# Interpolation

As a first step, Meteonorm calculates monthly average values for six key climate parameters
for a target location (latitude, longitude, and altitude):

* Global horizontal irradiance
* Air temperature
* Dew point temperature
* Wind speed
* Precipitation
* Number of days with precipitation

A standardized interpolation method is applied for all parameters, blending
quality-controlled measurements from ground weather stations with high-resolution gridded data.

Measurements from ground weather stations provide local accuracy while gridded data provides global coverage.
In non-polar regions, satellite data is used for global horizontal irradiance.
In polar regions, and for the other parameters, ERA5 reanalysis data is used.


## Distance-Based Blending

The core of the interpolation is a distance-based blending strategy. The influence of
ground stations versus gridded data depends on the 3D weighted distance between the target location and the
nearest station.

* **Nearest station is closer than $$d_1$$ km**: only data from the nearest station is used.
* **Nearest station is between $$d_1$$ and $$d_2$$ away**: data from up to six nearby stations
is blended using the _inverse distance weighting_ method described below.
* **Nearest station is between $$d_2$$ and $$d_3$$ km away**: station data is blended with gridded data. The weight of the station data decreases as the distance increases.
* **Nearest station is more than $$d_3$$ km away**: only gridded data is used.


| Parameter                       | $$d_1$$  | $$d_2$$ | $$d_3$$ |
| :-------------------------------| :------- | :------ | :------ |
| Global horizontal irradiance    | 2	     | 10      | 50      |
| Temperature                     | 2	     | 20      | 100     |
| Dew point temperature           | 2	     | 20      | 100     |
| Wind speed                      | 2	     | 20      | 100     |
| Precipitation                   | 2	     | 20      | 100     |
| No. of days with precipitation  | 2	     | 20      | 100     |

<p id="algorithms-interpolation-tab1">**Table 1:** _Distance thresholds (in km) used for blending gridded and ground station data._ </p>

The following schema illustrates the blending logic:

![algorithms-interpolation-fig1](./algorithms_interpolation_fig1.png)
**Figure 1:** _A visual representation of the blending logic based on the distance to the nearest station._


## Inverse Distance Weighting

When multiple ground stations are used, a target time series is interpolated using a modified
inverse distance weighting (IDW) method. This method takes into account both the horizontal
distances between target and stations, and the differences in altitude, applying a parameter-specific
vertical scale factor and gradient. For $$N$$ stations, the target value $$Y_x$$ is the weighted
average of the station values $$Y_i$$:

<span className="block text-center">$Y_x = \frac{
    \sum_i^N w_i \cdot \bigl(Y_i + (z_i - z_x) \cdot g_v  \bigr)
}{
    \sum_i^N w_i
}$</span>

where

* $$w_i = \frac{1}{\biggl(\frac{d_i}{R}\biggr)^2 + v^2 \cdot (z_i - z_x)^2}$$<br/>
  is the weight for station $$i$$ (note that we only consider stations with $$d_i \lt R$$),
* $$d_i$$ is the horizontal distance between the target and station $$i$$,
* $$R$$ is the maximum search radius (see [Table 2](#algorithms-interpolation-tab2)),
* $$v$$ is the vertical scaling factor ("altitude penalty", see [Table 2](#algorithms-interpolation-tab2)),
* $$z_i - z_x$$ is the difference in altitude between the target and station $$i$$,
* $$g_v$$ is the vertical gradient (see [Table 2](#algorithms-interpolation-tab2)).

| Parameter                       | Vertical Gradient ( $$g_v$$ )  | Vertical Scaling Factor ( $$v$$ ) | Max. Search Radius ( $$R$$ ) |
| :------------------------------ | :----------------------------- | :-------------------------------- | :--------------------------- |
| Global horizontal irradiance    | 0         	                   | 100                               | 150 km                       |
| Temperature                     | -0.5 °C / 100m	               | 200                               | 300 km                       |
| Dew point temperature           | -0.4 °C / 100m	               | 200                               | 300 km                       |
| Wind speed                      | +0.2 m/s / 100m	               | 200                               | 300 km                       |
| Precipitation                   | 0         	                   | 200                               | 300 km                       |
| No. of days with precipitation  | 0                              | 200                               | 300 km                       |

<p id="algorithms-interpolation-tab2">**Table 2:** _Parameter-specific factors for the IDW interpolation._ </p>