*2.2. Methods*

IASI will add unique capabilities to the present study because we were able to simultaneously retrieve *Ts* and *Td* (e.g., [28–32]) from this instrument. To this end, we developed two retrieval prototypes: one for simultaneous inversion of infrared observations (level 2 or L2 prototype), and the second for remapping L2 products on a regular grid (L3 prototype).

The layout of the overall scheme we developed is sketched in Figure 2. The procedure consists of three main steps identified in Figure 2 with grey boxes.

**Figure 2.** Schematic flow chart of the methodology developed in the present study to yield Level 3 monthly maps of the water deficit index [30,33,34].

The IASI Level 2 and 3 prototypes have been developed in previous studies. The most up-to-date versions of both schemes can be found in [30,33] for the L2 package and [34] for Level 3 Optimal Interpolation. The digital object identifier (doi) shown in Figure 2 allows the interested reader to open online references where the two schemes are analytically presented. For this reason, they are just summarized in the present paper. In contrast, the Pre-OI scheme is described in more detail, as it implements the equations and formulas needed for calculating the state vector and associated covariance matrix, which are passed to the OI scheme to compute the maps of the water deficit index.

### 2.2.1. The L2 Retrieval System

The L2 prototype, which we also call δ-IASI, consists of an optimal estimation scheme (e.g., [35]), which simultaneously inverts the full IASI spectrum to retrieve the state vector, which is made up of the surface emissivity (*ε*), the surface temperature (*Ts*), the atmospheric profiles of temperature (*T*), water vapour (*Q*), ozone (*O*), HDO (*D*), carbonyl sulfide or OCS, and scalar scaling factors for the column amount of CO2, CO, N2O, CH4, SO2, HNO3, NH3, and CF4. However, the parameters relevant to the present analysis are *Ts*, and the atmospheric profiles for *T* and *Q*. Our L2 prototype for IASI has been variously validated as far as the surface parameters and *T* and *Q* profiles are concerned. Validation for surface parameters can be found, e.g., in [32,33], whereas for *T* and *Q* they can be found in [30,36].

### 2.2.2. The L2 Pre-OI and the Definition of the Water Deficit Index

Regarding Figure 2, the Pre-OI acts on the IASI Level 2 data to extract the geophysical parameters close to the surface and compute the water deficit index and its variance to input the final optimal interpolation scheme. From the profiles of *T* and *Q*, we considered only the elements, which correspond to the lowermost atmospheric layer, say *T*1 (in units of K) and *Q*1 (in units of gr/Kg). The corresponding layer pressure was denoted with *P*1 (in units of hPa). From the L2 products, we also extracted the surface temperature, *Ts*. The three parameters ( *Ts*, *T*1, *Q*1) were piled up in a vector, *x*1 of size *n* = 3, whose covariance matrix was denoted with *S*1, whose size was *n* × *n*. Both *x*1, *S*1 are outputs of the IASI L2 system.

The computation of the dew point temperature, *Td*, involves the calculation of the actual and saturation water vapour pressures. These are referred to as *Pw* and *Pws* , respectively. From *Q*1, we can compute *Pw* according to:

$$P\_w = 10^{-3} P\_1 Q\_1 \frac{R\_w}{R\_{air}} = \beta P\_1 Q\_1; \beta = 10^{-3} \frac{R\_w}{R\_{air}} \tag{1}$$

with *Pw* in hPa and where *Rw* = 461.5 J K−<sup>1</sup> Kg−<sup>1</sup> and *Rair* = 286.9 J <sup>K</sup>−1Kg−<sup>1</sup> are the specific gas constants of water vapour and air, respectively. According to [37], *Pws* is computed with the formula:

$$P\_{\text{res}} = 10^{-2} \frac{\exp\left(a\_1 - \frac{a\_2}{t\_1 + a\_3}\right)}{\left(t\_1 + a\_4\right)^{a\_5}}\tag{2}$$

with *t*1 = *T*1 − 273.15 (temperature in degrees Celsius) and *Pws* in hPa. Equation (2) is valid for *t*1 > 0 (vapor pressure of water), and where *a*1 = 34.494, *a*2 = 4924.99, *a*3 = 237.1, *a*4 = 105, *a*5 = 1.57 are fit parameters that in case *t*1 are expressed in degrees Celsius. From (1) and (2), we obtain the fractional relative humidity:

