1. Introduction

An indication of the stability of a column of air is essential to be able to forecast a strong convection event. The combination of humidity and temperature profile information determines the (static) stability of an air column. At this time, radiosonde observations provide the only operational upper-air collocated measurements of temperature, humidity, and wind. Temperature sensors on commercial aircrafts provide temperature soundings but (currently) no humidity observations. These observations are distributed to the users using the Aircraft Meteorological Data Relay (AMDAR) system. Currently, humidity measurements from commercial aircraft are being carried out now in the United States using Tropospheric Airborne Meteorological Data Reporting (TAMDAR), although these observations are not operationally available at Koninklijk Nederlands Meteorologisch Instituut (KNMI; the Royal Netherlands Meteorological Institute). Moreover, information from these observations is restricted to flight routes and airports. The latter are operating at specific times during the day, which is a weakness of the system. Satellite temperature sensors, such as the Advanced Tropospheric Vertical Sounder [(ATOVS) on board the National Oceanic and Atmospheric Administration (NOAA) polar satellites] measure outgoing radiation at a number of wavelengths. These radiation parameters are linked to upper-tropospheric temperature (and humidity), but these observations need a model to be able to convert them into a temperature profile (Li et al. 2000). Another disadvantage is that satellite-derived temperatures are not frequent in time, in the case of polar-orbiting satellites, or have a low spatial resolution, in the case of a geostationary satellite.

Radiosonde profiles are obtained with a frequency of 2 or 4 times per day at specific times. A forecaster has to combine satellite information with numerical weather prediction model output at times for which radiosonde observations are not available. Although radiosondes are regarded as the best way of measuring profile information, they should not be treated as truth profile information. For example, a radiosonde balloon rises from the surface to 15–30 km in approximately 1–2 h. During its ascent, the wind speed and direction are deduced by its horizontal movement. A radiosonde is thus rarely a vertical profile. Moreover, there are known problems with certain types of radiosondes resulting from, for example, ice contamination of the humidity sensors or day/night inconsistencies (Leiterer et al. 1997; Lorenc et al. 1996). Radiosonde observations are also subject to biases and nonsystematic errors caused by sensor malfunctions. The quality of a radiosonde observation is also dependent on the age of the sensor and proper calibration before launch.

The global positioning system (GPS) signals transmitted by GPS satellites and received at the earth surface have traversed the atmosphere. The atmosphere delays and bends the signal because of differences in density and humidity. Each signal contains information about the temperature and humidity along the path between the GPS satellite and receiver. By collecting the signal from a large number of satellite–receiver pairs for a number of receivers, the atmospheric delay can be estimated very accurately, simultaneously with the position of the receivers and other parameters (Rothacher and Merwart 1996; Webb and Zumberge 1993). From this atmospheric delay an estimate of the total column water vapor can be estimated when surface pressure and temperature are available (Bevis et al. 1992). The spatial coverage is (up to now) restricted to land, apart from some sparse platforms at sea. However, the great advantage of GPS observations is that a very high temporal resolution can be achieved (up to minutes) with reasonable accuracy. Studies have shown that GPS observations can be used for nowcasting purposes (Mazany et al. 2002; de Haan et al. 2004; Elgered et al. 2004). Because GPS delays are determined using differences in observed and expected arrival time of a signal, it is intrinsically calibrated. This makes GPS-derived atmospheric information also a very good observation for climate purposes (Gradinarsky et al. 2002; Vedel and Stendel 2005, manuscript submitted to Climatic Change).

In this article a connection is made between GPS observations and the possibility of (strong) convection. Strong convective systems may suddenly develop. An external trigger may change a potential instable column of air into a completely unstable one. Entrainment, friction, and orographic effects can change the stability. The time scales at which this can occur are on the order of hours to several minutes. Because GPS delay signals contain moisture information and can be obtained with a high temporal resolution, these observations may reveal potential unstable air masses. During convection, air masses are constantly transported upward and downward, and the frequency at which this occurs may also be apparent in the GPS signals.

