TCThe CryosphereTCThe Cryosphere1994-0424Copernicus PublicationsGöttingen, Germany10.5194/tc-10-879-2016Analyzing airflow in static ice caves by using the calcFLOW methodMeyerChristianeMeyerUlrichPflitschAndreasMaggiValterhttps://orcid.org/0000-0001-6287-1213Universita di Milano-Bicocca, Dipartimento di Scienze Ambiente e Territorio e Scienze della Terra, Piazza della Scienza 1, 20126 Milan, ItalyUniversity of Bern, Astronomical Institute, Sidlerstrasse 5, 3012 Bern, SwitzerlandRuhr-University Bochum, Geography Department, Working Group Cave- and Subway-Climatology, Universitätsstrasse 150/Building NA, 44780 Bochum, GermanyC. Meyer (christianemeyer@gmx.ch)25April201610287989429July201530September201529March201613April2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://tc.copernicus.org/articles/10/879/2016/tc-10-879-2016.htmlThe full text article is available as a PDF file from https://tc.copernicus.org/articles/10/879/2016/tc-10-879-2016.pdf
In this paper we present a method to detect airflow through ice caves
and to quantify the corresponding airflow speeds by the use of
temperature loggers. The time series of temperature observations at
different loggers are cross-correlated. The time shift of best
correlation corresponds to the travel time of the air and is used to
derive the airflow speed between the loggers. We apply the method to
test data observed inside Schellenberger Eishöhle (ice cave). The
successful determination of airflow speeds depends on the existence of
distinct temperature variations during the time span of
interest. Moreover the airflow speed is assumed to be constant during
the period used for the correlation analysis. Both requirements limit
the applicability of the correlation analysis to determine
instantaneous airflow speeds. Nevertheless the method is very helpful
to characterize the general patterns of air movement and their slow
temporal variations. The correlation analysis assumes a linear
dependency between the correlated data. The good correlation we found
for our test data confirms this assumption. We therefore in a second
step estimate temperature biases and scale factors for the observed
temperature variations by a least-squares adjustment. The observed
phenomena, a warming and an attenuation of temperature variations, depending
on the distance the air traveled inside the cave, are explained by
a mixing of the inflowing air with the air inside the
cave. Furthermore we test the significance of the determined
parameters by a standard F test and study the sensitivity of the
procedure to common manipulations of the original observations like
smoothing. In the end we will give an outlook on possible applications
and further development of this method.
Introduction
Ice cave research in its historical dimension has a long history in
Europe , which dates back to the 16th century. Theories
about the origin of the cave ice are equally old, numerous, and
contradictory, depending on the scientific knowledge and ability to
conduct measurements in the respective century. In the
nineteenth century the first instrumental measurements were conducted
(compare ) before modern
ice cave research found its beginning with the works of,
e.g., , , , and
others. Evidently, right from the beginning the main focus was to
understand the processes and dynamics of the ice body and specific
cave climate elements; among those, the course of the air temperature
in the specific study sites as well as the airflow regime were a main
focus. Until today long-term measurements have been rare but do exist in
several European commercial caves, e.g., Scarisoara ice cave
, Dachstein Rieseneishöhle ,
Dobsinska ice cave , and Schellenberger Eishöhle
. In addition, short-term measurements are conducted by
speleological organizations and others at many sites, thus covering
numerous ice cave sites worldwide. The full potential of these study
sites and recorded data has not yet been exploited. Most climate studies in
ice caves concentrate on air, ice, and rock temperature, as
temperature loggers are available for relatively low prices and thus
also affordable for private studies by, e.g., speleological
organizations. Depending on the individual questioning, this may be
sufficient for a basic cave climate analysis. Besides financial
reasons, ice cave studies are facing two other problems in general:
the accessibility of the study site and the energy supply for
technical devices. The study sites are in many cases in remote places
in the high mountains, exposed to avalanches and winter conditions
often lasting several months. As a consequence, e.g., airflow
measurements using sonic anemometers are not always possible, though
an understanding of the airflow regime is indispensable for the
understanding of these complex systems (e.g., ). For the development but also
degradation of subterranean ice, the airflow regime is the main
influencing factor beside the time/amount of water and the thermal
conditions or the heat transfer between the different media (rock,
ice, water, air) . states that the
main factor that characterizes a cave in general is the air temperature.
Among the deduced topoclimatological factors, the airflow regime, which is first of all determined by the thermal
relation between the exterior atmosphere and the cave atmosphere, is
the most important physical factor to describe the topoclimate of a cave.
