Water vapor concentration structures in the atmosphere are well approximated horizontally by Gaussian random fields at small scales (

As a consequence, the atmosphere has an inherently stochastic component associated with the horizontal small-scale water vapor features, which, in turn, can make deterministic forecasting or nowcasting difficult. These results can be useful in areas where high-resolution modeling of water vapor is required, such as the estimation of the water vapor variance within a region or when searching for consistency between different water vapor measurements in neighboring locations. In terms of weather forecasting or nowcasting, the water vapor horizontal variability could be important in estimating the uncertainty of the atmospheric processes driving convection.

Meteorologists frequently need to determine the water vapor characteristics of an air parcel in the atmosphere. In the review from

One such examples is passive satellite instruments measuring in the infrared or the microwave region of the spectrum. These instruments measure air-mass regions at horizontal scales of tens of kilometers (i.e., surfaces of a few hundred square kilometers) and vertical thickness of a few kilometers. They actually integrate the radiation coming from all the air sub-parcels into which the measurement region can be subdivided. In fact, the horizontal variability of water vapor within the measurement region in the field of view of satellite instruments can have significant effects when calculating the radiances impinging on the instrument via a radiative transfer model (RTM). In this case, it is necessary to know the variance of the water vapor concentration within the remotely sensed air parcel

Another example is the calculation of instability indices for nowcasting purposes, particularly the convective available potential energy (CAPE). Operational meteorology usually takes as a first approximation of these instability indices the ones obtained from NWP model forecasts or measured from remote sensing satellites. To refine such indices with ground-based station measurements is not simple due to significant differences between different measurements or NWP models

While it is common to treat atmospheric water vapor as a fluid in turbulent motion in study fields for which measurements are made at small scales, this is usually not so in other areas where larger scales are measured or modeled. Ground station or lidar measurements usually do apply concepts from turbulence theory

It should be noted that turbulent behavior in the atmosphere can be grouped into two different categories depending on whether the measurement is done mostly in the vertical or the horizontal, each of them involving different scale lengths. The vertical measurements typically have much smaller scale lengths than the horizontal phenomena studied here. Measurements involving mostly vertical variability are typically the scintillation measurements done at low zenith angles (e.g.,

Achieving a complete characterization of water vapor concentration in an air parcel would require knowledge of the water vapor concentration at all “points” within such an air parcel. As this goal is not achievable in practice, to bridge this gap it would be extremely convenient to have an approximate model of the behavior of water vapor concentration in the atmosphere at smaller scales. A way this is solved in other areas of geophysical sciences is by using kriging

As it turns out, the atmosphere is a fluid in turbulent motion, from which it follows that Kolmogorov's theory of turbulence applies. This theory basically states that fluids in turbulent motion have parameter fields which on average follow a GRF. In this paper it will be shown how the water vapor concentration in the atmosphere at smaller horizontal scales, on average, does indeed follow this pattern. This will be done in two ways. The first evidence will be to calculate what is known as the structure function: that is, how the variances scale with horizontal distance. This will be done for several instruments and it will be shown that they do scale following the “two-thirds law” as expected from Kolmogorov's theory

In Sect.

In this section the basic theory of turbulence is presented along with the mathematical definition of what a Gaussian random field is. The concept of structure function will also be introduced, alongside an example for the atmosphere. The theory is presented for two-dimensional horizontal fields, but the atmosphere has in reality three dimensions. A few remarks regarding the third dimension are made in the last subsection.

Kolmogorov's theory of turbulence is the set of hypotheses stating that a small-scale structure is statistically homogeneous, isotropic and independent of the large-scale structure. The source of energy at large scales is either velocity (wind) shear or convection. This set of hypotheses together with the Navier–Stokes equations are the foundations of Kolmogorov's theory. From these hypotheses, the experimentally observed “laws” can be derived. These are the two-thirds law and the law of finite energy dissipation

Panel

A key concept in Kolmogorov's theory of turbulence is that of the structure function. This shows how the average of the squared difference of a fluid parameter between two spatially separated points behaves as a function of their distance. Usually, a log–log plot is used to make this representation. This is illustrated in the central panel of Fig.

Apart from the energy injection range, which, as we shall see later, is difficult to see with other instrumentation due to its narrow range, the conclusion that can be drawn from Fig.