2. Data

GPS is a constellation of 24 satellites (which transmit radio frequency signals) and (ground based or spaceborne) receivers. The civil use of GPS has increased rapidly in the last 10 years. The receiver constellation used in this study is set up for geodetic and surveying purposes, which require very accurate positional information (on the order of millimeters).

A GPS satellite transmits a radio frequency signal with a time tag and code specific for the transmitting satellite. By comparing the clock of the receiver and the time stamp of the signal, the excess path can be estimated when the positions of the satellite and the receiver are known. To obtain millimeter accuracy, however, parameters such as satellite and receiver clock errors and atmospheric correction (or delay) need to be estimated. By collecting all observations in a time window from a fixed network of receivers, the estimates of these parameters can be obtained by a least squares fit. The atmospheric delay is usually determined in the zenith direction and is called zenith total delay (ZTD). The ZTD is related to refractivity N as follows:

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where h_A is the height of the antenna and z is height. The ZTD is normally given in meters. The refractivity N at a certain height is a function of temperature T, pressure p, and water vapor pressure e, that is,

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where k₁ = 77.6 K hPa⁻¹, k₂ = 70.4 K hPa⁻¹, and k₃ = 3.739 × 10⁵ K² hPa⁻¹ are constants (Bevis et al. 1992). The ZTD can be split up into a dry or hydrostatic part [p–T term in Eq. (2), called zenith hydrostatic delay (ZHD)] and a wet part [e–T terms in Eq. (2), called zenith wet delay (ZWD)],

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The ZTD is an estimate of the path delay for signals with zenith elevation. GPS satellites are rising or setting continuously for each receiver. When a signal reaches a receiver at a low elevation angle, the path through the atmosphere will be long and may span over 200 km horizontally in the troposphere. At very low elevations the fraction between wet slant delay and wet zenith delay deviates substantially from the fraction between dry slant and dry zenith delay. Therefore, the dry and wet part should be treated separately when mapping a total slant delay to the zenith. The fractions between zenith dry (wet) and slant dry (wet) delay determine the so-called Niell mapping functions. Based on radiosonde observations, Niell (1996) determined a general relation between the fraction of slant and zenith delays with respect to elevation. Signals with an elevation of 50° and higher are more representative for the column of air above the receiver. Although the elevation dependence of the mapping functions is still present, the differences between dry and wet mapping functions are small at these high elevations (Niell 1996). The actual fraction between slant and zenith delay may deviate from the more or less climatological value proposed by Niell (1996). The cause of these deviations is gradients in humidity and pressure distribution (de Haan et al. 2002). Not surprising is that these differences are most pronounced at low elevations.

A network of GPS receivers is used in the current study. Data from the whole network are processed using the Bernese software (Rothacher and Merwart 1996) with final orbits. The GPS data used in this study have a temporal resolution of 30 s. In the processing, the minimum elevation angle is set to 10°. Double differences are used to eliminate clock errors. Using zero differencing (Alber et al. 2000), an estimate of the slant total delay (STD) is obtained. From these zero differences, the nonisotropic residual is estimated by

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where mf_h and mf_w are Niell dry and wet mapping functions, and M(α, β) is the multipath correction at elevation α and azimuth β (de Haan et al. 2002). ZHD_a is an a priori estimate of the dry delay, and ZWD_e is estimated in the processing.

Elosegui and Davis (2003) showed that there are some problems with the method proposed by Alber et al. (2000). The main problem lies in the fact that offsets at low elevations appear in the solutions at other locations of the network because of the zero-differencing technique. The offset is spread over the network, although the magnitude of the spread will be relative to the value of the mapping functions. The magnitude of the resulting offset is expected to decrease when the difference in elevation angle increases. The decrease is relative to the quotient of the value of the mapping functions.

