Theor. Appl. Climatol. (2008) 94: 225–239
DOI 10.1007/s00704-007-0356-7
Printed in The Netherlands
1
2
Department of Statistics and Actuarial Financial Mathematics, University of the Aegean, Karlovasi, Samos, Greece
School of Environmental Sciences, University of East Anglia, Norwich, U.K.
A comparison of temperature inversion statistics at a coastal
and a non-coastal location influenced by the same synoptic regime
A. E. Milionis1 , T. D. Davies2
With 13 Figures
Received 10 January 2007; Accepted 29 August 2007; Published online 11 January 2008
# Springer-Verlag 2008
Summary
The primary aim of this work is to examine to what extent
the climatology of atmospheric temperature inversions at
one location is site specific, and to what extent it reflects a
wider area for which the same synoptic conditions can be
assumed. To this end radiosonde data from a coastal and a
non-coastal location in eastern England separated by 210 km
and influenced by the same synoptic conditions are used.
Analysis of these data shows that there is a pronounced
difference between the inversion climatologies at the two
sites. The vertical distribution of base-heights of inversions
has a very distinct maximum at a height of about 200 m at
the location proximate to the coast. This maximum is not
present at the inland location, and the difference is due to
both sea-breezes and advection from the sea due to synopticscale wind field. Examining the vertical distributions of baseheights of inversions at the two locations under conditions
that either maximize or minimize the effect of sea-breeze it
is found that the differences in the two distributions are to a
certain extent deterministic (therefore predictable) rather
than random, as the dominant mechanisms which are responsible for these differences (diurnal and yearly cycles)
have an obvious regularity. Using standard statistical methods
it is further shown that, apart from this difference, nearly all
other inversion statistics for the two locations are similar
when the atmospheric layer from surface to 700 hPa is taken
into consideration. However, when only the first inversion
Correspondence: A. E. Milionis, Department of Statistics and
Actuarial Science, University of the Aegean, Karlovasi 82300,
Samos, Greece, e-mail: amilionis@aegean.gr, on leave from the
Bank of Greece, Department of Statistics
in each temperature profile is considered, the inversions
activity throughout the year, defined with the aid of an
index, in the two locations is not correlated, indicating that
for the lowest part of the surface-700 hPa region, local
factors overwhelm the synoptic conditions. Thus, these results provide evidence that the inversion climatology at one
location can be generalised over a wider area where the
same synoptic regime can be assumed. Given that, at least
to an extent, any differences in the characteristics of inversions due to local factors can be inferred once the underlying mechanisms are carefully studied, this work has
also important implications for micrometeorological studies
as for instance the local diffusion and transport of air
pollutans.
1. Introduction
The development of the climatology of (statically) stable layers in the lower atmosphere is
of importance since, amongst other reasons, it
is related to the ability of the atmosphere to
inhibit vertical motion. With a few exceptions
(Sivaramakrishnan et al. 1972; Nodzu et al.
2006) it is the isothermal lapse rate that is taken
as the limiting one, and only layers with negative
lapse rates, (the so-called temperature inversion
layers), are considered. That happens not so much
because there is any strong physical reason for
such a separation, but due to the fact that the first
stable layer encountered in a temperature profile
226
A. E. Milionis, T. D. Davies
is most often an inversion layer (Milionis and
Davies, 1992, 1994a). For this reason many studies on the effect of atmospheric conditions on air
pollution relate air pollution episodes with temperature inversions (e.g. Kukkonen et al. 2005;
Janhall et al. 2006; Malek et al. 2006; Kerminen
et al. 2007). Additionally, several of the existing
inversion climatologies are restricted to the surface inversions or the first elevated inversions only
(e.g. Hosler 1961; Tyson et al. 1976; PrestonWhyte et al. 1977).
In our previous research on inversion layers
(Milionis and Davies 1992, 1994a, 1994b, 2002,
2007) we used radiosonde data from the U.K. upper air station at Hemsby (52 390 N, 1 410 E, at
an elevation of 14 m) to: (a) establish the main
characteristics of the climatology of inversion
layers (Milionis and Davies 1992, 1994a, 2002,
2007) and (b) examine their effect on groundlevel air pollution (Milionis and Davies 1994b).
In order to explain some of the results we made
assumptions about the effect of the local topography, in particular the proximity of Hemsby to
the North Sea (the upper air station is located
1.5 km from the coast). These assumptions, although reasonable, need further justification, and
one way to test for that is to compare the inversion
statistics for Hemsby with those from non-coastal
stations. As the influence of local factors are
expected to affect the statistics of inversions mainly in the first kilometre or so above the ground,
this comparison is important particularly due to
the significance that the inversion layers of this
atmospheric region have in local-scale diffusion
and transport of air pollutants.
