interference and ranging bias when measuring the sea surface. Compared to the radar altimeter, a laser altimeter has much smaller footprints (approximately 17.5 m for ICESat-2 and 70 m for ICESat). Hence, it is a promising technology for obtaining the sea level in offshore areas and in the Arctic, especially in areas that are less than 3 km from the coastline, where radar altimeters cannot operate. For a satellite laser altimeter, more than half of all measurements are performed on the ocean surface, which makes this research very important and urgently needed.
Funding
National Natural Science Foundation of China (Grants 41506210, 41801261); National Science and Technology Major Project (Grants 11-Y20A12-9001-17/18), 42-Y20A11-9001-17/18); Postdoctoral Science Foundation of China (Grants 2016M600612, 20170034).
Acknowledgments
We thank the National Snow and Ice Data Center for distributing the ICESat/GLAS data and the Goddard Space Flight Center for distributing the MABEL data. The comments from the two anonymous reviewers improved the manuscript and we thank the reviewers and the editor very much. This is publication number 61 of the Sino-Australian Research Centre for Coastal Management.
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