SD-GCM Formulas

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Formulas in SD-GCM
In the SD GCM tool, users have access to three statistical downscaling (SD)
methods, which include the following models:
1. The Delta method
2. The Quantile Mapping (QM) method (Panofsky and Briar, 1968)
3. The Empirical Quantile Mapping (EQM) method (Boe et al., 2007)
A- Delta Method:
The Delta method is a statistical downscaling technique used in climate science
and hydrology to estimate local or regional climate variables, such as precipitation
or temperature, from the output of global climate models (GCMs) or regional
climate models (RCMs). This method is relatively simple and relies on the concept
of change or delta between the historical observed climate data and the modeled
climate data.
Here's a basic description of how the Delta method works:
- Calculate the difference or delta between the GCM/RCM model output and the
observed historical data for a specific time period (e.g., monthly or annually).

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These deltas represent the modeled change in climate conditions compared to
historical conditions.
- To downscale future climate projections, apply the calculated deltas to the
future GCM/RCM model data. Essentially, you are adding or subtracting the deltas
to the model output to adjust it to conditions more similar to historical
observations.
The Delta method is a useful tool for obtaining localized climate projections when
more complex and computationally intensive downscaling methods are not
feasible, but it should be used with an awareness of its limitations and
assumptions.
As presented in Eq. 1 and Eq. 2 the precipitation and temperature of GCM data are
downscaled
𝑃𝑆𝐷−𝐹𝑢𝑡
𝐷𝑒𝑙𝑡𝑎
= 𝑃𝐺𝐶𝑀−𝑅𝐶𝑃/𝑆𝑆𝑃 ×
𝑃
̅𝑂𝑏𝑠
𝑃
̅𝐺𝐶𝑀ℎ𝑖𝑠𝑡.
(1)
𝑇𝑆𝐷−𝐹𝑢𝑡
= 𝑇𝐺𝐶𝑀−𝑅𝐶𝑃/𝑆𝑆𝑃 + (𝑇
̅𝑂𝑏𝑠 − 𝑇
̅𝐺𝐶𝑀ℎ𝑖𝑠𝑡) (2)

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𝑃𝑆𝐷−𝐸𝑣𝑎𝑙
= 𝑃𝐺𝐶𝑀ℎ𝑖𝑠𝑡 ×
𝑃
̅𝑂𝑏𝑠
𝑃
̅𝐺𝐶𝑀ℎ𝑖𝑠𝑡.
(1-1)
𝑇𝑆𝐷−𝐸𝑣𝑎𝑙
= 𝑇𝐺𝐶𝑀ℎ𝑖𝑠𝑡 + (𝑇
̅𝑂𝑏𝑠 − 𝑇
̅𝐺𝐶𝑀ℎ𝑖𝑠𝑡) (2-1)
In this equation, we introduce and use various data components as follows:
1. Future Period Downscaled Data: 𝑃𝑆𝐷−𝐹𝑢𝑡
and 𝑇𝑆𝐷−𝐹𝑢𝑡
denote downscaled
precipitation and temperature data, respectively, specifically for the future period.
2. Evaluation Period Downscaled Data: 𝑃𝑆𝐷−𝐸𝑣𝑎𝑙
and 𝑇𝑆𝐷−𝐸𝑣𝑎𝑙
represent
downscaled precipitation and temperature data, respectively, designed for the
evaluation period.
3. GCM Scenario Data: 𝑃𝐺𝐶𝑀−𝑅𝐶𝑃/𝑆𝑆𝑃 and 𝑇𝐺𝐶𝑀−𝑅𝐶𝑃/𝑆𝑆𝑃 represent data derived
from Global Climate Models (GCMs) under various Shared Socioeconomic
Pathways (SSPs) or Representative Concentration Pathways (RCPs). These datasets
serve as scenario-based inputs.
4. GCM Historical Data: 𝑃𝐺𝐶𝐺ℎ𝑖𝑠𝑡 and𝑇𝐺𝐶𝑀ℎ𝑖𝑠𝑡 represent historical data obtained
from Global Climate Models (GCMs), specifically from each individual model.
5. Average Observed Mean Data: 𝑃
̅𝑂𝑏𝑠 and 𝑇
̅𝑂𝑏𝑠 signifies the average observed
precipitation and temperature over a specified period.

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6. Historical GCM Mean Data: 𝑃
̅𝐺𝐶𝑀ℎ𝑖𝑠𝑡 and 𝑇
̅𝐺𝐶𝑀ℎ𝑖𝑠𝑡. refers to the GCM's mean
simulation data for historical precipitation records.
B- Quantile Mapping (QM)
Quantile Mapping (QM) is a statistical downscaling method commonly used in
climate science and meteorology to bridge the gap between coarse-resolution
climate model output and finer-scale, local climate data. This method is
particularly useful when you want to make climate model projections more
relevant and accurate for specific regions or locations.
You calculate the cumulative distribution functions (CDFs) for both the observed
(high-resolution) data and the climate model (coarse-resolution) data. The CDF
essentially shows the probability of observing a particular value or less. For each
value in the climate model data, you find the corresponding quantile from the
observed data's CDF. This step involves matching the probabilities from the model
to the observed data. After mapping the quantiles, you adjust the climate model
data values based on the quantile mapping. The adjusted climate model data now
closely resemble the statistical characteristics of the observed data, including the
distribution of values and their variability. Quantile Mapping helps in addressing
biases and discrepancies between climate model outputs and observed data. By