Although the largest part of the published research work on inversion climatology refers to data
from a single location (e.g. Milionis and Davies
1992, 2007; Prezerakos 1998; Abdul-Wahab
2004; Kassomenos and Koletsis 2005), there are
several studies on the comparison of inversion
climatologies for different locations, in particular
coastal and non-coastal locations (Hosler 1961;
Preston-Whyte et al. 1977; Nodzu et al. 2006),
which are worthy to be briefly reviewed. Hosler
(1961), in his study of frequencies of surface and
near surface inversions over the contiguous U.S.A.,
concludes that inversion frequencies at stations
in coastal areas reflect marine influences. Low
level stability, depending on whether the nearby
water surface is colder or warmer that the adja-
cent land may be either inhibited or enhanced
due to advection. As Hosler found, for coastal
areas along the south-eastern States and Gulf of
Mexico, the formation of inversions is inhibited
particularly overnight, while the cold waters of
the North Atlantic Ocean cause higher frequencies of inversions at coastal stations, particularly during the day, due to the cool sea-breezes.
Preston-Whyte et al. (1977), studying the climatology of the lowest (i.e. first) elevated inversions over Southern Africa, found that the height
of the first elevated inversion is greater over
the plateau than over coastal areas. They argued
that the height at which elevated inversions occur depends not only on the prevailing synoptic situation, but also on the degree of surface
heating and upward convective and turbulent
mixing. Nodzu et al. (2006) using radiosonde
data for a period of 29 years from 14 upper air
costal and non-coastal stations in the Indochina
Peninsula investigate the interaction between temperature inversions (more precisely layers with
=z>10 K km1 , where represents the
potential temperature and z the height) and seasonal changes in convective activity during the
dry season to rainy season transition. They conclude that the 14 upper air stations can be classified according to three types of vertical thermal
stability, by examining the temporal variations of
the distribution of the base-height of inversions.
As a general rule, however, in all the above mentioned studies differences in the characteristics
of inversions for different locations are attributed to the combined effect of topography and
the synoptic conditions. Therefore, in that way
it is in general not possible to isolate the true
effect exclusively due to topography. The only
work in which the focus was exactly on the effect of topography is the one of Riordan et al.
(1986), who using data from two meteorological
towers, compared the strength and frequency of
inversions at two dissimilar sites, one of which
was on the shore of Lake Robinson in South
Carolina, U.S.A., and the other at a top of a hill
about 175 km away. Despite the differences in
the local conditions, they found many similarities between the inversion climatologies at both
sites; for example, high correlation in the dayto-day values of the strength of the inversions
in the predawn hours, which indicated an overall
synoptic control.
A comparison of temperature inversion statistics
In this work a comprehensive study on the
comparison of inversion statistics derived using
data from two different locations will be undertaken including all inversions up to 700 hPa at two
U.K. upper air stations (Hemsby and Crawley),
aiming to examine the effect of local factors ceteris paribus. To this end the two locations have
been chosen so as to assume that they are influenced by the same synoptic regime. In that way it
will be made possible to test previous assump-
227
tions about the topographic effects on the inversion Climatology derived from the Hemsby
record. Further, having identified possible influences caused by local topography, it is important
to examine to what extent differences in the inversion characteristics caused by such influences
could be potentially predictable. This is of much
importance in order to assess the extent to which
the conclusions from climatological analysis of
inversions at a site can be applied more widely.
Fig. 1. Location of Hemsby and
Crawley
228
A. E. Milionis, T. D. Davies
2. The data
Temperature, humidity, and wind data were used
from the upper air stations of Hemsby and
Crawley. Hemsby, as shown in Fig. 1, is situated
1.5 km from the coast in eastern England (52
390 N, 1 410 E) at an elevation of 14 m, in an extensive area of low relative relief. The upper air
station of Crawley is located about 210 km SW of
Hemsby, and its distance from the sea is about
32 km.
The raw radiosonde data were provided by the
U.K. Meteorological Office and cover a period of
five years (1976–80). The data used for this study
are based on the so-called significant levels and
have been derived from the raw radiosonde data
for the midday and midnight soundings (for further details on the data, see Milionis and Davies
1992, 1994a, 2002). Adjacent inversions with
different lapse rates were merged to give a deeper inversion. The isobar level of 700 hPa was
chosen as the reasonable upper limit for the study
of the tropospheric inversion climatology in the
two sites, as above that pressure surface both the
frequency of occurrence and the mean potential
temperature difference of inversions decrease very
rapidly (a detailed analysis and evidence is presented in Milionis and Davies 1992).
3. Methodology
In our recent analysis of the Hemsby record we
conclude that the prevailing weather exerts a
strong influence on the statistics on inversion
layers (Milionis and Davies 2007). Therefore, it
is important to control for this influence in order
to identify the effect of local topography. This
can be made possible by considering data from
two (or more) sites where there is justification for
assuming that they are influenced by the same
synoptic regime. Given that this assumption is
justified, then the two (or more) sites should be
chosen so that they have different local geography. The upper air station of Crawley fulfils these
requirements, as it can be assumed that it is largely influenced by the same synoptic regime, and
the influence of the sea is of less importance.
The first step for the study of a possible seabreeze effect on the statistics of inversions is
to examine the vertical distribution of the base
height of inversion layers for the two locations.
