Anomaly Indicators

The climatology is computed using a moving average window and averaging through multiple years with various time scales: monthly, bimonthly, decadal, and weekly. Anomalies are then detected using one or more of the following indices:

Z-score: The Standardized Z-score

The Z-score is calculated using the formula:

\[z\_score = \frac{(x - \mu)}{\sigma}\]

where:

x: The average (aggregated) value of the variable in the time series data based on the specified time step.
μ (mu): The long-term mean of the variable (the climate normal).
σ (sigma): The long-term standard deviation of the variable.

SMAD: Standardized Median Anomaly Deviation

The SMAD is calculated using the formula:

\[SMAD = \frac{(x - \eta)}{IQR}\]

where:

x: The average (aggregated) value of the variable in the time series data based on the specified time step.
η (eta): The long-term median of the variable (the climate normal).
IQR: The interquartile range of the variable. It is the difference between the 75th and 25th percentiles of the variable.

Beta Distribution

The beta distribution is implemented using the scipy.stats module in Python. The probability density function (PDF) of a Beta distribution is given by:

\[f(x; \alpha, \beta) = \frac{x^{\alpha-1} (1-x)^{\beta-1}}{B(\alpha, \beta)}\]

where:

( B(alpha, beta) ) is the Beta function, which normalizes the distribution.

In scipy.stats, the Beta distribution is implemented as scipy.stats.beta(a, b) where a and b correspond to the shape parameters α and β, respectively.

Gamma Distribution

The gamma distribution is implemented using the scipy.stats module in Python. The PDF of a Gamma distribution is given by:

\[f(x; k, \theta) = \frac{x^{k-1} e^{-x/\theta}}{\theta^k \Gamma(k)}\]

where ( Gamma(k) ) is the Gamma function.

In scipy.stats, the Gamma distribution is implemented as scipy.stats.gamma(a, scale=θ) where a corresponds to the shape parameter k, and scale corresponds to θ.

ESSMI: Empirical Standardized Soil Moisture Index

The index is computed by fitting the nonparametric empirical probability density function (ePDF) using the kernel density estimator (KDE):

\[\hat{f}_h = \frac{1}{nh} \sum_{i=1}^{n} K\left(\frac{x - x_i}{h}\right)\]

where the kernel function ( K ) is given by:

\[K(x) = \frac{1}{\sqrt{2\pi}} \exp\left(-\frac{x^2}{2}\right)\]

and:

(hat{f}_h): the ePDF
( K ): the Gaussian kernel function
( h ): the bandwidth of the kernel function as a smoothing parameter (Scott’s rule)
( n ): the number of observations
( x ): The average (aggregated) value of the variable in the time series data based on the specified time step.
( x_i ): the ( i )-th observation

The ESSMI is then computed by transforming the ePDF to the standard normal distribution with a mean of zero and a standard deviation of one using the inverse of the standard normal distribution function:

\[ESSMI = \Phi^{-1}(\hat{F}_h(x))\]

where:

(Phi^{-1}): the inverse of the standard normal distribution function
(hat{F}_h): the ePDF

The kernel density estimator and the inverse of the standard normal distribution function can be implemented using the scipy.stats module in Python. The KDE can be computed using scipy.stats.gaussian_kde, and the inverse standard normal distribution can be obtained using scipy.stats.norm.ppf.

SMDS: Soil Moisture Drought Severity

The SMDS is calculated using the formula:

\[SMDS = 1 - SMP\]

where the Soil Moisture Percentile (SMP) is given by:

\[SMP = \frac{\text{rank}(x)}{n + 1}\]

where:

SMP: Soil Moisture Percentile. It is the percentile of the average value of the variable in the time series data.
SMDS: Soil Moisture Drought Severity. It represents the severity of the drought based on the percentile of the average value of the variable in the time series data.
rank(x): The rank of the average value of the variable in the time series data.
n: The number of years in the time series data.
x: The average (aggregated) value of the variable in the time series data based on the specified time step.

SMCI: Soil Moisture Condition Index

The SMCI is calculated using the formula:

\[SMCI = \frac{(x - \text{min})}{(\text{max} - \text{min})}\]

where:

x: The average (aggregated) value of the variable in the time series data based on the specified time step.
min: The long-term minimum of the variable.
max: The long-term maximum of the variable.

SMCA: Soil Moisture Content Anomaly

The SMCA is calculated using the formula:

\[SMCA = \frac{(x - \text{ref})}{(\text{max} - \text{ref})}\]

where:

x: The average (aggregated) value of the variable in the time series data based on the specified time step.
ref: The long-term mean (( mu )) or median (( eta )) of the variable (the climate normal).
max: The long-term maximum of the variable.

SMAPI: Soil Moisture Anomaly Percentage Index

A method for detecting anomalies in time series data based on the Soil Moisture Anomaly Percent Index (SMAPI) method.

The SMAPI is calculated using the formula:

\[SMAPI = \left( \frac{(x - \text{ref})}{\text{ref}} \right) \times 100\]

where:

x: The average (aggregated) value of the variable in the time series data based on the specified time step.
ref: The long-term mean (( mu )) or median (( eta )) of the variable (the climate normal).

SMDI: Soil Moisture Deficit Index

The SMDI is calculated recursively using the formula:

\[SMDI(t) = 0.5 \times SMDI(t-1) + \left( \frac{SD(t)}{50} \right)\]

where:

( SD(t) ) is the Soil Moisture Deficit at time ( t ), defined as follows:

\[\begin{split}SD(t) = \begin{cases} \frac{(x - \eta)}{(\eta - \text{min})} \times 100 & \text{if } x \leq \eta \\ \frac{(x - \eta)}{(\text{max} - \eta)} \times 100 & \text{if } x > \eta \\ \end{cases}\end{split}\]

( x ) The average (aggregated) value of the variable in the time series data based on the specified time step.
( eta ) is the long-term median of the variable (the climate normal).
( text{min} ) is the long-term minimum of the variable.
( text{max} ) is the long-term maximum of the variable.
( t ) is the time step of the time series data.