The nonisotropic residuals are mapped onto the zenith by the wet Niell mapping functions, although, as said before, the difference between the dry and wet mapping function at these elevations is small. Time series of these (mapped) residuals will be analyzed for each satellite separately. The errors resulting from the problems addressed by Elosegui and Davis (2003) are small because data are only used with an elevation angle larger than 50°. This minimal elevation angle diminishes the influence of advection of moisture gradients.

Two periods with GPS observations are used in the study presented here: 23 October–7 November 2000 and 1–24 May 2003. The period in 2000 was governed by a large number of cold fronts (see also de Haan et al. 2004) passing the Netherlands. The latter period was marked by a number of thunderstorms. In this latter period, an extensive measurement campaign called the Baltex Bridge Campaign-2 (BBC2) was organized in the Netherlands. Dedicated radiosonde measurements and a large number of other observations were conducted during this period. The main emphasis of this campaign was cloud research. The primary location for this campaign was Cabauw, where the radiosondes were launched and where a GPS receiver was installed. In the first period, October–November of 2000, no dedicated radiosondes were available, and the GPS receiver at Cabauw was not installed at that time. Figure 1 shows the location of Cabauw and part of the GPS receiver network used to estimate the delays, as well as the radiosonde launch sites at De Bilt, Netherlands, and Brussels, Belgium. Table 1 shows the radiosonde sites and receiver locations.

The radiosonde data obtained at Cabauw have a high temporal resolution (2 s), and the launch times were chosen based on the weather forecast. A radiosonde balloon rises with a vertical speed of approximately 5 m s⁻¹, which implies that the vertical profile resolution for the Cabauw radiosondes is approximately 10 m. In De Bilt, Brussels, and Göteborg, Sweden, radiosonde launches are performed every 12 h (at 0000 and 1200 UTC). The distance between the GPS sites and the radiosonde launch sites is small (within 10 km), except for between Cabauw and De Bilt (distance of 31 km) and between Onsala, Sweden, and Göteborg (distance of 83 km). Notice also that, apart from Onsala–Göteborg, height differences between GPS and radiosonde sites are small. Radiosonde observations from De Bilt have a temporal resolution of 10 s (vertical profile resolution of 50–60 m). The radiosonde observations from the other two locations (Brussels and Göteborg) were obtained at so-called significant levels, which reduces the vertical resolution drastically while retaining the main temperature and humidity inversions. On all radiosonde observations, a visual quality check was performed to remove spurious data. Moreover, for the radiosonde observations from Cabauw an extra quality check is performed by comparing the observed temperature and humidity data of the first 200 m of the profile obtained from the measurement tower at Cabauw.

3. Method

In this section, the method to link GPS observations to the stability of an air column is presented. A measure of static (in)stability is convective available potential energy (CAPE). CAPE is an estimate of the maximum available kinetic energy an air parcel can achieve given a certain temperature and humidity profile.

Linear or first-order stability is obtained by analysis of small displacements of an air parcel. Suppose an air parcel is adiabatically displaced by z′ in the vertical direction. Because of mechanical equilibrium, the pressure of the parcel will automatically adjust to the environment. Following Holton (1992) and Salby (1996), Newton's second law for the parcel implies

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where the primes denote the parameter corresponding to the displaced parcel, ρ is density, t is time, and g is gravitational acceleration. Using the hydrostatic and mechanical equilibrium, this yields

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Applying the universal gas law and introducing the potential temperature θ [defined by θ = T(p/p₀)^κ, where p₀ is usually chosen to be 1000 hPa and κ = 0.286], this results in

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Now θ′ is the potential temperature at the displacement z′. The parcel is transported adiabatically and so θ is conserved, implying that

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Combining this with Eq. (7) yields

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where

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where N_BV is known as the Brunt–Väisälä or buoyancy frequency. This frequency is approximately 1.2 × 10⁻² Hz for average tropospheric conditions, which is equal to a period of around 8 min. The stability is determined by the sign of N²_BV: positive values determine a stable environment with oscillation at frequency N_BV; negative values are related to unstable environments.