If there are differences in these two distributions,
it is important to examine possible variations in
these differences in the vertical distributions of
the base-heights of inversions that may be caused
by factors which either inhibit or enhance the
sea-breezes. If such variations do exist, stronger
evidence for the sea-breeze effect will be documented as further insight on the underling physical mechanisms will be provided. Moreover, it
is of great importance to examine further these
variations and find out, whether or not, they are
of deterministic (hence, predictable) or random
character. Indeed, if the variations in the differences in the vertical distributions of the baseheights of inversions in the two locations are
(at least to an extent) deterministic, then knowledge of the pattern of these variations will make
it possible (again to a certain extent) to infer the
character of the one distribution from the other.
To follow the proposed approach it is necessary to examine the vertical distributions of the
base heights of inversions at the two locations
separately for midday and midnight, summer and
winter. This is because, on average, there is a
positive day-time temperature differential between the land and the sea in summer, so the seabreeze circulation is expected to be more common than during winter. On the other hand, at
midnight, the temperature differential between
the land and the sea often decreases or even reverses its direction. Hence, the difference in the
vertical distribution of base heights of low level
inversions at the two locations should be maximised in summer at midday and be minimised in
winter at midnight. In fact if the sea-breeze is
indeed the dominant factor which is responsible
for any differences in the vertical distribution of
base-heights in the two locations, the two distributions should look very similar in winter at
midnight.
Further, as it is assumed that Hemsby and
Crawley are under the same synoptic regime, it
must be examined if different weather patterns
affect the statistics of inversions at Crawley in
much the same way as at Hemsby. To this end,
it makes sense to examine the statistics of inversions: (a) in each location separately under different weather patterns; (b) at both locations for
each of the "major" weather patterns. As far as the
definition of the major weather patterns is concerned, the well-known Lamb’s weather classification (LWC) which refers to the daily weather
A comparison of temperature inversion statistics
229
Table 1. The weather types for each weather class
Anticyclonic
Anticyclonic
Anticyclonic
Anticyclonic
Anticyclonic
Anticyclonic
Anticyclonic
Anticyclonic
Anticyclonic
Anticyclonic
(A)
West (AW)
East (AE)
South (AS)
North (AN)
North West (ANW)
North East (ANE)
South West (ASW)
South East (ASE)
Cyclonic
Westerly
Cyclonic (C)
Cyclonic West (CW)
Cyclonic East (CE)
Anticyclonic South (CS)
Cyclonic North (CN)
Cyclonic North West (CNW)
Cyclonic North East (CNE)
Cyclonic South West (CSW)
Cyclonic South East (CSE)
Westerly (W)
South Westerly (SW)
over the British Isles (Lamb 1972) can be used.
This classification scheme, although subjective, it
is consistent with the so-called objective classifications schemes (e.g. the so-called Jenkinson objective version of LWC (Jenkinson and Collison
1977), and on some occasions reflects better the
true synoptic conditions (see for example Jones
et al. (1993) for a comparison of the two weather
classification schemes). According to the LWC,
a particular weather type is one of the possible
combinations between eight directional types,
each corresponding to a 45 directional sector
and two non directional types, namely; cyclonic
and anticyclonic. Purely directional or purely anticyclonic or cyclonic types are included in the
classification, so altogether there are 26 weather
types. Days where no particular weather type
could be recognized are termed unclassifiable.
The classification refers to the average weather
over a period of 24 h. The most frequent weather
types are westerly, anticyclonic, and cyclonic and
will be the basis for the three weather classes
that we shall use to examine inversion statistics.
Hybrid directional anticyclonic and directional
cyclonic types will be grouped together with the
pure anticyclonic or cyclonic, respectively, to
form the anticyclonic and cyclonic classes used
in this study. The South-West (SW) type will be
grouped together with the westerly type as they
are associated with very similar air masses (P. M.
Kelly, Climatic Research Unit, University of East
Anglia, personal communication) to form the
Westerly class. Table 1 shows the weather types
which comprise each class.
If the assumption that, apart from the geographic influences, which mainly affect the
base-heights of inversions, the inversion climatology developed at one site can be applied to the
whole geographic region which is under the same
synoptic regime, then the following two hypotheses should be valid:
Hypothesis (a): for the same weather class the
statistics of inversions at the two locations must
not differ;
Hypothesis (b): the three weather classes must
cause the same differentiation in the inversion
statistics at the two locations.
The inversion characteristics which will be
considered in this study are: the strength T (difference in the dry-bulb temperatures between the
top and the base of the inversion), the depth H
(difference between the height of the top and the
height of the base of the inversion), the difference in dew point temperature between top and
base of elevated inversions (Td), the lapse rate,
and the difference in the potential temperature
between the top and the base of the inversion
(), which will be called ‘‘intensity’’ of inversion hereafter. It can be shown that the intensity
of inversions is directly proportional to their ability to inhibit the vertical movement of pollutants
(see Milionis and Davies 1992). At this point it
should be noted that the stability of an air parcel
depends on the difference in the density of this
air parcel as compared to the density of the surrounding air. As virtual temperature is defined as
the temperature that dry air must have to equal
the density of moist air for the same pressure, it
is evident that variations in virtual temperature
can be used equivalently to variations in density.
For this reason it would be better to use differences in the virtual potential temperature,
instead of differences in potential temperature.