$$
\sigma h = \frac{P\_{\text{uv}}}{P\_{\text{uv}}} \tag{3}
$$

From (1) and (2), we can also compute the vapour pressure deficit or *VPD* = *Pws* − *Pw*. Finally, the dew point temperature, *Td* can be calculated by using the well-known Magnus formula (e.g., [38]):

$$t\_d = \frac{c\chi}{b - \chi'}, \ x = \ln(rh) + \frac{bt\_1}{c + t\_1} \tag{4}$$

where *td* is in degrees Celsius (we will use *Td* when referring to degrees Kelvin units), and *b* = 17.62 (dimensionless), *c* = 243.12 C. Finally, the IASI-based water deficit index, *wdi*, is defined according to:

$$wdi = T\_s - T\_d = t\_s - t\_d \tag{5}$$

Equation (5) stresses that the index can be computed indifferently with both temperatures in K or C degrees, although the computation of the dew point temperature has to be performed in C, according to Equation (4), before converting it to K.

For the application of the optimal interpolation to the mapping of the water deficit index, we also need the variance of the index, *<sup>σ</sup>*2*wdi*. Considering the chain of equations from (2) to (5), we can formally write *wdi* as a function *wdi* = *f*(*Ts*, *T*1, *Q*1), from which, using the usual rule of variance propagation (see, e.g., [39]), we obtain:

$$
\sigma\_{wdi}^2 = \mathbf{g}^\mathbf{f} \mathbf{S}\_1 \mathbf{g} \tag{6}
$$

with the superscript *t* indicating the transpose operation, and

$$\mathbf{g} = \left(\frac{\partial f}{\partial T\_s}, \frac{\partial f}{\partial T\_1}, \frac{\partial f}{\partial Q\_1}\right)^t \tag{7}$$

We stress that the parameters defined by Equations (5) and (6) have to be computed for the IASI retrievals and the ECMWF background, as shown in the diagram of Figure 2. For the background, the covariance matrix is assumed to be diagonal, as we use background derived from climatology (see [29]) for which we do not consider correlation among air temperature, humidity, and surface temperature.

Considering that

$$f(T\_s, T\_1, Q\_1) = T\_s - T\_d = t\_s - t\_d = t\_s - \frac{c\chi}{b - \chi} \tag{8}$$

we have

$$\frac{\partial f}{\partial T\_s} = \frac{\partial f}{\partial t\_s} = 1$$

$$\frac{\partial f}{\partial T\_1} = \frac{\partial f}{\partial t\_1} = \frac{\partial f}{\partial x}\frac{\partial x}{\partial t\_1} = -\frac{cb}{(b-x)^2} \left( -\frac{a\_2}{(t\_1+a\_3)^2} + \frac{a\_5}{(t\_1+a\_4)} + \frac{bc}{(t\_1+c)^2} \right) \tag{9}$$

$$\frac{\partial f}{\partial Q\_1} = \frac{\partial f}{\partial x}\frac{\partial x}{\partial Q\_1} = -\frac{cb}{(b-x)^2}\frac{1}{Q\_1}$$

The parameter *wdi*, when referring to a surface covered by vegetation or crops, can help to understand the water stress or deficit during long-lasting droughts or heatwaves. This is because vegetation releases water into the atmosphere through transpiration. The process involves the vaporization of liquid water in plant tissues and the consequent release of vapour into the atmosphere (for example, see [40])). Similar to direct evaporation, transpiration depends on the amount of energy available: solar radiation, wind, and vapour pressure gradient at the surface–atmosphere interface. Consequently, solar radiation, air temperature and humidity, and wind velocity must be considered when evaluating and assessing a satellite-based index to quantify water deficit.

In Equation (5), the role of energy supply is modelled with *Ts*. The sun's radiation will cause a rapid increase in the surface temperature of the land. On the other hand, *Td* will take into account both air temperature and air humidity. The effect of wind is more difficult to introduce. However, drought and heatwave conditions minimize the spatial gradient and wind intensity. The air subsidence and low intense pressure gradients characterize meteorological conditions that favour summer heatwaves.