For this reason proposes to classify
the different types of cave topoclimate using the diverse types of
airflow regimes. propose, for
the specific case of ice caves in temperate regions, to classify on
the basis of two criteria: cave air dynamics and the type of ice.
They explain this by the importance of the airflow regime as the
“dominating process at the origin of cave ice” in, e.g., static or
dynamic ice caves, just to mention the best known ice cave
types. Numerous case studies highlight the role of airflow for the
development of ice caves, (e.g., ).
For these reasons we present here calcFLOW, a practical attempt to use the database which is
available for the majority of ice caves, i.e., air temperature
measurements for computing air fluxes.
In this paper we present the basic principles and the methodology of the calcFLOW method and
apply it to Schellenberger Eishöhle (Germany). The results allow the
interpretation of observations that have so far not been well understood, but also reveal
principle shortcomings of the setup of the loggers that limit the analysis.
They will be useful to install a refined network of
temperature loggers inside the cave. We are convinced that also other
observation campaigns may benefit from analysis by the calcFLOW method.
In the last part of this paper possible further applications of the
calcFLOW method are discussed. All calculations were
conducted by using the GNU Octave open-source
software
https://www.gnu.org/software/octave/
.
Study site and data
defined ice caves as caves containing
ice all year around. One can further distinguish different types based
on the origin of the ice, the main ice building processes, and the type of the ventilation
. Ice caves occur mainly at elevations below the
0 ∘C isotherm (in the Alps at about 2000 m elevation)
due to the availability of water, but they
may also occur in permafrost regions .
Boundary criteria, which additionally limit the existence of ice caves are
the airflow system, the number of surface openings, and the cave morphology.
One common type are the static ice caves. Like in our example, this kind of
ice cave only has one natural entrance, which is situated in the upper or
middle part of the cave, and therefore acts like a cold air trap.
In summer, when outside temperatures
are above the cave air temperatures, the cooler air stays in the cold air
trap and is only slowly warmed by the surrounding rock. Stable temperature stratification
occurs when deep temperatures are preserved over summer.
The open phase or so-called “winter situation”, when air exchange with the
external atmosphere occurs, is limited to external temperatures below the cave
air temperatures. When outside
temperatures drop below the current cave air temperature,
the colder air replaces the warm air inside the cave. The
cold air enters the cave along the floor of the cave passages, while
the warm air is pushed out along the ceiling towards the
cave entrance. The temperatures observed close to the cave floor and
at the ceiling therefore may differ greatly. For this reason care has
to be taken in the selection of the positions for the temperature
loggers to capture the airflow of interest. By the mixing of cold and warm
airflows and by the contact of the inflowing
cold air with the cave walls and cave ice, the inflowing air will
gradually warm up, and on the other hand, the cave is cooled down from the
entrance towards its inner reaches. As a consequence the
stratification of the cave air is disturbed.
Instead, the air temperature positively correlates with the distance
the air traveled inside the cave. Temperatures recorded along the floor
of descending passages that track the inflowing cold air will show an inverted
gradient compared to temperatures observed during the closed phase.
As soon as the outside temperatures rise above the cave
temperature and the inflow of cold air stops the
stratification of the air is restored.
To illustrate the calcFLOW method we apply it to temperature data collected in
Schellenberger Eishöhle located on Untersberg (Germany).
Untersberg is an isolated mountain in the most northern part
of the Berchtesgaden Alps (Northern Limestone Alps) at the border between
Austria and Germany (Fig. ).
Schellenberger Eishöhle is a big alpine cave (total length: 3621 m,
total depth: +39, -221 m), including a static ice cave part which has been run as
a show cave since 1925. Apart
from the 500 m long ice cave part, there is one major non-ice
part, which forks off close to the entrance in a northeasterly direction and
leads through several deep shafts to the deepest point of the cave
(-221 m). The cave is situated at 1570 ma.s.l. at the foot
of the eastern walls of Untersberg (the cave entrance is marked in Fig. ).
