#### 2.1. Refraction Correction

The ATLAS ATL03 data product provides the geolocated photon detections [

20]. These data are the input to the higher-level geosphysical data products that use customized algorithms for specific surfaces (land ice, sea ice, vegetation, etc.). However, none of the higher-level products contain the required algorithmic steps to produce valid bathymetric measurements. More precisely, the ICESat-2 products do not currently account for the refraction and the corresponding change in the speed of light that occurs at the air–water interface. This produces both horizontal and vertical errors in the geolocation recorded in ATL03, resulting in locations that are deeper and further off nadir than the true measurement. The impact of refraction on the ICESat-2 collection is depicted in

Figure 2.

The input to the refraction correction algorithm developed and used in this work consists of the following:

1. Geolocated seafloor photon returns, subset from the ATL03 geolocated photon returns through either manual or automated analysis to segment seafloor points;

2. A water surface model, W_{ij}, derived from the water surface photon returns;

3. Estimates of the refractive indices of air, n_{1}, and of water, n_{2};

4. Angle of incidence, θ_{1}, for each photon, obtained from the elevation of the unity pointing vector for the photon. More specifically, θ_{1} = π/2 – (ref_elev), where ref_elev is one of the parameters available in the ATL03 output. (Note, this is actually a simplification, which does not account for the curvature of the Earth across ATLAS’s swath, as will be discussed later.)

5. Azimuth of the unit pointing vector for the photon, κ, obtained from the parameter, ref_azimuth, which is one of the output parameters from the ATL03 algorithm.

The approach used in the refraction correction is able to compute horizontal and vertical offsets that can be applied to the uncorrected bottom return photon’s coordinates, as illustrated in

Figure 3. The advantages of this method are that it is both computationally efficient and geometrically intuitive. While the NASA ATL03 algorithm works in an inertial frame, our refraction correction is derived in a spacecraft-centered coordinate system defined as follows:

Z is the vertical direction (parallel to and opposite the local direction of gravity), and

Y is perpendicular to

Z (i.e., in the horizontal plane) and oriented along the azimuth of the pointing vector. Note that while the

Z axes are the same in

Figure 2 and

Figure 3, the

Y axes may differ. They are the same in the case that the azimuth of the photon pointing vector is exactly aligned with the across-track direction. If desired, it is possible to consider this as a 3D coordinate system by defining

X to be mutually perpendicular to

Y and

Z and oriented so as to complete a right-handed coordinate system (i.e., out of the page in

Figure 3). Once obtained, the horizontal offset Δ

Y can be projected onto the local (

E,

N) coordinate axes using the azimuth of the unit pointing vector,

κ, which is one of the output parameters (ref_azimuth) from the ATL03 algorithm [

20].

The refraction correction can be considered a simple rotation and scaling of the in-water range vector, but it is most easily derived and intuited geometrically with reference to

Figure 3. In the figure,

${\theta}_{1}$ is the angle of incidence;

${\theta}_{2}$ is the angle of refraction;

D is the uncorrected depth, which is obtained by differencing the

Z coordinates of the water surface model and uncorrected bottom return photons;

S is the slant range to the uncorrected bottom return photon location;

R is the corrected slant range;

P is the distance between the uncorrected and corrected photon return points in the

Y–

Z plane, and other angles and distances are as labeled. It can be readily recognized that

$\varphi ={\theta}_{1}-{\theta}_{2}$, and

$\beta =\gamma -\alpha $. From Snell’s Law, the angle of refraction,

${\theta}_{2}$, is

Due to the change in the speed of light that occurs at the air–water interface (which is not accounted for in the current ATL03 geolocation procedure), the relationship between

R and

S is

The remaining relationships needed to compute (Δ

Y, Δ

Z) can be obtained from the law of sines, the law of cosines, and simple trigonometric functions, referencing three triangles in the lower half of

Figure 3: triangle DTS (right triangle), triangle RPS (scalene triangle), and the triangle with Δ