However, as virtual temperature is defined as:
v ¼ (1 þ 0.61r), where r is the mixing ratio, the
230
A. E. Milionis, T. D. Davies
calculation of virtual temperatures requires humidity measurements and, as is well known, errors in the humidity measurements are common
and much more serious than errors in temperature measurements in radiosonde data (see WMO
1975, 1983 for further details). Further, in almost
all similar studies where use is made of radiosonde data, it is the potential temperature instead
of the virtual potential temperature which is considered. For these reasons potential temperatures
will be also be used in this work.
Formal testing for the existence of possible
differences in the statistics of inversions described above can be conducted by using standard statistical methods. To test hypothesis (a)
a t-test can be used for the differences in the
mean values of the statistics which correspond
to the characteristics of inversions described
above. To test hypothesis (b) the one way analysis of variance for the differences in the mean
values of the same inversion characteristics for
the three weather classes will be used. It is noted
that the mean values of all the above mentioned
inversion characteristics refer to averages over
the entire sample period and over the entire surface-700 hPa region.
4. Results and discussion
Figure 2 shows the distribution of inversion
layers as a function of base-height, considering
atmospheric layers of 200 m width, for both loca-
Fig. 2. Fraction of inversions per mille as a function of
base-height. All elevated inversions
tions for the general case (i.e. both midday and
midnight inversions and for the whole 1976–80
time period). It is very clear from Fig. 2 that the
maximum of the distribution around 200 m at
Hemsby, does not exist at Crawley. Apart from
this, the two curves have a generally similar character. The most obvious reason for this difference
in the two distributions is the sea-breeze effect
which is present only at Hemsby (although there
may be a small number of occasions where seabreeze penetrates as far inland as Crawley).
According to the methodology outlined above,
further evidence can be provided by examining
the two distributions separately for summer and
winter, midday and midnight. Hence, the distributions of inversions as a function of baseheights for Hemsby and Crawley were examined
for the following cases:
(i)
(ii)
(iii)
(iv)
summer (May–August), midday
summer, midnight
winter (November–February) midday
winter, midnight
The results are shown in Figs. 3–6. Examination of these figures shows that the number
of inversions at Hemsby for case (i) is at a maximum around 200 m (Fig. 3) and this maximum
is much more pronounced than in the general
case (see Fig. 2), a fact which provides further
evidence that the sea-breeze effect is the causal mechanism. Figure 4, which corresponds to
case (ii), shows that this local maximum, although relatively weaker, still exists at midnight.
Fig. 3. Fraction of inversions per mille as a function of
base-height for the period May–August. Midday inversions
A comparison of temperature inversion statistics
Fig. 4. Fraction of inversions per mille as a function of baseheight for the period May–August. Midnight inversions
Fig. 5. Fraction of inversions per mille as a function of
base-height for the period November–February. Midday
inversions
Although a search of the literature indicates that
at midnight, land-breezes have been observed
rarely in the U.K., it is rather unlikely that seabreezes occur at midnight. Hence, the local
maximum around 200 m in Fig. 4 needs further
investigation. Another physical process that may
be responsible for the local maximum around
200 m is advection from the sea, due to easterly
winds driven by synoptic scale processes. Such
processes must also be responsible for the local
maximum at 200 m at Hemsby for case (iii)
(Fig. 5), as the sea-breeze circulation is much
less common during the winter than during the
summer. Finally, for case (iv) (Fig. 6) the two
231
Fig. 6. Fraction of inversions per mille as a function of
base-height for the period November–February. Midnight
inversions
distributions look very similar and the maximum
at 200 m at Hemsby has diminished. This is a
very important finding, as the sea is, on average,
warmer than the land for the conditions of case
(iv) and the sea-breeze effect is, by and large,
eliminated. The remarkable similarity of the
two distributions, once the sea-breeze effect disappears, is a very clear indication of the overall
synoptic control.
It is important to examine further the possible influence of the second physical mechanism
(i.e. advection of cold air from the North Sea
due to synoptic scale processes) which may also
contribute for the maximum of the distribution
around 200 m at Hemsby. This can be made possible by examining the distribution of inversion
layers at Hemsby as a function of both baseheights and wind direction. Both sea-breeze and
advection must be combined with on-shore
winds, so the vertical distribution of the baseheights of inversions at Hemsby were examined
separately for easterly winds (direction (45–
135 ), which are clearly on-shore winds, and
westerly winds (direction 225–315 ), which are
clearly off-shore winds. Again, midday and midnight inversions were considered separately
for both the May–August and the November–
February periods.