Another measure of stability is CAPE, which is closely related to buoyancy through

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where z_LFC is the level of free convection and z_LNB is the level of neutral buoyancy (Holton 1992; Salby 1996). Between these levels, the Brunt–Väisälä frequency is negative and CAPE is positive.

To provide insight to the frequency range related to convection, the time scale of free convection is very informative. A measure of the free-convection time scale is defined as

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where z_i is the height of the lowest inversion and w* is the scaling vertical velocity. The time scale t*, which represents the time needed for a thermal to rise in a convective boundary layer is on the order of 5–10 min. Furthermore, because CAPE represents the maximal available potential energy, the maximal vertical velocity a parcel can achieve is

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The value of w_max is overestimated because not all available potential energy will be transformed into kinetic energy. If one assumes that w* is one-half of the value of w_max and that CAPE is approximately 100 m² s⁻², the time scale will be around 12 min. Moreover, for cumulus convection with a characteristic height of 1 km and a mean vertical velocity of 1 m s⁻¹, the time scale is approximately 16 min.

The relation between ZTD and the Brunt–Väisälä frequency is not straightforward. Recall that the ZTD is the integral value of the refractivity N [see Eq. (1)]. Replacing temperature T by potential temperature θ and water vapor pressure e by specific humidity q, the derivation of N with respect to z results in

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where ϵ = 0.621 98 (Sonntag 1994). Note that not all temperatures are replaced by potential temperatures. The term N²_BV/g is on the order of 10⁻⁵, and the other term [(1 + κ)/p](dp/dz) is approximately 10 times that value. This term is always negative because pressure is decreasing with increasing height. The last term on the right-hand side in Eq. (14) is neglected because it is about a factor of 10⁻³ smaller than the other terms. In general, the refractivity decreases with height. When the Brunt–Väisälä frequency is negative (i.e., unstable environment), the derivative of the refractivity will be larger (less negative). When the derivative becomes positive, an inversion in the refractivity profile appears. Such an inversion may cause multipath effects of radio frequency signals (radar, GPS). Note that in hydrostatic equilibrium the pressure term can be written as

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where R_d = 287.05 J kg⁻¹ K⁻¹ is the specific gas constant dry air and δ = 0.6077. For temperatures ranging from 270 to 290 K, this value will be from around 1.18 × 10⁻⁴ to 1.25 × 10⁻⁴ m⁻¹. Values of q are very variable: q lies generally in the range from 0 to 0.006 kg kg⁻¹. The derivative is much smaller, with (absolute) maximum values of around 1 × 10⁻⁵ kg kg⁻¹ m⁻¹.

By rewriting Eq. (14) and using the definition of N_dry = k₁p/T, the following equation is obtained:

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The order of the left-hand side is 10⁻⁵ m⁻¹. The terms on the right-hand side are on the order of 10⁻⁵ m⁻¹, as can be seen from Fig. 2a. In this figure the pressure term and the Brunt–Väisälä frequency of the right-hand side of Eq. (16) are shown together with the left-hand side for a high-resolution radiosonde observation at Cabauw. Figure 2b shows the values of q and dq/dz for this radiosonde launch. Below a height of 5 km the derivative of q varies very rapidly, which results in an equivalent variability of (1/N_dry)(dN/dz). Above 5 km, the influence of dq/dz diminishes because of small values of (and thus small changes in) q. Furthermore, note that the pressure term is gradually decreasing and that the change in (1/N_dry)(dN/dz) is very similar to the Brunt–Väisälä frequency above approximately 6–7 km. Above 12 km, N_BV varies because of the variability in the derivation of the logarithmic of the (dry) temperature.

Thus, the Brunt–Väisälä term cannot be neglected, and the above equations reveal the relation between the refractivity N and the Brunt–Väisälä frequency N_BV. The variability in dq/dz also has a large influence on the derivative of the refractivity, and, because in convective situations dq/dz will rapidly change with time, the derivative of the refractivity will also change rapidly. The ZTD is the double integral of a function that merely depends on the Brunt–Väisälä frequency and the derivative of the specific humidity.

