Calculate iceThickness from seaIceFreeboard

Calculate ice thickness from ice freeboard, ice depth, and the densities of the snow, ice, and surface water.

This will return ice thickness in a variable named (by default) DerivedObsValue/iceThickness. The derived error standard deviations are combined in quadrature into the variables observation error data, as well as the constituent random and systematic components going into RandomErrorStandardDeviation/iceThickness and SystematicErrorStandardDeviation/iceThickness respectively.

Variables used

• ice freeboard \(f\) (\(m\))

• water density \(\rho_w\) (\(kg/m^3\))

• snow density \(\rho_s\) (\(kg/m^3\))

• ice density \(\rho_i\) (\(kg/m^3\))

• snow depth \(d\) (\(m\))

• instrument-sourced random error standard deviation of ice density measurements \(\sigma_{\rho_{i}}\) (\(kg/m^3\))

• instrument-sourced random error standard deviation of ice freeboard measurements \(\sigma_f\) (\(m\))

• instrument-sourced systematic error standard deviation of snow density measurements \(\sigma_{\rho_{s}}\) (\(kg/m^3\))

• instrument-sourced systematic error standard deviation of snow depth measurements \(\sigma_d\) (\(m\))

Output Variables

• calculated random error standard deviation of the derived ice thickness \(\sigma_{tR}\) (\(m\))

• calculated systematic error standard deviation of the derived ice thickness \(\sigma_{tS}\) (\(m\))

• ice thickness \(t\) (\(m\))

Parameters

Inputs:

• ice freeboard variable: Sea ice protruding above the water surface level variable (default ObsValue/seaIceFreeboard)

• ice density variable: Density of the ice variable (default ObsValue/iceDensity)

• snow depth variable: Depth of the snow on top of the ice variable (default ObsValue/totalSnowDepth)

• snow density variable: Density of the snow on top of the ice variable (default ObsValue/snowDensity)

• sea water density variable: Density of the sea water below the ice variable (default ObsValue/seaWaterDensity)

Inputs if calculating error:

• calculate error standard deviations: (default true)

• ice freeboard error standard deviation variable: (default ObservedErrorStandardDeviation/seaIceFreeboard)

• ice density error standard deviation variable: (default ObservedErrorStandardDeviation/iceDensity)

• snow depth error standard deviation variable: (default ObservedErrorStandardDeviation/snowDepth)

• snow density error standard deviation variable: (default ObservedErrorStandardDeviation/snowDensity)

Outputs:

• ice thickness variable: Thickness of the sea ice including both above and below the water surface level variable (default DerivedObsValue/iceThickness)

• ice thickness systematic error standard deviation variable: (default SystematicErrorStandardDeviation/iceThickness)

• ice thickness random error standard deviation variable: (default RandomErrorStandardDeviation/iceThickness)

Example yaml block

obs filters:
- filter: Variable Transforms
  Transform: Calculate iceThickness from seaIceFreeboard
  ice freeboard variable: ObsValue/seaIceFreeboard
  sea water density variable: ObsValue/seaWaterDensity
  snow depth variable: ObsValue/totalSnowDepth
  snow density variable: ObsValue/snowDensity
  ice density variable: ObsValue/iceDensity
  ice freeboard error standard deviation variable: \
      ObservedErrorStandardDeviation/seaIceFreeboard
  ice density error standard deviation variable: \
      ObservedErrorStandardDeviation/iceDensity
  ice thickness variable: iceThickness
  ice thickness systematic error standard deviation variable: \
      SystematicStandardDeviation/iceThickness
  ice thickness random error standard deviation variable: \
      RandomStandardDeviation/iceThickness

Method

The ice thickness is calculated using the formula from (Ricker et al. 2014). In that paper they refer to the ice freeboard as the radar freeboard, this is because they assume these two quantities are equal. We maintain the distinction to allow us flexibility when calculating the ice freeboard from the radar freeboard.

\[t = \frac{f \rho_{w} + d \rho_{s}} {\rho_{w} - \rho_{i}}\]

The filter will also (optionally) propagate the uncertainties through Gaussian error propagation to give the random error standard deviation on the thickness, assuming the ice freeboard and ice density uncertainties are random and uncorrelated:

\[ \begin{align}\begin{aligned}\sigma_{tR}^2 &= \sigma_{f}^{2} \left( \frac{\partial t}{\partial f} \right)^{2} + \sigma_{\rho_{i}}^{2} \left( \frac{\partial t}{\partial \rho_{i}} \right)^{2}\\\sigma_{tR} &= \sqrt{ \left( \sigma_{f} \frac{\rho_{w}}{\rho_{w} - \rho_{i}} \right)^{2} + \left( \sigma_{\rho_{i}} \frac{f \rho_{w} + d \rho_{s}} {\left(\rho_{w} - \rho_{i}\right)^{2}} \right)^{2} }\end{aligned}\end{align} \]

The filter will also optionally give the systematic error, assuming that the errors on snow depth and snow density are systematic:

\[ \begin{align}\begin{aligned}\sigma_{tS}^2 &= \sigma_{d}^{2} \left( \frac{\partial t}{\partial d} \right)^{2} + \sigma_{\rho_{s}}^{2} \left( \frac{\partial t}{\partial \rho_{s}} \right)^{2}\\\sigma_{tS} &= \sqrt{ \left( \sigma_{d} \frac{\rho_{s}} {\rho_{w} - \rho_{i}} \right)^{2} + \left( \sigma_{\rho_{s}} \frac{d} {\rho_{w} - \rho_{i}} \right)^{2} }\end{aligned}\end{align} \]