Aircraft are equipped with sensors for flight efficiency and safety. For this purpose, an aircraft measures the speed of the aircraft, its position, and ambient temperature and pressure. For a few decades a selection of these observations are transmitted to a ground station using the AMDAR system. An atmospheric profile can be generated when measurements are taken during take-off and landing. See for more details. Recently, a new type of aircraft-related meteorological information has become available, which originates from observations inferred from a tracking and ranging radar used for air traffic control. These data are called Mode-S EHS because they use the Selective EnHanced Surveillance Mode of the radar

http://mode-s.knmi.nl

A Doppler weather radar is capable of determining one component of the velocity of scattering particles. Only the velocity component along the line of sight, the so-called radial velocity, can be determined. A Doppler radar is commonly associated with measurements of frequency shifts because of the low velocities of hydrometeors. However, these shifts cannot be observed directly. The phase of the scattered electromagnetic waves is employed to determine the Doppler frequency shift instead. During pulse-pair processing, the velocity is effectively deduced from the phase jump of the received signal. The unambiguous velocity interval of the instrument, especially for C-band radars, is enhanced by applying a dual-pulse repetition frequency (PRF). The two Royal Netherlands Meteorological Institute (KNMI) radars are C-band with a wavelength of 5.3 cm. The high PRF is chosen to be four-thirds of the low PRF, resulting in an unambiguous velocity of 4 times the low PRF unambiguous velocity, which is 23 m s-1 for the lowest elevations and 47 m s-1 for the highest elevations. The PRF also determines the unambiguous range of the radar, which is 240 km for the lowest elevations and reducing to 120 km for the highest elevations. The radar beam will have an increasing height with increasing distance to radar due to the curvature of the earth.

A sodar (sonic detection and ranging) is a ground-based remote-sensing instrument for measuring wind and turbulence in the lower atmosphere. A mono-static sodar is operated and maintained by KNMI at Amsterdam Airport Schiphol (AAS) since March 2006. A sodar emits short acoustic pulses into the atmosphere and receives atmospheric echoes generated by small-scale density fluctuations that are associated only with thermally driven turbulence, which is not always present. The transmitted signals can be phase shifted to point the beam in different directions. At Schiphol, three are in use for the instrument, and one antenna is oriented vertically. The zenith angle of the other beams is dependent on the transmit frequency and varies between 10 and 30∘. The distance of the measuring volume is determined from the propagation time of the acoustic wave and the estimated acoustic velocity. Since the temperature inhomogeneities move with the wind, a Doppler frequency shift is observed that makes it possible to derive the wind speed relative to the beam axis. By measuring the Doppler shift for different beam directions, the full 3-dimensional wind at specific altitudes can be determined. Thereby it is assumed that the flow is horizontally homogeneous over the area containing the different measuring volumes.

The non-hydrostatic HARMONIE Hirlam ALADIN Research on Mesoscale Operational NWP in Euromed; model is the follow-up of the hydrostatic HIRLAM (HIgh Resolution Limited Area Model) model; HARMONIE explicitly resolves convective processes. The model grid size of the HARMONIE model version (cy38h1.2) operational at KNMI is 2.5 km, and the HARMONIE model has been available since early 2012 at KNMI. The model domain covers mainly western Europe and part of the North Atlantic and the number of grid points is 800 × 800, meaning that the domain covers a 2000 × 2000 km2 area. The number of model levels equals 65 with higher density in the lower part of the troposphere. The operational HARMONIE model version at KNMI is nested in the European Centre for Medium-Range Weather Forecasts (ECMWF) model. Note that lateral boundary conditions are ECMWF forecasts, generated from an (global) analysis using a large variety of observation types (SYNOP, radiosonde and satellite information). Table  lists the HARMONIE version used in the study and its main characteristics. In this study we will only use the 3 h forecast data.

Table  describes the main characteristics of the observation data sets used in this study. Figure  shows an example of the wind data.

The period used in this study runs from 1 January to 30 September 2013 because a rerun of HARMONIE data with version 38h1.2 is used and no more data were available overlapping the Mode-S EHS data set.

Observations are regarded at the same when the time difference is less than 5 min. Note that the model has a 3 h cycle (a new run is started every 3 h), which reduces the collocation time window to 10 min every 3 h because we use the 3 h forecast only in this study. We did not interpolate the model to the observation time and the interpolation in space was chosen to be bilinear.