Atmospheric time scales related to convection are around 12 min (frequencies of 5 h⁻¹). The part of the GPS signal with a convective origin will most likely have the same frequency. When GPS signals are influenced by convection and this influence is additional to other fluctuations of the signal, the total power of the signal will increase. Time series used in this study are nonisotropic estimates from satellite–receiver combinations and are treated separately. Only observations with an elevation larger than 50° will be used; the corresponding nonisotropic residual is mapped onto the zenith using the Niell mapping function. A quantitative value related to the signal is defined as the integral of the quadratic absolute value of the Fourier transform, that is,

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where H( f ) is the Fourier transform of the projection to the zenith of the nonisotropic residual signal δ_non-iso(t). The quantity P is related to the power of the signal, with units of meters squared per second.

4. Results

In this section the values of CAPE are compared with the spectral power of a GPS signal. The results are presented in two steps. First, special attention is given to a single day to set the framework and to investigate the time dependence of the low-frequency spectral power. Next, the results are presented in more general terms for the periods of October–November of 2000 and May of 2003.

Power spectrum plots are derived from nonisotropic residual time series observed in a 1-h window at Cabauw. The residual signal is shown in the top panels of Fig. 3 for three different hours, the azimuth–elevation plot is shown in the middle panel, and the corresponding power spectra are shown in the bottom panels. Note that the elevation shown is 50° and higher. The highest frequency that can be observed is the so-called Nyquist frequency, which is 1/(2 × 30) s⁻¹ in this case, which corresponds to a cycle of 60 h⁻¹. In the first and last hour, only one satellite with an elevation higher than 50° is visible for the full 60 min; the other hour has time series from two satellites and their power spectra, indicated by the dashed and solid lines. The value of CAPE, as observed by the radiosonde launched at Cabauw in the same hour, is shown in the bottom panels, for the second and third hour window.

The difference for values of CAPE in spectral density distributions is remarkable. Moreover, two different satellites show almost the same spectral frequency distribution, although the signal itself differs. The power spectrum corresponding to the largest value of CAPE has a maximum of around 200 mm². This maximum lies around 1 h⁻¹. There is a second broader maximum around 2 h⁻¹. The power spectrum related to CAPE values of 14 m² s⁻² show a less extreme maximum at higher frequencies.

A time series of P is shown in Fig. 4. Also shown in this plot are the values of CAPE as observed by radiosonde launches at Cabauw (open circles, launched at irregular intervals) and De Bilt (crossed circles, launched every 12 h at 0000 and 1200 UTC). The P values are obtained every 15 min, through Fourier analysis of nonisotropic residual data observed in a window of 1 h. The highest values of the P are observed during daytime, which can be related to the fact that convection is triggered by solar heating. There are two radiosondes launched right around this first maximum. At 1200 UTC, the radiosonde launch of De Bilt shows a CAPE of approximately 250 m² s⁻². At the same time the P is steadily increasing again and thereafter remains at a local maximum for a few hours. At 1500 UTC the CAPE observed by the radiosonde launched at Cabauw is 363 m² s⁻². At the same time the values P are again large. Right after the launch a peak in P is observed. Notice that at 0000 UTC CAPE from De Bilt is low while the value of P at Cabauw is relatively high. This difference can be due to the distance between De Bilt and Cabauw, which is approximately 30 km.

The trend in the P signal is most likely due to changes in vertical distribution of the humidity in the column of air above the GPS receiver. The frequencies are in atmospheric terms high (although not extreme), but in terms of the GPS signal they are not high at all. High-frequency response might contain information on the turbulence of the atmosphere (Cornman et al. 2004; Kleijer et al. 2004). The low frequencies seem to have a convective origin, which can be explained by the fact that the ZTD is a function of the N_BV and dq/dz [see Eq. (16)].