The access to the cave is by a 4 m high and 20 m wide portal,
which leads to Josef-Ritter-von-Angermayer-Halle, the largest room in the cave
with a length of 70 m and
a width of 40 m, that is illuminated by daylight. The floor
of this hall,
17 m below the entrance level, completely consists
of a major ice monolith, which is surrounded by the cave trail. The
two passages Wasserstelle and Mörkdom connect to the deepest part of
the ice cave called Fuggerhalle, 41 m below entrance
level. They are also partly covered with ice. Temperature loggers were
placed in Angermayerhalle (T1 and T4), along one of the passages
leading downwards (Wasserstelle: T2), and in Fuggerhalle (T3, see
Fig. ). The loggers recorded temperature data with an
interval of 10 resp. 15 min. These temperature measurements were recorded
for a first cave climate study of Schellenberger ice cave (compare
) and the logger setup was not optimized for the
application of the calcFLOW method. Therefore synchronizing the sampling rates of the different loggers was not emphasized.
Analyzing the observed temperature data, several questions arose. The two
loggers in Angermayerhalle show quite different temperature behavior that
could not easily be explained. Moreover the logger in Fuggerhalle recorded
temperatures that seemed to be too warm for the lowest part of the ice cave where the
coldest air was expected. The development of the calcFLOW method was motivated
by these observations and led to reasonable explanations for the observed phenomena.
The model
As described in Sect. , two different stages of a static ice cave have to be
distinguished: an open and a closed phase. During the closed phase, or so-called “summer
situation”, the air temperature in the cave
is below the temperature outside and no interaction between the inside and outside atmosphere
by gravitational air mass transport takes place.
In this case the undisturbed air inside the cave shows stratification due to its specific weight,
the densest (coldest) air occupying the deepest ranges of
the cave. As long as the slow warming of the cave during the closed
phase is ignored, the difference in temperature observed by two
loggers at different locations in the cave is constant over time and
may be described by a simple bias:
TB(t)=TA(t)+b,TA and TB being the temperatures observed at time t by the
loggers at locations A and B inside the cave. b is the temperature
bias observed between both loggers and is considered to be constant over time in this simple model.
The phenomenon of stratification of air in static ice caves during the closed phase
is a basic principle and is not discussed further here.
Ground map and side view of Schellenberger Eishöhle with positions of all temperature loggers.
Instead we focus on the open phase, the so-called “winter
situation” that is most relevant for the cooling of the cave and
therefore for the existence of the cave ice.
During the open phase, loggers at different locations in the cave will
record a completely different scenario than during the closed
phase. We expect a temperature bias, but now with inverted sign, the
cave being warmer the further inside the logger is placed (see Sect. ). We
furthermore expect the variations in air temperature that are driven
by the weather and the day/night cycle outside the cave to be
measurable also inside the cave, but attenuated, due to mixing of the
inflowing air with the more stagnant air inside the cave. Thirdly, we
assume that the cold inflowing air needs some time to travel from
logger A to logger B. Our model for the air temperature
measurements taken by different loggers during the open phase of
a static ice cave includes all three parameters: bias, scale factor
(attenuation of temperature variations), and travel time of the air from
logger A to logger B. The model for the open phase therefore
reads
TB(t)-T‾B=s⋅(TA(t-Δt)-T‾A).TA, TB, and t are defined as above. The model is augmented by
a scale factor s and the travel time Δt of the air moving
from logger A to logger B.
In fact Δt is the parameter ultimately of most interest to calculate the speed of air flow between loggers.
T‾A and T‾B
are the mean temperatures measured by loggers A and B. The terms
TB(t)-T‾B and TA(t-Δt)-T‾A describe
the temperature variations around the means recorded by the two loggers,
that are attenuated by factor s at logger B due to the mixing of the
inflowing air with stagnant air along the way from logger A to
logger B. The bias b=T‾B-T‾A is hidden in
the difference between the mean temperatures at A and B.
We express the temperature modeled for logger B as a function of the
temperature measured by logger A:
TB(t)=s⋅(TA(t-Δt)-T‾A)+b∗,b∗=T‾B=T‾A+b.
The parameters b∗ and s of this simple model may be
estimated from the observed temperature data by a standard least-squares adjustment process . To keep things simple, the
single temperature measurements are assumed to be independent of
each other and not affected by colored noise (i.e., their errors are
assumed to be normally distributed).
To set up the design matrix A of the adjustment process we
have to compute the partial derivatives of the modeled temperatures at
logger B with respect to the unknown parameters b∗ and s:
A=∂TB(t1)∂b∗∂TB(t1)∂s⋮⋮∂TB(tn)∂b∗∂TB(tn)∂s,∂TB(t)∂b∗=1,∂TB(t)∂s=TA(t-Δt)-T‾A.