It should also be noted that these considerations apply when averaging a big sample of measurements. In practice, when taking a smaller sample, the structure function will vary significantly from one region to another. One of such deviations from the average structure function is the location in the vertical axis of the inertial range. If the data that fit the two-thirds law are high along this axis, it would indicate a high turbulence or a high-horizontal-concentration variability regime. If the data are placed in a lower place along the ordinate axis, they would nominally indicate a lower turbulence or concentration variability. The exact position where these points lie in the structure function graph will depend on the degree of turbulence of the region being analyzed. Another deviation from the average structure function is the frontier between the inertial and the synoptic range, which can, as we shall see later, vary significantly from one region to another. But, for the atmosphere on average this frontier lies around 6 to 10 km in the horizontal.

Kolmogorov's theory also implies that in the inertial range the parameter under study follows a GRF on average. To have a feeling of how a GRF looks, the far left image in Fig.

In this paper the only parameter we will focus on is one scalar field, namely the atmospheric water vapor concentration. This study could easily be extended to more parameters, like temperature. A field is defined by

A random field is one in which the value of the parameter is random and follows a certain probability distribution. In the case of Gaussian random fields (e.g.,

The notation of the covariance can now also be simplified to

In the real atmosphere, all this is, of course, a simplification. Nevertheless, we will show that this approximation holds relatively well for small scales of observed water vapor concentrations. In summary, a Gaussian random field (GRF) will be understood in this paper as a random field satisfying Eqs. (

In this subsection very brief mention will be made of how the vertical dimension can be dealt with. These considerations are particularly important for satellite remote sensing in the thermal infrared or microwave spectral region.

Adding the vertical dimension to the two horizontal ones would complete the study in the three-dimensional space. In meteorology and satellite remote sensing it is convenient to divide the atmosphere into vertical layers. Satellites, in particular, can be considered instruments observing the atmosphere divided into several layers of finite thickness. For example, in the thermal infrared, to a very rough first approximation, for any one spectral channel the measured satellite radiance can be considered an average emittance over several layers. Therefore, the particulars of the structure function observed by the satellite will depend on the vertical correlation between these layers. If the layers have completely independent statistical properties, then the covariances of each layer can be averaged independently to give the combined covariance. This would constitute the typical

In summary, if the vertical layers are statistically independent, the satellite-observed structure function will have a much lower

In this paper, one NWP model (ECMWF) and data from several satellite and radiosonde instruments have been used. These datasets are detailed in the following subsections.

The radiosonde measurement data are from the EUMETSAT EPS/MetOp campaigns made in 2007 and 2008 at Lindenberg (Germany) and Sodankylä (Finland) observatories

Sonde measurements sample the atmosphere every second as the radiosonde ascends in the air. This effectively means measuring the troposphere in layers around 0.6 to 0.1 hPa thick in the pressure levels used in this study. They range from 950 to 200 hPa, respectively. The humidity measurements from the first radiosonde of the sequential sonde pair is vertically interpolated to the vertical pressure grid of the second sonde. By doing this, both water vapor measurements from each pair of sequential sondes can be compared directly. To calculate the structure function, the normalized differences between water vapor partial pressure measurements from each radiosonde at the same pressure level are calculated. All water vapor units are converted using the Hyland and Wexler

For temperature, the temperature difference is directly calculated,

The Spinning Enhanced Visible Infrared Imager (SEVIRI) is an imager instrument on board the Meteosat Third Generation (MSG) geostationary satellite

The SEVIRI image date and time for the determination of the structure function have been selected randomly and correspond to 20 August 2019 at 10:00Z. The corresponding “air-mass RGB” image can be seen in Fig.

SEVIRI/MSG “air-mass RGB” image of the date and time selected (20 August 2019 at 10:00Z). Highlighted in red is the analyzed region. In cyan, the location of the ECMWF profile selected for RTM calculations is shown.