The results are shown in Figs. 7–10. As can
be seen, for the period May–August there is
a very pronounced maximum in the distribution
232
A. E. Milionis, T. D. Davies
Fig. 7. Fraction of inversions per mille for Westerly and
Easterly winds as a function of base-height for the period
May–August. Midday inversions, Hemsby
Fig. 8. Fraction of inversions per mille for Westerly and
Easterly winds as a function of base-height for the period
May–August. Midnight inversions, Hemsby
Fig. 9. Fraction of inversions per mille for Westerly and
Easterly winds as a function of base-height for the period
November–February. Midday inversions, Hemsby
Fig. 10. Fraction of inversions per mille for Westerly and
Easterly winds as a function of base-height for the period
November–February. Midnight inversions, Hemsby
of the base-heights at 200 m for both midday and
midnight with easterly winds at Hemsby. This
is in sharp contrast with the corresponding distributions for westerly winds, where there are
very low frequencies at that height (Figs. 7 and
8). However, the maximum for easterly winds
at midnight is clearly of lesser magnitude than
the one at midday. For the period November–
February the very distinct maximum at about
200 m for easterly winds, which was present during the period May–August, has now become a
local maximum of much smaller relative magnitude at midday (Fig. 9), and has almost completely disappeared at midnight (Fig. 10). On
the other hand, for westerly winds, as during
the period May–August, the number of inversions with base-height around 200 m is relatively
low. Consequently, the character of the Figs. 7–
10 confirmed the conclusions drawn from the
study of Figs. 2–6.
All the previous Figures have shown that, apart
from the sharp difference in the distributions of
base-heights of inversions around 200 m in
Hemsby and Crawley, which was attributed to
the sea-breeze effect and advection owing to synoptic scale processes, the two distributions are
very similar. Moreover, the way that this difference varies as a function of time is, to a certain
extent, deterministic, hence, predictable. This is
an encouraging first indication that the inversion
climatology which is developed at one location
can be assumed for a wider geographic region,
once the influence of local geography has been
isolated. Further, as the prevailing weather is an
A comparison of temperature inversion statistics
233
Table 2. Analysis of variance for the characteristics of elevated inversions (Hemsby)
Weather class
A
C
W–SW
F-stat.
Inversion characteristics
Intensity (K)
Depth (m)
Mean
s.d.
Mean
5.00
3.60
4.41
49.76
2.8
2.1
2.4
260.5
166
209.5
138
243.8
137
18.36
s.d.
Strength ( C)
Td ( C)
Lapse rate ( C=m)
Mean
s.d.
Mean
s.d.
Mean
s.d.
2.21
1.36
1.81
36.41
2.0
1.3
1.9
3.51
1.83
2.8
20.75
5.1
3.8
4.5
1.28
1.12
1.13
2.48
1.6
1.8
1.6
Table 3. Analysis of variance for the characteristics of elevated inversions (Crawly)
Weather class
Inversion characteristics
Intensity (K)
A
C
W–SW
F-stat.
Depth (m)
Strength ( C)
Td ( C)
Lapse rate ( C=m)
Mean
s.d.
Mean
s.d.
Mean
s.d.
Mean
s.d.
Mean
s.d.
4.81
3.69
4.38
34.20
2.70
2.04
2.40
252.1
217.9
234.2
10.02
151
139
136
2.11
1.34
1.87
29.03
1.9
1.4
1.9
4.01
2.61
3.93
12.84
5.4
4.4
5.6
1.28
1.07
1.28
1.80
2.1
2.2
2.0
important factor that affects the statistics of
inversions (Milionis and Davies 2007), it is important to examine whether or not the major
weather classes defined previously affect the statistics of inversions in the two locations in the
same way. To this end, a one way analysis of
variance will be performed to examine the differences in the mean values of the various statistics of inversions for the three weather classes.
Remarks about the assumptions for the application of the method, as well as further details are
given in Milionis and Davies (2007). In this
work, together with the mean values and the corresponding standard deviations for the inversion
statistics for the three weather classes, the values
of the F-statistic will be presented. An F-value
with one asterisk (‘ ’) will indicate that the result
from the F-test is significant at 10% level, two
asterisks (‘ ’) will indicate that the result is
significant at 5% level, and no asterisk will indicate that the quoted F-value is not significant at
10% level.
Table 2 shows the results of the one way analysis of variance for the statistics of elevated inversions for Hemsby, while Table 3 shows the
corresponding results for Crawley. From Tables 2
and 3 it is apparent that the differences in the
mean values of the intensity of elevated inversions for the three weather classes for both
Hemsby and Crawley are statistically significant,
with the anticyclonic class having the highest
value of mean intensity and the cyclonic class
the lowest. The same conclusion is drawn for
the depth as well as the strength of elevated inversions for both locations. For the lapse rate the
differences are not significant at the 5% level for
both locations. This is reasonable as, for both
locations, both the strength and the depth are
progressively lower for the anticyclonic, westerly, and cyclonic classes, so their ratio does not
change significantly in the three weather classes.
For the differences in the mean values of Td,
although for the three weather classes the differences are significant, a more careful examination
reveals that this result for Crawley is only due to
the fact that for the cyclonic class the value of
Td is much lower than for the other two weather classes. Indeed, a significant value of the Fstatistic only confirms the existence of significant
differences among the means and it does not follow necessarily that every subset of two means
shows a significant difference. The latter may
be investigated separately by means of the socalled multiple comparison tests. From the several existing tests of such kind (see for example
Hsu 1996; Montgomery 2000) the so-called
Fisher’s Least Significant Distance (LSD) test
(Fisher 1966) will be used in the present case.