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preserving the statistical properties of the observed data, it provides more
reliable projections of future climate conditions at finer spatial and temporal
scales.
Some advantages of Quantile Mapping include its simplicity and ability to correct
for both mean biases and variability biases. However, it's essential to note that
this method assumes a stationary relationship between the historical and future
climate, which may not always hold true in the face of significant climate change.
In such cases, more complex downscaling methods or additional considerations
may be necessary.
This concept is calculated according to Equation 3, applicable to precipitation data
(or any other variable).
𝑃𝑡
𝐸𝑣𝑎𝑙
= 𝐼𝑛𝑣𝐶𝐷𝐹𝑃𝑡
𝑜𝑏𝑠
(𝐶𝐷𝐹𝑃𝑡
𝐻𝑖𝑠𝑡
(𝑃𝑡,𝐻𝑖𝑠𝑡)) (3-1)
𝑃𝑡
𝑃𝑟𝑒𝑑𝑖𝑐𝑡
= 𝐼𝑛𝑣𝐶𝐷𝐹𝑃𝑡
𝑜𝑏𝑠
(𝐶𝐷𝐹𝑃𝑡
𝐻𝑖𝑠𝑡
(𝑃𝑡,𝑅𝐶𝑃/𝑆𝑆𝑃)) (3-2)
Where 𝐼𝑛𝑣𝐶𝐷𝐹𝑃𝑡
𝑜𝑏𝑠
represents the inverse cumulative distribution function (CDF)
based on observational (or station) data.

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𝐶𝐷𝐹𝑃𝑡
𝐻𝑖𝑠𝑡
represents the cumulative distribution function (CDF) based on historical
period of GCMs model.
𝑃𝑡,𝑅𝐶𝑃/𝑆𝑆𝑃 represent each value of precipitation time series in RCPs/SSPs scenarios
of GCMs model.
𝑃𝑡,𝐻𝑖𝑠𝑡 represent each value of precipitation time series in historical period of
GCMs model.
The formula provided above remains consistent for any variable; only the
distribution utilized can vary. In SD-GCM, we employ a Gamma distribution for the
Multiplicative Correction Factor and Normal distributions for the Additive
Correction Factor.
C- Empirical Quantile Mapping (EQM)
Quantile Mapping (QM) and Empirical Quantile Mapping (EQM) are both
statistical downscaling techniques used in climate science and meteorology and
the formula is similar, but they have distinct differences in their approaches:
To use QM, we should select a distribution for data like Normal, Gamma...
distribution but EQM is a specific implementation of quantile mapping that relies
solely on empirical data without assuming any specific probability distribution for

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the data. The key feature is that EQM directly maps quantiles from the model
data to the quantiles of the observed data, without making any distributional
assumptions. EQM is nonparametric in nature. It doesn't rely on fitting a
particular probability distribution to the data but instead uses the data itself to
perform the mapping.
In summary, while both Quantile Mapping (QM) and Empirical Quantile Mapping
(EQM) are methods for adjusting climate model data to match observed data,
EQM is a specific implementation of QM that is nonparametric and relies entirely
on empirical data for the mapping, making it a simpler and more data-driven
approach. QM, on the other hand, is a broader framework that can involve
different types of quantile mapping, including parametric methods that assume
specific probability distributions for the data. The choice between QM and EQM
depends on the specific research objectives and the characteristics of the data
being used.
D- Compose with Wavelet

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We have recently added a new icon to the tool, enabling you to integrate wavelet
analysis with the existing methods. To gain a deeper understanding of how this
feature operates, please consult the flowchart provided below:
Wavelet
Wavelet
Wavelet
Delta or
QM or
EQM
Delta or
QM or
EQM
An
Dn
An
Dn
An
Dn
Inverse
Wavelet
RCPs/SSPs/Historical Data
Historical Data
Observation Data
Downscaled An
Downscaled Dn
Downscaled Data
End
As depicted in the flowchart, wavelet transformation can be integrated with any of the
three methods mentioned above. Given the variability in wavelet transform levels and
numbers, you have the option to check a broad array of techniques by combining them
with the three exponential scaling methods.
Wavelet Parameters
In the context of wavelet analysis, "wavelet type," "number," and "level" refer to key
parameters that determine how wavelet transformations are applied to a signal or data
set:
1. Wavelet Type:

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- The "wavelet type" specifies the shape of the wavelet function that will be used for
the analysis. Different wavelet types are designed to capture different types of patterns
or features in the data. Common wavelet types include Daubechies (db), Symlet(sym),
and Coiflet(coif), among others. The choice of wavelet type depends on the
characteristics of the data and the specific goals of the analysis.
2. Number of Wavelets:
- The "number of wavelets" typically refers to the number of wavelet functions used
in a wavelet transform. In discrete wavelet transforms, this often means how many
scaling and wavelet functions are employed at each level of the transformation. For
example, a common choice is to use two wavelets: one for approximation (scaling) and
one for detail (wavelet) at each level. However, in some cases, more wavelets may be
used to capture finer details or different frequency components.
3. Level of Decomposition:
- The "level of decomposition" or "number of decomposition levels" indicates how
many times the data will be iteratively transformed using wavelets. Each level of
decomposition divides the data into different frequency components or scales. A higher
level of decomposition provides more detailed information about the data's frequency
content but may also result in a larger number of transformed coefficients.

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These parameters are essential for customizing wavelet analysis to suit the specific
characteristics of the data and the goals of the analysis. The choice of wavelet type,
number, and level can have a significant impact on the results and the ability to extract
relevant information from the data. It's often necessary to experiment with different
parameter settings to find the most suitable configuration for a particular analysis.

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