# Calculating glacier ice volumes and sea level equivalents

## How do we estimate the global volume of ice?

The IPCC and other outlets frequently indicate how much sea levels will rise under given climate change scenarios . Other times, you might see that such and such volume of ice (km3) is equivalent to so many millimetres of sea level rise (sea level equivalent (SLE); the amount of sea level rise on full melting of the ice). But how are these calculations made?

Table 1. Sea level equivalent (SLE) from various land ice sources. From IPCC AR5 (Vaughan et al, 2013).

 Ice on land Sea level equivalent (m) Antarctic Ice Sheet 58.3 Greenland Ice Sheet 7.36 Glaciers and ice caps 0.41

The first complexity is in calculating the volume of ice in the world. This is complex, as we often do not have a complete picture of the bed of the ice sheet or glacier. For Antarctica, BEDMAP2 provides the most complete and up-to-date estimate of ice volume, and it is derived by combining thousands of radar and seismic measurements of ice thickness . In fact, BEDMAP 2 is derived from 25 million measurements. Fretwell et al. 2013 estimated that the Antarctic Ice Sheet comprised 27 million km3 of ice, with a sea level equivalent of ~58 m. BEDMAP 2, showing that the bedrock on which the West Antarctic Ice Sheet rests is well below sea level. The Antarctic Ice Sheet has a sea level equivalent of 58.3 m (i.e., if all the ice in Antarctica melted, this is how much global sea levels would rise).

For glaciers, ice thickness datasets are sparse .  While we have a good estimate of global glacier ice surface area  from satellite measurements, direct observations of glacier ice thickness are available for only around 1100 glaciers . These include radar measurements (both airbourne and from the ice surface) and seismic measurements. Unfortunately, these methods are time consuming and costly, and glaciers are often remote and difficult to access. This figure shows the global distribution of glaciers. The diameter of the circle shows the area covered. The area covered by tidewater glaciers is shown in blue. The number refers to the RGI region. From the IPCC AR5 Working Group 113. The sea level equivalent of glaciers is important because glaciers are responding rapidly to climate change and are likely to be a large contributor to global sea level rise over the next century.

Yet glacier volume and ice thickness data are essential parameters required for understanding sea level rise, water resource management and assessment of future glacier changes . Glaciers and small ice caps are also contributing rapidly to present-day sea level rise [6, 7]. Ice thickness data and bed topography are essential data required for numerical modelling of glaciers .  Other methods employed to calculate glacier volume include volume-area scaling [8-12], and calculations of ice thickness based on ice surface slope [13, 14] that assume a given basal shear stress. These methods, when correctly applied, do well against direct observations of ice thickness [8, 10].

## Calculating the mass of ice

Calculating the sea level equivalent for a given volume of ice requires some simple maths and a knowledge of the densities and properties of ice and sea water.  Ice volumes are usually given in km3. The mass of ice is usually given in metric gigatonnes (Gt). 1 Gt = 109 tonnes (where 1 tonne = 1000 kg).

Table 2. Densities of ice and water (at 1 atmospheric pressure and 4.3°C)

 Density of glacier ice 916.7 kg/m3 or 0.9167 Gt/km3 Density of pure water 1000 kg/m3 or 1.000 Gt/km3 Density of sea water 1027 kg/m3 or 1.027 Gt/km3

We can calculate the mass of something if we know the volume and the density:

Density = Mass / Volume

Mass = Volume x Density

Volume = Mass / Density

Because ice and water are different densities, 1 km3 results in different masses. However, remember that 1 Gt of ice = 1 Gt of water! They take up different volumes but have the same mass. So, 1 Gt (whether ice or water) is equal to:

• 1.091 km3 ice
• 1.000 km3 pure water
• 0.9737 km3 sea water

We can convert a given volume of ice (in km3) to a mass of ice (in Gt) by using the following equation:

Mass of ice (Gt) = Volume of ice (km3) x Density of ice (Gt/km3)

If we have a calculated ice volume of 500 km3, then:

Mass of ice = 500 x 0.9167

The mass of ice here is 458.30 Gt. If it all melted, you would have 458.30 Gt of water.