234
A. E. Milionis, T. D. Davies
For the application of this test all differences of
pairs of class-means are compared with LSD.
The latter is defined as:
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
1
þ
LSD ¼ ta=2 Sw
N1 N2
where, ta=2 is the critical value of the t-distribution at significance level a,
Sw is the within samples estimate of the population variance,
N1 , N2 are the samples sizes of the two classes.
When the difference between the two means
is found greater than LSD then the null hypothesis (i.e. that the two means do not differ significantly is rejected at 100(1 a)% confidence
level. The application of this test to the differences in the mead values of Td for Crawley
and the 5% level of significance gives:
(a) for the pair C, A: difference in means ¼ 1.95,
LSD ¼ 0.52;
(b) for the pair C, W–SW: difference in means ¼
1.32, LSD ¼ 0.59; and
(c) for the difference A, W–SW: difference in
means ¼ 0.09, LSD ¼ 0.55
Hence, the difference in the mean values of
Td for the anticylonic and W–SW classes are
not statistically significant while the difference in
the mean values of each of these classes with the
cyclonic class are both significant, as both are
greater than the corresponding LSD value, conforming our previous assertion about the source
of the significant result for Td in Crawley. This
is the only difference in the two locations in
terms of the influence of the prevailing weather.
We note, however, that it is possible that these
differences in the Td values for the two locations
do not reflect physical reality (as, for example, the
possible influence of the North Sea at the Hemsby
data), but are simply a result of errors in the measurement of humidity. As mentioned in the previous section, the most serious errors in radiosonde
data occur in the measurement of humidity.
It is also important to examine whether or
not, for each weather class, the differences in
the mean values of the statistics of elevated inversions are statistically significant. The results
for this comparison are shown in Table 4. This
table quotes the values of the t-statistic for the
means of each inversion statistic and for each
Table 4. Results of the t-test for the differences in the mean
values of the inversion characteristics in Hemsby and
Crawley for the three main weather classes (mean values
and standard deviations are shown in Tables 2 and 3)
Weather Inversion characteristics
class
Intensity Depth Strength Td
A
C
W–SW
1.46
0.79
0.23
1.07
1.09
1.10
0.27
1.27 0.57
2.02 3.44 4.04 Lapse
rate
1.60
0.45
1.80 weather class. Differences significant at the
10% level are denoted with one asterisk (‘ ’).
Differences significant at the 5% level are denoted by two asterisks (‘ ’). The results show
that for the anticyclonic class the differences in
the mean values of the inversion statistics are not
significant for all characteristics except the difference in the dew point temperature, where the
mean value is lower at Crawley and the difference is statistically significant at the 5% level.
Exactly the same results are observed for the
other two weather classes. As noted previously,
the lower Td values in Crawley may be due to
measurement error in humidity.
The above analysis shows that the statistics
which correspond to the main characteristics of
elevated inversions (up to the 700 hPa pressure
surface) in Hemsby and Crawley are not only
differentiated in much the same way by the prevailing weather conditions, but also under the
same prevailing weather the mean values of all
the statistics of elevated inversions (except for
Td) at the two locations do not differ. We
should note at this point that all the mean values
used in the previous statistical analysis of the
characteristics of inversions have been calculated
averaging over the entire period that the data
cover. Therefore, any differences that might occur
at a particular period of the year are smoothed
out. The level of analysis based on the mean
values, calculated as described above, suffices
as a first step for a climatological study. However,
it is also necessary to examine what happens for
shorter periods (e.g. months). Of particular interest is to examine the degree of co-movement
from period to period, i.e. to examine the extent
to which the annual variations of the activity of
inversions in the two locations follow each other.
To proceed further, instead of examining each
A comparison of temperature inversion statistics
inversion statistic separately, the intensity of
inversions can be combined with their frequency
to create the so-called ‘‘activity’’ of inversions.
An index which expresses the activity of inversions can be defined in the following way
(Milionis and Davies 1994a, 2007): If N is the
total number of available vertical temperature
profiles (with or without inversions), n the corresponding number of profiles with one inversion
at least, and K the corresponding total number of
inversions, then the activity index AI is defined
as the product of (mean intensity of inversions)
times (average number of inversions per profile
with one occurrence at least) times (proportion of
temperature profiles with occurrence of one inversion at least), i.e.:
PK
PK
#i
i¼1 #i K n
¼ i¼1
AI ¼
n N
K
N
235
Fig. 12. Comparison of the annual cycle of the activity of
surface inversions
AI has two advantages: (1) it is computationally
simple; (2) it is independent of the time period
to which it is referring, a desirable property for
a quantity which is a function of ‘‘intensity’’.
Using this index the annual cycle of the activity
of inversions – an important element in an inversion climatology – will be compared for the two
locations.