Convection is often driven by solar heating and thus has a diurnal signal. Whenever P is related to convection, a diurnal signal should be present. Figure 5 shows the diurnal cycle for P over the 24-day period for Cabauw in May of 2003. The value of P is determined every 15 min in May of 2003; hourly bins of GPS ZTD data are used. The mean and standard deviation are determined using these P values. The mean and standard deviation with respect to the time of day are shown in Fig. 5; statistics at times at which less than 10 values of P were observed are omitted. From 0500 UTC onward, the value of P steadily increases, and it reaches its maximum around 1300 UTC, which is close to the time of day at which in the Netherlands the convective activity is generally the largest.

In Fig. 6 the value P is plotted versus the CAPE as observed by the radiosonde launches for the periods of October–November of 2000 and May of 2003. The value of P is determined in a 1-h window starting at the radiosonde launch time. The relation between the two parameters is evident: observations of large values of CAPE are coincident with large values of P. The relation can be traced back to the definition of CAPE, which is related to a (negative) Brunt–Väisälä frequency. A negative N_BV yields a smaller decrease in height for the refractivity N and will influence the observed ZTD in a similar way. Data points with high CAPE and low P are not observed. Note that CAPE represents the upper bound of the available potential energy. For small values of CAPE there seems to be a larger spread in P. This spread can be caused by fluctuations in moisture [last term of Eq. (16)] while the profile itself is still more or less stable.

The number of date points is 66, and the correlation between CAPE and P is approximately 0.57. Table 2 shows the correlations separated by GPS receiver and radiosonde location combination. Except for Brussel and Cabauw, all pairs have a correlation between 0.6 and 0.7. The number of comparisons for Brussel is small (10), and there is an outlier with a low CAPE and high P. Removing this outlier results in a correlation of 0.59. Furthermore, the maximum CAPE used in the comparisons of Brussels is 450 m² s⁻², whereas all other comparisons show some CAPE values larger than 500 m² s⁻². The Cabauw–Cabauw correlation is close to 0.6.

These correlations are not very high; keeping in mind that CAPE is obtained from a profile at a certain time and P is determined using Fourier analysis of a time series of 1 h, however, the magnitude of the correlations is remarkable: it indicates that there is a relation between the two parameters. The fact that that there seems to be an inverse relationship between correlation and distance between radiosonde and GPS is striking and cannot be explained with the current dataset.

The number of observation pairs is smaller than expected. Recall that the radiosonde launch times were, apart from Cabauw, at 0000 and 1200 UTC. The number of satellites with elevations above 50° is small between 1100 and 1200 UTC, causing the gaps in the diurnal plots (Fig. 5). Because the GPS satellites orbits are chosen such that the observed elevation angles are repeated every day, only more satellites can fill this gap. Another solution would be to include also lower elevation angles. However, then the simple mapping of the nonisotropic residual to the zenith may fail because of differences between the dry and wet Niell mapping functions.

5. Conclusions and recommendations

In this article, a method for detection of convection from GPS delay signals is presented. A network of GPS receivers was used to determine the GPS atmospheric delay very accurately. Two periods, October–November of 2000 and May of 2003, for which GPS data with a temporal resolution of 30 s were available were investigated.

The GPS signals were obtained from slant delay estimates with elevations higher than 50° and were projected onto the zenith using the wet Niell mapping function. The Fourier spectrum of frequencies is related to convective available potential energy retrieved from radiosonde observations. The power of the signal has a diurnal signal consistent with the diurnal signal of convective activity in the Netherlands and is shown to have a correlation of 0.57 with values of CAPE. The GPS constellation of satellites and a GPS network have proven to be very reliable. When using the algorithm described in this article, however, there are hours during the day in which the number of observations with an elevation higher than 50° is small, and thus an accurate frequency spectrum cannot be obtained.

In this study the GPS frequency spectrum is only compared with CAPE values from radiosonde observations. By investigating other observations related to convection, such as lidar or Doppler radar, more knowledge can be gathered on the value of GPS frequency spectrum analysis.