#
`freeboard2thickness` documentation

`freeboard2thickness` is part of Antarctic Mapping Tools for Matlab (Greene et al., 2017). Click here for a complete list of functions in AMT.

`freeboard2thickness` estimates ice thickness from surface height above sea level, assuming hyrostatic equilibrium.

## Contents

## Syntax

T = freeboard2thickness(F) T = freeboard2thickness(...,'rhoi',iceDensity) T = freeboard2thickness(...,'rhow',waterDensity) T = freeboard2thickness(...,'rhos',snowDensity) T = freeboard2thickness(...,'Ts',snowThickness)

## Description

`T = freeboard2thickness(F)` estimates the ice thickness `T` in meters from height above the geoid F in meters.

`T = freeboard2thickness(...,'rhoi',iceDensity)` specifies ice density in kg/m^3. Default ice density is 917 kg/m^3.

`T = freeboard2thickness(...,'rhow',waterDensity)` specifies water density in kg/m^3. Default water density is 1027 kg/m^3.

`T = freeboard2thickness(...,'rhos',snowDensity)` specifies snow density in kg/m^3. Default snow density is 350 kg/m^3, however the default snow thickness is 0 m, so the snow density value will only affect calculations if snow thickness is specified.

`T = freeboard2thickness(...,'Ts',snowThickness)` specifies snow thickness in meters. Default snow thickness is 0 m.

## Example 1:

The surface of an iceberg is 80 cm above sea level and it's pure ice. How thick is the iceberg?

T = freeboard2thickness(0.8)

T = 7.4691

And you can go back the other way with the sister function:

F = thickness2freeboard(7.47)

F = 0.8001

## Example 2:

The surface of an iceberg is 80 cm above sea level, including 40 cm of snow which has a density of 300 kg/m^3. How thick is the iceberg?

T = freeboard2thickness(0.8,'Ts',0.4,'rhos',300)

T = 4.8255

## Example 3: When PIGs float

This example looks at a flowline down Pine Island Glacier. To get the coordinates of an actual flowline based on surface velocities you can use the `flowline` function, but for now let's just enter some manual points and plot them with a MODIS Mosaic of Antarctica background image. I'm densifying the points along the flowline to 100 m spacing with the `pspath` function:

[lat,lon] = pspath([-75.3585 -75.2041 -75.0089 -74.8032],... [-98.0220 -99.4048 -100.7342 -101.5879],... 100); % (100 m spacing along track) plotps(lat,lon,'b-','linewidth',2) modismoaps axis off

If you have the Bedmap2 plugin for AMT making elevation profiles along a line is pretty easy:

figure bedmap2_profile(lat,lon);

This might seem a bit reduntant, but let's plot the Bedmap2 surface in red. We'll also need to plot it as a function of distance along the lat,lon path the same way `bedmap2_profile` plotted the distance above. So use `pathdistps` to get the cumulative distance along track:

% Get surface elevation from Bedmap2: sfz = bedmap2_interp(lat,lon,'surface'); % Calculate distance along the flowline in kilometers: d = pathdistps(lat,lon,'km'); % Plot the interpolated Bedmap2 surface as a thick red line: plot(d,sfz,'r','linewidth',2)

Now we can estimate the base of the ice using the hydrostatic assumption. Start by assuming the ice shelf is made of pure ice with a density of 917 kg/m^3.

% Calculate ice thickness from freeboard: thickness = freeboard2thickness(sfz); % The base elevation is the surface minus thickness: base = sfz-thickness; % Plot the hydrostatic ice base as a thick red line: plot(d,base,'r','linewidth',2) % Find the index of the last finite surface elevation measurement: ind = find(isfinite(base),1,'last'); % Place a text label: text(d(ind),base(ind),' 0 m FAC','fontweight','bold','color','r')

That's not a fantastic match! As I recall, to get ice shelf base elevations, Bedmap2 used the hydrostatic assumption *after* accounting for some amount of firn air content in the ice column. We can do the same by setting the "snow" thickness to 20 m and set the density of the snow to 0 kg/m3. I think Ligtenberg's data show that firn air content is something like 20 m for PIG. So let's try this again with 20 m FAC:

thickness2 = freeboard2thickness(sfz,'Ts',20,'rhos',0); base2 = sfz-thickness2; plot(d,base2,'b','linewidth',2) text(d(ind),base2(ind),' 20 m FAC','fontweight','bold','color','b')

## Glacier thinning and grounding line retreat

bed = bedmap2_interp(lat,lon,'bed'); base = sfz - freeboard2thickness(sfz,'Ts',15,'rhos',0); % find the index where the ice base lifts off the bed: ind = find(base>=bed,1,'first') ; % Plot a red circle at hydrostatic grounding line: plot(d(ind),base(ind),'ro','markersize',10)

Here's a crude approximation of grounding line retreat as a function of glacier thinning. We'll loop through, finding the grounding line location just as above, but for an ever-thinning PIG:

% Steps of 10 cm surface elevation change from 0 to 40 m: dh = 0:0.1:40; % Preallocate grounding line location vector: gl = nan(size(dh)); % Make a new profile figure figure bedmap2_profile(lat,lon); % make a colormap: colors = jet(length(dh)); % Loop through each thinning value: for k = 1:length(dh); % Surface elevation with k amount of thinning: sfzi = sfz - dh(k); % Hydrostatic base elevation with thinning: basei = sfzi - freeboard2thickness(sfzi,'Ts',15,'rhos',0); % Plot surface and ice base elevations: plot(d,sfzi,'-','color',colors(k,:)) plot(d,basei,'-','color',colors(k,:)) % find the index where the ice base lifts off the bed: ind = find(basei>=bed,1,'first'); % log the grounding line location: gl(k) = d(ind); end

Now we can plot grounding line location as a function of surface elevation change.

figure plot(gl,-dh) axis tight box off xlabel({'grounding line location';'distance along flowline (km)'}) ylabel 'ice shelf surface elevation change (m) '

I should stress that the analysis above is mighty crude. I would not trust these exact numbers because we've completely neglected stress within the ice, grounding zone dynamics, bed uncertainty, ice thickness uncertainty, etc., but a basic principle of glaciology is shown: Starting from the top right corner of the plot, as the ice thins, the grounding line retreats. And sometimes, a little bit of thinning can cause a lot of grounding line retreat.

## Citing AMT

If this function or any other part of Antarctic Mapping Tools is useful for you, please cite the paper that describes AMT.

Greene, C. A., Gwyther, D. E., & Blankenship, D. D. Antarctic Mapping Tools for Matlab. *Computers & Geosciences*. 104 (2017) pp.151-157. doi:10.1016/j.cageo.2016.08.003.

## Author Info

This function was written by Chad A. Greene of the University of Texas Institute for Geophysics (UTIG), April 2017, for inclusion in the Antarctic Mapping Tools package.