Figures 11–13 show the annual cycle of the
AI-index for elevated, surface and all (elevated þ surface) inversions respectively, for both
Hemsby and Crawley. For the surface inversions
Fig. 13. Comparison of the annual cycle of the activity of
all (surface þ elevated) inversions
Fig. 11. Comparison of the annual cycle of the activity of
elevated inversions
only midnight soundings were taken into consideration. From these figures it is clear that,
although there are some differences in the activity of inversions in individual months, overall
there is quite a good agreement in the annual
variations of the AI-index. This is confirmed by
the values of the correlation coefficients between
the annual cycles of Hemsby and Crawley, which
are shown in Table 5. These coefficients express
the linear co-movement of the activity of inversion for the two locations. All three correlations
236
A. E. Milionis, T. D. Davies
Table 5. Values of the correlation coefficients between the
annual cycles of the activity of inversions at Hemsby and
Crawley
All inversions
Surface inversions
Elevated inversions
Correlation coefficient
t-statistic
0.655
0.804
0.802
4.64
5.70
5.68
t-Values have
beenffi calculated using the formula (Spiegel
pffiffiffiffiffiffiffiffiffi
r ðN2Þ
p
ffiffiffiffiffiffiffiffiffiffi
1988): t ¼
where r the correlation coefficient, and
ð1r2 Þ
the N the sample size
The critical t-value for N 2 ¼ 10 degrees of freedom and
5% significance level is 2.23
are statistically significant at 5% level. For both
surface and elevated inversions the annual cycle
of one station can explain about two thirds of the
annual cycle of the other station. However, combining surface and elevated inversions together,
the coefficient of determination is reduced to
40%. It is also of interest to note that the values of the average monthly activity of inversions
for Hemsby and Crawley are very close (4.86
and 4.95 K, respectively). We must also remark
that, given the strong influence of the prevailing
weather on the activity of inversions and the fact
that for the study period the proportions of the
various weather types are not representative of
the average climatic conditions (Jones and Kelly
1982), the annual cycle discussed above should
not be taken as representative of a longer period.
Our study of the annual cycles was undertaken
only for the examination of the co-movement of
the activity of inversions in the two locations.
The above analysis shows that the annual variation of the activity of temperature inversions
throughout the surface-700 hPa layer is, by and
large, independent of location, for those sites that
are influenced by the same synoptic regime. At
the same time, however, there is a substantial part
in the annual variation of the activity of inversions in one location that cannot be explained
by the corresponding variation in the other location, particularly when both elevated and surface
inversions are considered together. We reckon
that this unexplained part should be attributed
primarily to the activity of inversions within
the atmospheric boundary layer (ABL) which
roughly corresponds to the atmospheric region
surface-850 hPa. This region accommodates approximately 62% of the total number of in-
versions and 68% of the inversion activity as
compared to the whole surface-700 hPa region
for Hemsby. The corresponding figures for
Crawley are nearly identical (63% and 67% respectively). That means that for the region surface-850 hPa on the one hand the difference in the
two distributions of base-heights of inversions
will be accentuated further (as now the total
number of inversions has been reduced), but on
the other hand the conclusions from the statistical analysis based on the mean values of the inversion characteristics, by and large, will be the
same as for the whole surface-700 hPa region.
Further, the analysis regarding the annual cycle
of the activity of inversions will be repeated this
time using only the first inversion found in a
temperature profile, which is the most important
for the local-scale diffusion of air pollutants
(see for instance Zannetti 1990) and, as a rule,
is found up to the 850 hPa pressure surface. When
only the first inversion of the temperature profile
is taken into account for the calculation of the
A-index the correlation coefficient for the monthly activity of inversions in the two locations
is reduced to 0.40 and for 10 degrees of freedom
corresponds to a t-statistic of 1.38 (see legend of
Table 5 for the details of the calculation of the
t-statistic), which is clearly not significant at
the 10% level. Therefore, although on average
the activity based on the first inversion of the
temperature profile at Hemsby and Crawley does
not differ, the way this activity is distributed during the year is not the same, indicating that the
effect of local factors overwhelms that of synoptic conditions. This is reasonable as the main
mechanism (sea-breeze) responsible for the differences in the annual variation of the activity of
inversions associated with the ABL in the two
locations is directly related to the annual cycle.
However, as the mechanism to which these differences in the annual variation of the activity
of inversions within the ABL are attributed has
an apparent regularity, further investigation (not
undertaken in this work) may reveal that these
differences, to an extent, may be potentially predictable, as was the case with the differences in
the base-heights of inversions.
Before closing this section some clarifications
on the use of the term ‘‘inversion climatology’’
are noteworthy. It is well known that the mean
value of a sample, as a sample estimator of the
A comparison of temperature inversion statistics
true population value, is not only an unbiased
estimator (i.e. its expected value coincides with
the true value in the population) but also it is the
best unbiased estimator (i.e. it has the minimum
variance in the class of unbiased estimators).
Further, the standard error of the sampling distribution of the mean value is given by the ratio of
the standard deviation over the square root of the
sample size. For the sample sizes, mean values
and standard deviations presented in the results
of this section it can be easily verified that the
improvement in the standard error of the mean
values would be, for most of the cases, approximately a tenth of a degree Celsius=Kelvin, or
about six meters for inversion depths, if we use
20–25, instead of five years of data, assuming
that the data series are covariance stationary.