We apply the triple-collocation method to find quantitative information on the observation error. This method exploits a data set consistent of three co-located measurements of the same parameter. In this paper we use two different wind data sets; the first consist of Mode-S EHS wind vector observation, sodar, and NWP at Schiphol airport, and the second data set consists of Mode-S EHS, radar radial wind, and NWP. The triple-collocation method determines simultaneously the linear calibration coefficients and the error of the three data sets under consideration. See and for more details. Below, a brief description of the method is given.

Assume we have three sets of data X1, X2, and X3, collocated in space and time, where X1 has the highest (expected) accuracy and X3 the lowest (expected) accuracy. Since the truth is unknown we take data set X1 as the unbiased reference (or truth). Assume furthermore that two other data sets have a linear relationship with this truth, that is x1=t+ϵ1,x2=a2t+b2+ϵ2,x3=a3t+b3+ϵ3, where t is the truth and ϵi is the accompanying error, which also contains the representation error, where ai and bi stand for the trend and bias calibration. Note that each data set is calibrated against the one with the highest resolution. After calibration, the data sets are transformed into an unbiased data set, which have an expected value of the error ϵi equal to zero, that is 〈ϵi〉=0, where 〈〉 denotes the expected value. Assume furthermore that the variance of the errors, 〈ϵi2〉, is independent of the truth t and has a Gaussian signature. As stated by this is true for the zonal and meridional wind components but not for wind speed and direction. In this paper we use additionally the radial wind component, which is a projection of wind vector on a (varying) azimuth angle, and thus the variance is expected to be independent of the true wind vector with a Gaussian error distribution.

The representativeness of the three observations most likely differ; there is a residual correlation error r2 of the scales that are represented by the high-resolution observations but are lacking in the relatively low-resolution NWP wind retrievals. Using all above-stated assumptions we are able to find estimates for the unknowns ai and bi; that is, with Mi, Mij and cij the first and second (mixed) moment, and co-variance, defined as Mi=〈xi〉,Mij=〈xixj〉 and cij=Mij-MiMj, the unknowns become a2=c23/c31 and a3=c23/(c12-a2r2)b2=M2-a2M1 and b3=M3-a3M1. Using these values we find that σ12=c11-c31c12/c23+r2σ22=c22/a22-c31c12/c23+r2σ32=c33/a32-c31c12/c23. A gross error check is performed to remove spurious outliers: the absolute difference of any two observations should be smaller than 4 times the square root of the sum of the (estimated) standard deviations. The only unknown now still is the residual correlation error r2. This correlation can be determined by a scale analysis of Mode-S EHS and NWP, following .

The data sets we use in this study consist of wind vector data (Mode-S EHS/sodar/NWP, N=2429 before gross error check) and radial wind (Mode-S EHS/radar/NWP, N=8132). For both data sets we need to determine the residual correlation error r2. Figure a shows the power spectral density (PSD) for the zonal component of the wind from Mode-S EHS (solid line, top) and HARMONIE (dashed bottom line). This graph is constructed using 9 months of Mode-S EHS collocations with HARMONIE. The PSD is calculated using Mode-S EHS data from aircraft, which reported wind for more than 100 km in length at a stable altitude. Note that the data set used to calculate the PSD differs from the triple-collocation data set, because the triple-collocated data set rarely contains points at a stable altitude over a length of more than 100 km. The thin dashed line shows the -5/3 Kolmogorov spectral decay. The PSD from Mode-S EHS lies close to this line, while the HARMONIE PSD is clearly lower displaying the lack of energy in the model at these scales. The area between these PSD represents the variance lacking in the model that is present in the observations; this area is approximately r2=0.312 m2 s-2. Figure b shows a similar plot but now for the simulated radial wind. We have used the distribution of azimuth angles in the radar radial wind data set to create a radial wind data set from NWP and Mode-S EHS. The area between the PSD for radial wind is around r2=0.285 m2 s-2.

estimated the representation error for buoys and scatterometer winds with respect to the ECMWF model to be r2=0.75 m2 s-2. This value is much higher than is found for HARMONIE in this study. The difference can not fully be explained by the difference observation (10 m vs. upper-air wind), nor in model resolution, nor in the fact that HARMONIE is a convection resolving model. Moreover, since HARMONIE uses ECMWF boundaries, upper-air characteristics from HARMONIE are linked to the ECMWF model. The main reason for the difference may lie in the fact that the used archived full-resolution model data, which cover an area of a few hundred kilometres and thus scales of more than approximately 400 km are lacking in the determination of the representation error. Furthermore, the HARMONIE model adds underdetermined turbulence on scales which are not initialized and are not observed by independent observations. This would add more (unrealistic) energy to the HARMONIE PSD, which results in an underestimation of the representation error. The difference in representation error needs further research but is not discussed here.