## Calculating sea level equivalent

To convert a mass of ice into the total amount global sea levels would rise if the ice all melted (i.e., the sea level equivalent), we need to know how much area the oceans cover. This is usually given as 3.618 x 108 km2. A 1 mm increase in global sea level requires 10-3 m3 (10-12 km3) of water for each square metre of the ocean surface, or 10-12 Gt of water.

We can calculate the volume of water required to raise global sea levels by 1 mm:

Volume = area x height

Area = 3.618 x 108 km2

Height = 10-6 km (1 mm)

Volume (km3) = (3.618 x 108 km2 ) x (10-6 km) = 3.618 x 102 km3 = 361.8 kmwater.

We can convert km3 of water to Gt of water as we did above; 1 kmwater = 1 Gt water.  In the same way, 1 Gt of ice = 1 kmwater. So, 361.8 Gt of ice will raise global sea levels by 1 mm. 361.8 Gt of ice is equivalent to 394.67 km3 ice.

If we took our 458.30 Gt of ice (as calculated above), then we could calculate the global sea level equivalent by:

SLE (mm) = mass of ice (Gt) x (1 / 361.8)

SLE = 458.30 x (1 / 361.8)

SLE = 1.27 mm

However, we should note that some of the world’s glaciers have parts that are below sea level. This ice will not affect sea level if it melted. The volume of glacier ice below the surface of the ocean should therefore be subtracted from the total volume of glaciers and ice caps when calculating sea level equivalents .

Also remember that ice that is floating (like ice shelves, sea ice and icebergs) does not contribute to sea level rise upon melting. Only land ice above sea level will contribute to sea level rise.

## Useful websites

1. Vaughan, D.G., et al., Observations: cryosphere. Climate change, 2013: p. 317-382.
2. Fretwell, L.O., et al., Bedmap2: improved ice bed, surface and thickness datasets for Antarctica. The Cryosphere, 2013. 7: p. 375-393.
3. Gärtner-Roer, I., et al., A database of worldwide glacier thickness observations. Global and Planetary Change, 2014. 122(0): p. 330-344.
4. Pfeffer, W.T., et al., The Randolph Glacier Inventory: a globally complete inventory of glaciers. Journal of Glaciology, 2014. 60(221): p. 537.
5. Andreassen, L., et al., Ice thickness measurements and volume estimates for glaciers in Norway. Journal of Glaciology, 2015. 61(228): p. 763-775.
6. Hock, R., et al., Mountain glaciers and ice caps around Antarctica make a large sea-level rise contribution. Geophysical Research Letters, 2009. 36: p. L07501.
7. Radic, V. and R. Hock, Regionally differentiated contribution of mountain glaciers and ice caps to future sea-level rise. Nature Geosci, 2011. 4(2): p. 91-94.
8. Grinsted, A., An estimate of global glacier volume. The Cryosphere, 2013. 7(1): p. 141-151.
9. Bahr, D.B., Estimation of glacier volume and volume change by scaling methods, in Encyclopedia of Snow, Ice and Glaciers. 2014, Springer. p. 278-280.
10. Bahr, D.B., W.T. Pfeffer, and G. Kaser, A Review of Volume‐Area Scaling of Glaciers. Reviews of Geophysics, 2014.
11. Bahr, D.B., W.T. Pfeffer, and G. Kaser, Glacier volume estimation as an ill-posed inversion. Journal of Glaciology, 2014. 60(223): p. 922-934.
12. Radić, V. and R. Hock, Regional and global volumes of glaciers derived from statistical upscaling of glacier inventory data. Journal of Geophysical Research: Earth Surface, 2010. 115(F1): p. F01010.
13. Marshall, S.J., et al., Glacier Water Resources on the Eastern Slopes of the Canadian Rocky Mountains. Canadian Water Resources Journal / Revue canadienne des ressources hydriques, 2011. 36(2): p. 109-134.
14. Cook, A.J., et al., Glacier retreat on South Georgia and implications for the spread of rats. Antarctic Science, 2010. 22(03): p. 255-263.
15. Haeberli, W. and A. Linsbauer, Brief communication” Global glacier volumes and sea level–small but systematic effects of ice below the surface of the ocean and of new local lakes on land”. The Cryosphere, 2013. 7(3): p. 817-821.