Hence, statistically, it is perfectly safe to derive
statistics of inversion characteristics from the
available data. It has been established in the international literature to refer to such statistics
using the term ‘‘inversion climatology’’. Hence,
the word ‘‘climatology’’ should not be taken at its
face value on this occasion. Additionally, the particular data period we used, coincides with the
one we used in our previous studies for Hemsby
(e.g. Milionis and Davies 1992, 2002, 2007). In
that way it was possible to examine the validity
of previous results and assumptions, which were
based on the Hemsby record, and to advance our
understanding about the characteristics of temperature inversions.
5. Summary and conclusions
There is little doubt that an inversion climatology at one site is useful for several reasons.
However, with data from one location only it is
not possible to assess the extent to which the
characteristics of inversion layers reflect the synoptic conditions and the extent to which they
reflect local influences, as the two effects cannot
be separated. In this work it is attempted to partial out the effect of local factors from that of
synoptic conditions. The effect of local factors
can be revealed controlling for the effect of the
synoptic conditions, by considering two locations
with different topography, but being influenced
by the same synoptic conditions. On the other
hand, the effect of the synoptic conditions can
be revealed by the identification of the conditions
237
which eliminate the effect of local factors and
examining the inversion characteristics at the
two locations under such conditions.
Following this approach the comparison of the
basic inversion statistics at Hemsby and Crawley
has shown that:
(a) There is a pronounced difference in the vertical distribution of occurrence of inversions
at the two locations: only the distribution at
Hemsby has a local maximum around 200 m.
This difference is attributed to the influence
of the North-Sea. Advection of cooler maritime air, due to the sea-breeze effect, as well
as due to synoptically driven easterly winds,
are the possible processes that are responsible for this local maximum of occurrence of
inversions.
(b) Further confirmation of the conclusions in
(a) is provided by considering the vertical
distributions of the base-heights of inversions
in the two location under conditions for
which the sea-breeze is either maximized
or minimized. It is shown that in winter at
midnight, when the sea-breeze effect is eliminated, the two distributions are remarkably
similar indicating the overall synoptic control. In contrast, in summer at midday, when
the sea-breeze effect is at its maximum, the
local maximum in the distribution of baseheights at 200 m in Hemsby is much more
pronounced than in the general case.
(c) The prevailing weather type, which was assumed to be the same in both locations,
affects the statistics of the inversion characteristics in both locations in much the same
way. This conclusion was drawn by showing
that (i) the main weather classes (i.e. anticyclonic, cyclonic and westerly) differentiate
the statistics of inversions in the same way
for both locations, and (ii) under the same
weather class the mean values of the inversion statistics do not differ in the statistical
sense, with the exception of Td, where it is
possible that the statistically significant difference in the mean value for all weather
classes is a result of measurement error in
humidity.
(d) As the mean values of the inversion characteristics in (c) are calculated averaging over
the entire time period, the month-to-month
238
A. E. Milionis, T. D. Davies
variation (annual cycle) of the activity of inversions in the surface – 700 hPa atmospheric
layer in the two locations was also examined.
The annual cycle of the activity of inversions
is qualitatively very similar in both locations
and the variations in the monthly activity of
inversions activity in the two locations are
strongly correlated, with about two thirds
of the variation in activity of both surface
and elevated inversions within the year in
one station being associated with the corresponding variations in the other station.
(e) However, there is still an unexplained part in
the within the year variation of the inversion
activity in the two locations, which is attributed mainly to inversions in the lower part
(approximately from the surface up to the
850 hPa level) of the surface-700 hPa atmospheric region. This is confirmed by taking
into account only the first inversion of each
profile for the calculation of the activity of
inversions. In that case the inversion activity
variations within the year in the two locations are not correlated.
Given the difference in the local character
of the two locations this work, on the one hand
provides strong evidence that an inversion climatology at one location can be regarded as representative of a wider area for which the same
synoptic regime can be assumed but, on the other
hand, it indicates that it is necessary to impose
constraints; these are associated with local factors which, near the ground, overwhelm the overall synoptic control. However, even if there are
differences in inversion characteristics due to differences in local topography, such differences
could, to a certain extent, be inferred if the underlying mechanisms follow a regular pattern.
This was evidenced for the differences in the
base-heights of inversions as the underlying mechanism was directly related to the yearly and daily
cycles. Therefore, at least to a certain extent,
the main features of an inversion climatology at
one location can be reconstructed once we know
the same features at a different location influenced by the same synoptic regime and the
mechanisms which are responsible for any differences in the climatology of inversions in the two
locations, if these mechanisms have a deterministic character. This important finding has useful
implications, for micrometeorological applications, such as the study of local-scale atmospheric diffusion, where data for atmospheric stability
should be collected from the same location as the
sources of the pollution. Ideally, for such applications tower data are much more appropriate
than radiosonde data. However, it is quite often
that the atmospheric scientist has to face lack
of availability of the most suitable data. Even
in such cases this work provides encouraging evidence on how an inversion climatology for a
different location can be usefully employed.
The conclusions drawn in this work validate the
assumptions made in previous research on inversion climatology at Hemsby, in regard to the influence of the North Sea as well as the synoptic
conditions (Milionis and Davies 1992, 2007).
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