The representative error has some relation to the azimuth angle (zonal component of the wind is equal to a radial wind observed with an azimuth angle of 90∘); see Fig. . Each point in this figure is based on the mean value of PSD determined from the data set of wind vectors mapped to the radial component using a prescribed azimuth. Not surprisingly the residual error exhibits a bi-periodic behaviour, which is due to the fact that the errors of opposite vectors are identical. The periodic behaviour of the residual error may be due to the fact that u wind is way stronger than v wind component in Northern Hemisphere. It may well be that the model underestimates the wind shear – thus influencing the zonal and meridional representation error differently. showed that the ECMWF model has a smaller mean and variability in wind shear compared to radiosonde, with different factors for zonal and meridional wind shear in the free troposphere (2.5 and 3 respectively). Also shown in this figure is the residual correlation when converting the wind vector to a radial wind component using the distribution of azimuth angles observed in the Mode-S EHS/radar data set. The resulting residual correlation error lies close to the mean value of the azimuthal residual errors.

Now that we have estimated the residual error we can use the triple-collocation method to determine the observation errors. Figure  shows the observation error of Mode-S EHS and sodar for different azimuth angles. From the original wind vector a radial component is constructed using a prescribed azimuth angle. Each error estimate is determined using 10 subsets of the data set, and consequently an uncertainty of the estimated error can be determined. This uncertainty is denoted by the shaded areas in Fig. .

Both wind observation errors have a clear azimuth dependence and exhibit again a bi-periodic behaviour; the errors of Mode-S EHS are between 1.2 and 1.5 m s-1, while sodar errors are within approximately 0.9 to 1.3 m s-1. The amplitude of Mode-S EHS radial wind errors is smaller than the sodar amplitude. The size and signature of the amplitude of the sodar might be related to the observation method exploiting (only) three beams. The minimum of the sodar radial wind error is at 0 and 180∘, corresponding to the meridional component, while the maximum error is at 90 and 270∘, corresponding to the zonal component. For Mode-S EHS the errors in zonal and meridional component are more or less equal; the maxima and minima are attained at approximately 45 and 225∘, and 135 and 315∘ respectively.

The trend and bias with respect to the first data set are simultaneously estimated by the triple-collocation algorithm. Obviously, the bias has no effect on the estimated observation errors; however, it can be informative because it gives information on the mean difference with the truth. The trend displays the scaling of the data set with the truth. Figure  shows the trend and bias for the radial wind component of the Mode-S EHS/sodar/NWP data set. The trend of Mode-S EHS is close to 1, indicating that radial wind observations are of the same order. The trend of NWP lies clearly below 1 (0.96±0.02); the radial NWP wind is overestimated when compared to sodar (and Mode-S EHS) radial winds. Note that the trend has a bi-periodic behaviour. The bias shows a different signal, both periodic in azimuth with the same phase. Since the signal is equal for both NWP as Mode-S EHS, the origin must lie in the sodar measurement and might be related to the method of observation using three beams or in an azimuth offset. This needs further research.

Next we discuss the consistency between the estimated Mode-S EHS error from both data sets by inspection of the zonal and meridional estimates. The radial wind is equal to the zonal component of the wind for an azimuth angle of 90 and 270∘. Similarly, the radial wind for an azimuth of 0 or 180∘ equals the meridional component. By selecting in the Mode-S EHS/radar/NWP data set azimuth angles between 75 and 105∘ (and 255 to 285∘) we can make an estimate of the zonal component of the wind error of Mode-S EHS at levels higher than the sodar. The result is shown in Table .