The

The structure function could be determined directly from the measured radiances, but these do not constitute an atmospheric parameter. It is therefore best to convert these radiances into a measurement of water vapor such as the HLWV defined above. A thorough retrieval, such as optimal estimation, within each Meteosat pixel could be derived. But, since the structure function is robust to any such estimations and it seems more illustrative for the reader to use a simple regression, only a first-order approximation of the HLWV will be performed. To estimate the HLWV from the

The RTM requires that only scenes unaffected by clouds are analyzed. Since the

The Ocean and Land Color Instrument (OLCI) is a push-broom imaging spectrometer that measures solar radiation reflected by the Earth. OLCI is on board the polar sun-synchronous Sentinel-3 Earth observation satellite series dedicated to ocean and land observation

From this instrument a retrieval of total column water vapor (TCWV) can be performed. The method used in this paper is based on the Copernicus Sentinel-3 OLCI Water Vapor (COWa) product. It uses an optimal estimation method to retrieve TCWV from the Oa17, Oa18, Oa19 and Oa20 OLCI bands. Because of this, it provides both a measurement of the TCWV and its uncertainty. The properties of the OLCI channels allow for accurate determination of the TCWV over land in clear-sky scenes. The TCWV estimation over ocean is far more uncertain and is not used in this paper. The method is fully described in

OLCI COWa TCWV field from 31 August 2016 at 09:45Z. The blue circles centered on the green dots are regions analyzed in Figs.

An image on 31 August 2016 at 09:45Z covering southeastern Germany and the western Czech Republic has been selected. This region is located over land and consists mostly of clear-sky pixels, making it an ideal candidate for the accurate measurement of TCWV with the OLCI. TCWV for this field is represented in Fig.

A comparison of the measured structure functions and small-scale horizontal variability with other sources can be instructive. For this reason, the same region as the one selected for OLCI is also selected for an NWP model. It should ideally be a high-resolution regional model. Since such a model was not available at the time of writing of this paper, the NWP model used here is the operational global ECMWF one. The data are retrieved from ECMWF's archive and obtained with a regular latitude–longitude grid of 0.125

ECMWF forecast TCWV field from 31 August 2016 valid at 10:00Z (10 h step from an analysis at 00:00Z). The contour of the OLCI observation from Fig.

In this section the methods applied to the data are discussed. In a first subsection, the way to calculate the structure function from the data is explained. Two different types of structure functions are calculated. The first one is denominated “pixel-centered structure function”. It is a structure function that is calculated on each and every pixel of the image. The second one is an “average structure function”, and, as the name indicates, it is a structure function calculated by averaging many pixel-centered structure functions.

In a second subsection, the calculations to derive the typical mathematical properties of GRFs are explained. Also, a histogram is obtained from the measurements, which should follow a Gaussian distribution. Several synthetic GRFs are generated, which are later compared to the measurements.

The water vapor structure function is calculated from the data in two different ways. One of them is centered in a particular pixel of the field or image, which will be called the pixel-centered structure function. The second one is an average over the whole satellite image or NWP field, which will be denoted as the average structure function. The way to calculate them is described below.

The goal is to have a structure function centered on each and every pixel within the satellite image or NWP field. Depending on the source of the data, the water vapor structure function is calculated for different parameters: TCWV for OLCI and ECMWF, HLWV for MSG. Note that for radiosondes (Fig.

Pixel-centered structure function for the OLCI COWa TCWV shown in panel

Pixel-centered structure function for the OLCI COWa TCWV shown in panel

After this, the distance between these two points is calculated. The square of the relative difference is calculated and its value is accumulated into its corresponding distance bin. A record of the number of occurrences in each bin is kept. To achieve relevant statistics, especially for the short distance ranges, the number of cases needs to be increased. This is done by shifting the pixel where the origin of distances is located around a

To reduce the uncertainties in the structure function or to have a global picture of it, it is convenient to obtain an average of all the pixel-centered structure functions of a given instrument. This is done in three steps.

The first step is to bring all cases to the same vertical axis in the structure function. This is achieved by averaging the structure value,

All components of the structure function are normalized by the

These rescaled structure function components are now averaged and brought back to the average level,

Finally, the logarithm of this quantity is taken to have the final value for the average structure function,

Values of

Examples of various average structure functions are shown in Fig.

Average structure functions for MSG/SEVIRI (red), Sentinel-3/OLCI (blue) and the ECMWF forecast (magenta). Also plotted is the plain structure function from radiosondes (green). Linear fits below

To demonstrate that water vapor structures at small scales do resemble GRFs we must first verify that measurements on an individual pixel do follow a Gaussian distribution by looking at its histogram. As a second step, water vapor measurements must visually resemble a GRF. For this, two synthetically generated GRFs have been created which can be compared to a spatial zoom into an OLCI TCWV measurement region. How these plots have been generated is described below.