Y and Δ

Z as two of the sides and

P as the remaining side (right triangle). From triangle DTS, we have

Applying the law of sines to triangle RPS and solving for α yields

Applying the law of cosines to triangle RPS gives

From the right triangle with Δ

Y, Δ

Z, and

P as sides

A final step is to project the horizontal offset, Δ

Y, onto the (

E,

N) axes using the azimuth of the laser pointing vector,

κ:

After alignment of local East and North directions to the map projection (e.g., Universal Transverse Mercator) Easting and Northing axes (using the convergence angle), Δ

E and Δ

N from Equations (10)–(11) serve as the corrections to the unrefracted measurement’s horizontal coordinates. Simply stated, the refraction correction algorithm initiates with the input parameters

n_{1},

n_{2},

W_{ij}, and for each photon,

E,

N,

Z,

${\theta}_{1}$, and

κ. These variables systematically apply to Equations (1)–(11) to determine the corrections (Δ

E, Δ

N, and Δ

Z) to the unrefracted photon coordinates. The corrections are added to the original coordinates (

E,

N, and

Z) to produce a corrected position,

E’ =

E + Δ

E,

N’ =

N + Δ

N, and

Z’ =

Z + Δ

Z, where prime denotes the corrected coordinate. From [

21], the index of refraction of seawater (

S = 35 PSU) at atmospheric pressure, a temperature of 20 °C, and a wavelength of 540 nm is 1.34116, while the corresponding value for fresh water (

S = 0) is 1.33469. These values are taken to be the default settings for

n_{2} in seawater and fresh water, respectively. Meanwhile, the default value of the refractive index of air,

n_{1}, is 1.00029.

With the center beams at near-nadir, assuming a mean orbital altitude of 496 km and 3.3 km separation of the beams on the ground [

22], the incidence angle for the outer beams is only 0.38°, and the horizontal offset for the refraction correction is 0.003

D. This equates to only 9 cm at a water depth of 30 m, which can be considered negligible with a beam footprint of 17 m. However, since ATLAS has the capability for up to 5º off-nadir pointing, the horizontal offset, due to refraction, cannot, in general, be considered negligible (This study used only data from a near-nadir orientation, as the off-nadir pointing has, to date, not been tested during science mode operations). In cases where a first-order approximation to the refraction correction is acceptable and it is determined that the planimetric component of the refraction correction can be safely ignored, the corrected elevation of a seafloor photon return can be approximated as

Alternately, when greater accuracy in the refraction correction is needed, a correction for the curvature of the Earth over ATLAS’s swath must also be included, since Earth’s curvature contributes to the nonparallelism of ATLAS’s nadir direction and the Earth surface’s normal at the point of intersection of the incident photon(s). This Earth curvature correction, which is applied to

${\theta}_{1}$, is computed as follows:

where

H is the orbital altitude of ICESat-2 (e.g., 496 km as a mean value), and

R_{e} is the radius of the Earth (e.g., 6371 km as a mean value).

The ATLAS refraction correction developed in this work was implemented in MATLAB and validated using data acquired by Quantum Spatial, Inc. over Lake Tahoe, California, using a high-accuracy, airborne bathymetric lidar system: the Riegl VQ-880-G. Like other airborne bathymetric lidar systems (e.g., [

23,

24]), and unlike ATLAS, the VQ-880-G is a linear-mode lidar system, rather than a single photon. Despite this obvious difference, the VQ-880-G data provide a nearly-ideal validation data set for two reasons: (1) The VQ-880-G processing workflow is similar to the one described above, in which subaqueous returns are first geolocated as if they were subaerial returns (i.e., as if there were no water present), and then a refraction correction is applied; and (2) Riegl VQ-880-G bathymetric lidar data have been extensively investigated [

24,

25,

26,

27], such that new refraction correction techniques can be validated by comparing the results against those obtained by Quantum Spatial. Tested on the Lake Tahoe data set, the refraction correction algorithm described above yielded root mean square (RMS) differences from those obtained by Quantum Spatial of 0.02 m vertical and 0.14 m horizontal, even with a simplifying assumption of a flat water surface.