It is also important to stress that evaporation and transpiration co-occur, and it is not easy to distinguish between the two processes. For this reason, we mention evapotranspiration when referring to the water exchange between vegetation and the air. In addition to water availability in the topsoil, evaporation from the cultivated terrain depends, as already mentioned, on the amount of impinging solar radiation. The solar energy at the surface decreases during crop growth because its foliage or canopy shadows the area below from the sun's rays as the crop develops. Therefore, water is predominately lost by soil evaporation when the crop is small, or when the leaves are not well developed. However, transpiration becomes the main process once the crop and leaves are well developed and completely cover the soil.

With this in mind, the parameter *wdi* can help to identify different regimes of water deficit:


### 2.2.3. The 2-D OI scheme

It is worth noting that when *wdi* is determined by L2 satellite observations, as in our case, we obtain data that are sparse and not homogeneously covering a given spatial region. Therefore, to better compare with other data sources and perform a correct collocation with stations at the ground, we used a resampling tool, which can remap *wdi* data to a regular grid. To this end, we used the tool developed in [34]. The technique is based on a 2-dimensional (2D) optimal interpolation (OI) scheme, and is derived from the broad class of Kalman filter or Bayesian estimation theory. For further details, we refer the interested reader to [34].

The steps involved in the mapping on a regular grid are exemplified in Figure 3 using the IASI retrieval for *wdi* for July 2017. Figure 3a,c show the IASI data points for *wdi*, and its square root of the variance (standard deviation) as estimated by the L2 retrieval scheme and the Pre-OI step (see Figure 2). These values are accumulated considering all the IASI overpasses for July 2017. As said before, the IASI scan pattern is made up of footprints with circular diameters of about 12 km at nadir, and the scanning lines are 50 km apart along the flight direction of the satellite. The IASI scan pattern over the target area for a single overpass is shown in Figure 4 for the benefit of the reader. Comparing Figure 3 with Figure 3a,c, it can be seen that after one month, the IASI clear sky footprints (we stress that we use only observations in a clear sky, which is diagnosed based on a stand-alone algorithm for cloud detection, e.g., [41]) are densely distributed over the area much more than the single IASI scan pattern overpass. The monthly ensemble of satellite overpasses improves the sampling of spatial data, and therefore allows, for example, a better comparison with in situ observations. We use the ensemble of multiple observations to build a map with a better spatial sampling. Towards this objective, we use the 2-D OI method, which remaps the data into a grid with a finer mesh than the original data.

**Figure 3.** July 2017. IASI L2 products for *wdi* (panel (**a**)) and its standard deviation (panel (**c**)) over the target area. The figure also shows the ECMWF background field (both mean (panel (**b**) and standard deviation (**d**)) at its native spatial resolution of 0.125◦ × 0.125◦.

**Figure 4.** Target region showing the IASI footprint scan pattern (red ovals) for one single overpass. The IASI morning overpass for May, the first of 2020 is shown in the figure.

The final mesh we use has a spatial sampling of 0.05◦ × 0.05◦. Another important aspect of OI remapping is the use of background fields. These fields are built up by using the time and space co-located ECMWF (European Centre for Medium-Range Weather Forecasts) analysis. The ECMWF fields are available on a grid-mesh of 0.125◦ × 0.125◦, and, for the case at hand, the values for *wdi* and its square root of the variance, i.e., standard deviation, are exemplified in Figure 3b,d, respectively. Based on the coarse ECMWF background, the un-gridded L2 IASI observations and the 2-D OI yields the results are shown in Figure 5; that is, the maps of *wdi* (panel (a)) and its standard deviation (panel (b)) at a sampling of 0.05◦ × 0.05◦. In this process, we lose temporal resolution, but we obtain a map with improved spatial sampling and precision, as shown by the standard deviation map, which, apart from boundary effects, is one ◦C or less.

**Figure 5.** Level 3 map at a grid step of 0.05◦ for the index *wdi* obtained from the source data shown in Figure 3 (panel (**a**)) and its standard deviation (panel (**b**)). The map is exemplified for July 2017.
