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Anomalies¶

Definition: the anomaly of a variable is the variation in the signal relative to the climatological signal.

There is an error in this formulation because you cannot subtract the climatology from the global time series because they are on different temporal domains.

\begin{aligned} \text{Climatology}: && && \boldsymbol{\bar{y}} &= \boldsymbol{\bar{y}}_c(t) && && t\in\mathcal{T}_\text{Reference}\subseteq\mathbb{R}^+ \\ \text{Data}: && && \boldsymbol{y} &= \boldsymbol{y}(\mathbf{x},t) && && t\in\mathcal{T}_\text{Globe}\subseteq\mathbb{R}^+ && && \mathbf{x}\in\Omega\subseteq\mathbb{R}^{D_s} \end{aligned}

(1)

From a code perspective, this can be stated where the global data is

# global data
data_globe: Array["Nx Ny Nt"] = ...
# climatology reference period
data_climatology: Array["Nc"] = ...
# IMPOSSIBLE to subtract one timeseries from another (even with broadcasting)
data_anomaly: Array["Nx Ny Nt"] = data_globe - data_climatology

TODO: Need to figure out how this works.

PsuedoCode

https://xcdat.readthedocs.io/en/latest/examples/climatology-and-departures.html

Example 1: Monthly Mean¶

This example was taken from the xarray documentation.

# calculate monthly mean
climatology: xr.Dataset = ds.groupby("time.month").mean("time")

# calculate anomalies
anomalies: xr.Dataset = ds.groupby("time.month") - climatology

Example 2: Monthly Standardization¶

We can also calculate the standardized monthly means. This implies calculating the monthly mean and standard deviation.

# calculate monthly mean
climatology_mean: xr.Dataset = ds.groupby("time.month").mean("time")
climatology_std: xr.Dataset = ds.groupby("time.month").std("time")

# create standardization function
std_fn = lambda x, mean, std: (x - mean) / std

# calculate anomalies
anomalies: xr.Dataset = xr.apply_ufunc(
    std_fn,
    ds.groupby("time.month"),
    climatology_mean,
    climatology_std
)

Example 3: Seasonal¶

Anomalies in EO

Anomalies¶

Example 1: Monthly Mean¶

Example 2: Monthly Standardization¶

Example 3: Seasonal¶