Heterogeneity-index#

An index reflecting the heterogeneity of the input field (typically SST). It was first proposed by Liu & Levine (2016) and then updated by Haëck et al. (2023).

Definition#

The Heterogeneity-Index (HI) is defined as the weighted sum of three components, each computed from the values of the input field inside a moving window.

Components#

  1. Standard-deviation (noted σ or V)

    Computed as the uncorrected sample standard-deviation.

    \[σ = \sqrt{ \frac{1}{N} \sum_i (x_i - \bar{x})^2 }\]
  2. Skewness (noted γ or S)
    \[γ = \frac{1}{Nσ^3} \sum_i (x_i - \bar{x})^3\]

    Note

    Note that we take the absolute value of the skewness when merging the components into the HI (or whenever it may be needed). Nevertheless we retain its signed value when computing the components, in case it might contain useful information.

  3. Bimodality (noted B)

    The principle behind this component is very much similar to that of the Cayula & Cornillon algorithm (CCA). Because we use relatively small windows (and thus have few input field values to construct an histogram), the CCA method might be difficult to apply. We instead do the following:

    • Construct an histogram (h) of the input field values. By default we use bins of width 0.1°C.

    • We compare that histogram to a gaussian distribution of the same characteristics as the input field distribution (ie \(g_k = 1/\sqrt{2\pi σ} \exp\left(-(x_k-\bar{x})^2 / 2σ^2\right)\))

    • The comparison is done with the L2 norm (sum of the squared differences): \(B = \sum_k (h_k - g_k)^2\).

Normalizing coefficients#

To compute the HI proper, we normalize each component by a coefficient so that they all have equivalent statistical weights. To that end, the coefficient is taken as the inverse of the components standard deviation over the available data. In other words, we normalize each component by its variance.

  • \(\tilde{σ} = aσ\), with \(a = 1 / \operatorname{std}(σ)\)

  • \(\tilde{γ} = bγ\), with \(b = 1 / \operatorname{std}(γ)\)

  • \(\tilde{B} = cB\), with \(c = 1 / \operatorname{std}(B)\)

Finally, to bound somewhat the range of HI values, we apply a final coefficient d so that 95% of the HI values are inferior to 9.5:

\[HI = d \left( aσ + bγ + cB \right)\]

Linear packing and shifting histogram bins#

For some datasets, the SST might be stored compressed with linear-packing.

Note

Very simply, instead of storing a variable in a 32 bits float, because we know the range of that variable (let’s say the values lie between 0 and 100), we store it on a smaller variable type such as a 16 bits integer (SHORT) or even 8 bits integer (BYTE). Let’s simplify and take an unsigned type. The integer values will lie between 0 and \(2^{16}-1 = 65 535\) for a USHORT. By multiplying those integer values by some factor we can obtain the range we want for our variable. Here a factor of 0.00153 will give a maximum value slightly above 100.

We end up with float values between 0 and 100 and a discretization interval equal to the scale factor (here 0.00153).

It can be the case that when computing the bimodality and creating the histogram of the input field values, the bins width coincides with the discretization interval (or a multiple of it). This can create spurious artifacts in the histogram. To avoid this we can shift the bins appropriately (half the scale factor for instance).

By default, for Dask and Numpy inputs we apply no shift. For Xarray inputs, we look into the input array metadata to see if a scale factor was applied. If nothing is found a warning will be emitted. The information is contained in the metadata attribute, in the key “encoding”. This information can be lost if the data-array undergoes some modifications.

Functions#

Supported types and requirements#

Supported input types: Numpy, Dask, Xarray

Requirements:

  • numpy

  • numba

  • scipy for computing the HI normalization coefficient

  • xarray-histogram for computing the HI normalization coefficient for Xarray data.

References#

[haeck_2023]

Haëck, C., Lévy, M., Mangolte, I., and Bopp, L.: “Satellite data reveal earlier and stronger phytoplankton blooms over fronts in the Gulf Stream region”, Biogeosciences 20, 1741–1758, DOI:10.5194/bg-20-1741-2023, 2023.

[liu_2016]

Liu, X. and Levine, N. M.: “Enhancement of phytoplankton chlorophyll by submesoscale frontal dynamics in the North Pacific Subtropical Gyre”, Geophys. Res. Lett. 43, 1651–1659, DOI:10.1002/2015gl066996, 2016.

On this page