These statistics show consistency between both triple-collocation data sets when taking into account the observed uncertainty of the estimates. Due to the small numbers of Mode-S EHS observations satisfying the azimuth angle conditions, the uncertainty for the estimates based on the Mode-S EHS/radar/NWP data set is larger than for the other triple-collocation data set (for the latter all observations can be used obviously). The uncertainty range of the Mode-S EHS/radar/NWP estimates overlaps the uncertainty of the Mode-S EHS/sodar/NWP. The Mode-S EHS error is approximately 1.4 to 1.5 m s-1 near the surface. Note that the zonal Mode-S EHS observations are found at a higher altitude, which, as we will see later, influences the magnitude of the error slightly.

The meridional component statistics, shown also in Table , are obtained by selecting angles smaller than 15∘ and larger than 345∘ azimuth, and between 165 and 195∘ azimuth. The estimate of the observation error is larger than the one for the zonal component, but the difference is of the order of 0.1 m s-1. Again the uncertainty of the Mode-S EHS/radar/NWP observation error is larger than the uncertainty from the other data set, and again there is a clear overlap of the uncertainty intervals.

We now focus on the radial wind component of the Mode-S EHS observation error. The result of the triple collocation for all radial wind component is shown in Fig. . The error bar denotes again the spread of the triple-collocation standard deviation estimates by dividing the data sets into 10 subsets and estimating the observation error for each subset.

The lowest data point originates from the triple collocation of Mode-S EHS/sodar/NWP, where we created radial wind observation from the wind vectors using the azimuth distribution as observed by the other data set. The other data points in Fig.  are based on triple collocation of radial wind observations from the Mode-S EHS/radar/NWP data set. Apart from levels higher than 600 hPa the estimated error is slightly below 1.5 m s-1. The higher levels deviate from 1.5 m s-1, which is related to the distribution of the azimuth angles used to estimate the Mode-S EHS observation error, as can be seen in Fig. , where the azimuth distribution is shown in bins of 30∘ for the different altitude bins. It is clear from this figure that the two highest estimates (higher than 600 hPa in Fig. , the triangles in Fig. ) have a clear different signature in azimuth distribution than the lower four estimates. Because the azimuth distributions differ substantially, the reconstructed radial wind data set of the highest levels is not consistent with that of the lowest levels. The distribution of the azimuth angles will influence the magnitude of the observation error estimate (see Fig. ). In order to have a better comparison between error estimates at different levels, a resampling of the data sets is performed, such that the azimuth distribution for each level matches the azimuth distribution of the whole data set. We used the distribution in 30∘ bins as a reference.

Figure  shows the resampled distributions. When two or more observed azimuth values are present in the original bin, the resampled bin is filled by randomly sampling (with replacement) from this bin until the number found is exactly equal to that of the reference bin. Note that when a bin contains none or only one azimuth value this bin is skipped. When this occurs, it will influence the number data points in the other bins, because the total number of data points of all bins is kept constant. The consequence is that the azimuthal distributions will differ from the reference distribution, as can be seen in Fig.  for the azimuth distribution of the lowest level (960 hPa, open square) and the highest level (538 hPa, solid triangle). All other new azimuth distributions match the reference very well (Fig. ).

The resulting estimates of Mode-S EHS observation error after resampling are shown in Fig. . Again, each data set is subdivided into 10 subsets, which are subsequently used to estimate the uncertainty of the observation error estimate using triple collocation. In general the estimates are slightly smaller than without resampling, while the estimate uncertainty is slightly larger, especially for the highest level. The overall increase in uncertainty is related to oversampling of the data set. For example, the increase of uncertainty observed for the top level is due to the relatively small number of data points in the altitude bin for this level. It appears that the observation error decreases with altitude above 800 hPa. The numbers used in the triple-collocation increase also slightly because of the resampling, which selects multiple data points from an undersampled (original) bin.

Finally, we present the trend and bias for the resampled radial wind data sets (see Table ). Again the trend of Mode-S EHS is around 1 with a small uncertainty obtained by splitting the data set into 10 subsets. The bias is between -0.1 and 0.4 m s-1 with an uncertainty of around 0.2 m s-1. The trend in NWP is smaller than 1, apart from the highest level. The bias is in general positive between 0 and 0.4 m s-1 with an uncertainty of around 0.2 m s-1 (again except the highest level). These numbers are in agreement with the previous trend and bias estimates presented in Fig. .