A square of

Normalized histogram of TCWV differences calculated in boxes within the complete OLCI measurement region (blue) and a Gaussian function with a standard deviation obtained from the data (red).

To produce a representation of the product noise or uncertainty, a GRF with its standard deviation equal to the average OLCI COWa TCWV uncertainty (around 0.33 mm) and also with no spatial correlation (

Spatially independent (

To generate a synthetic GRF which follows the two-thirds law, an algorithm following the isotropic spectral method

To appreciate the small-scale features of the OLCI COWa TCWV fields, a zoom has been performed in a randomly selected region centered on (long, lat)

In this section, the results are shown. First, the structure functions will be discussed, and, in a later section, qualitative and quantitative views of the GRFs will be shown.

The average structure functions for several meteorological satellites and the ECMWF NWP model are shown in Fig.

Figure

Another distinct feature of this figure is the wide range of horizontal variability of water vapor, i.e., the different displacement of the curves in the vertical axis. The radiosonde data measure in very thin layers of at most 0.6 hPa in depth. All the other instruments measure in extremely thick layers or even the complete atmospheric column. This implies that the diminishing of the variance (smaller

Even though they are indeed measuring the same parameter at the same location in space and time, the contrast between the OLCI and ECMWF NWP curve is significant. Since the ECMWF NWP is a global model with a coarser resolution, it lacks the information at scales smaller than 6 km. Also, the horizontal variability of the ECMWF model is significantly smaller than the one from OLCI. This can also be verified visually by comparing the direct products from Figs.

Finally, a feature that also stands out is the presence of a small region in which the horizontal variability decreases with increasing distance, which is present in the OLCI pixel-centered structure functions (Figs.

The pixel-centered structure functions obtained from OLCI, two of which are shown in Figs.

The first property for a random field to be Gaussian is that individual pixels must follow a normal distribution. This is verified in the histogram plotted in Fig.

To show that the OLCI TCWV does indeed behave like a GRF, several different panels are represented in Fig.

In the left panel, a somehow especially restricted GRF has been generated. Its particularity is that it has no spatial correlation (

The measured TCWV can be compared with a synthetically generated GRF that does follow a spatial correlation under the two-thirds law. This is shown in the central panel of Fig.

The average structure function quite accurately follows a two-thirds law at small scales for several instruments (Fig.

To confirm this, the histogram of individual pixels is shown to follow a Gaussian distribution (Fig.

These assumptions can be applied only to scales below approximately 6 km on average. They do not apply at scales greater than this value, since the structure function deviates from the two-thirds law (Fig.

As a consequence of these results, the water vapor concentration in the atmosphere is inherently turbulent and chaotic at these small scales. There will always be a random component which will be impossible to measure on a full spatial scale in general and, in particular, that of a typical satellite infrared sounder instrument with a footprint of 15 km. Water vapor near the surface is also a critical parameter for the triggering of convection. This means that nowcasting will always have an inherently stochastic component associated with it. Because of this, it is highly likely that the best approach to making forecasts for nowcasting would be to have a probabilistic method.

Global NWP models do not seem to be able to accurately reproduce the intensity of the horizontal variability of water vapor at small scales (Fig.

All these results can be of practical importance to estimate the horizontal variability of water vapor within a region. It can also be applied in making several neighboring measurements consistent, since the random differences between two measurements can be estimated. The uncertainty usually present in the estimation of the atmospheric processes involved in convection could potentially also benefit from the proper characterization of water vapor horizontal variability.

The code is simple enough to be independently replicated.

Data from the MSG satellite, SEVIRI instrument are available from EUMETSAT's archives (

XC was responsible for writing the paper, calculation of structure functions, establishment of Kolmogorov's theory, and calculation of MSG HLWV. CCH provided OLCI data and guidance as well as calculating structure functions. SDSM calculated averaged structure functions. BS provided data and guidance on how to use sonde data. TR provided data and guidance on how to use sonde data.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the special issue “Analysis of atmospheric water vapour observations and their uncertainties for climate applications (ACP/AMT/ESSD/HESS inter-journal SI)”. It is not associated with a conference.

We thank Miguel Angel Martínez Rubio for assisting in the conversion of SEVIRI/MSG radiances into column water vapor content.

This paper was edited by Andreas Zahn and reviewed by two anonymous referees.