The optimal solutions b^∗ and s^ of the sought-for
parameters are found by solving the equation
b^∗s^=(ATPA)-1ATPTB,
where TB is the column vector of temperatures measured at
logger B. The weight matrix P is the identity matrix, as
long as all temperatures are observed with comparable quality
(otherwise it is a diagonal matrix with the diagonal elements equal to
the inverse of the square of the assumed a priori errors). With the
estimated parameters b^∗ and s^ the difference
between observed and modeled temperatures at logger B, determined by
the sum of squares of the residuals, is minimized.
To determine the third unknown parameter Δt in the same way,
we would have to compute the partial derivative:
∂TB∂Δt=∂TB∂TA∂TA∂Δt=s⋅∂TA∂Δt.
Neither an a priori value for s nor
∂TA/∂Δt are known. We therefore
propose to determine the time shift Δt independently by
cross-correlation of the time series of observed temperatures
TA and TB.
The correlation between cave and outside temperatures to our knowledge
was first studied by , who did not take into account
time shifts between different logger sites.
The idea behind the correlation analysis presented here is that a weather-induced
temperature pattern is visible at all measuring stations inside the
cave and that it is sufficiently unique to produce a distinct maximum
of correlation when cross-correlating the observed temperature time
series of two different loggers. For this purpose one of the time
series is shifted in time until maximum correlation is reached. The
time shift corresponding to optimal correlation of both time series is
equal to the travel time of the air between the two temperature
loggers. To determine the airflow speed, the length of the passage
between the two loggers has to be divided by the travel time of the
air. An analogous method is used, e.g., in hydrology to determine the
travel time of a flood pulse or, when applied to karst springs, the
time delay between rainfall and discharge (see, e.g., ).
In case of hydrology the medium is water, not air,
and the observable is the flow rate, not the temperature.
Pearson's correlation coefficient between two linearly correlated time
series X and Y of n samples each is computed by
r=∑i=1n(xi-x‾)(yi-y‾)∑i=1n(xi-x‾)2∑i=1n(yi-y‾)2,
where x‾=1/n∑i=1nxi and
y‾=1/n∑i=1nyi are the mean values of
the corresponding time series. The correlation coefficient r will
take values between -1 and 1. A value of 1 validates the
assumption that Y=b+s⋅X with bias b and scale s. Note that
this assumption exactly corresponds to our simple model
introduced above, and therefore r may additionally serve to validate
the applicability of the model.
Application to data
To illustrate the methods introduced in Sect. , we apply them to
temperature measurements recorded in the static ice cave
Schellenberger Eishöhle.
In Fig. , temperature observations of the four different loggers
are displayed for a period of 6 days. During this period, a gradual
cooling can be observed during the first 5 days, interrupted by a
warm spell on 30 January. On 1 February warmer weather sets in, resulting in
a rather abrupt rise in cave temperatures.
As mentioned in Sect. , the loggers recorded temperature
observations at either 10 or 15 min intervals.
For our analysis observations at common 30 min intervals
were chosen. It turned out that for the determination of wind speeds,
a higher sampling rate would have been beneficial. It therefore is planned
to synchronize and increase the sampling rate in the future.
In a first step, time shifts between one of the loggers in
Angermayerhalle (T1) and all the other loggers (T2, T3, and T4) were
determined for an example epoch early in the afternoon of 30 January,
applying the correlation analysis. In a second step, temperature
biases and scale factors between the corresponding loggers were
determined from the same set of data according to the least-squares
formalism introduced in Sect. .
Temperature observations of loggers T1 (Angermayerhalle, lower
part), T2 (Wasserstelle), T3 (Fuggerhalle), and T4 (Angermayerhalle, upper
part) available for analysis.
Observed temperatures (left panels) and correlation functions (right
panels) during a period of large temperature variations, well suited for
correlation analysis. Data of loggers T2, T3, or T4 are cross-correlated with
the data of logger T1 using a correlation length of 101 (a, b),
51 (c, d), and 25 (e, f) samples.
Observed temperatures (left panels) and correlation functions (right
panels) during a period of small temperature variations, apparently not so
well suited for correlation analysis. Data of loggers T2, T3, or T4 are
cross-correlated with the data of logger T1 using a correlation length of
101 (a, b), 51 (c, d), and 25 (e, f) samples.
Raw data (a) of loggers T2, T3, and T4 were shifted in time
relative to logger T1 to be correlated (b). The time period shown
in (a) corresponds to the search window, while a time span of 51
samples was used for correlation analysis and to adjust temperature biases
and scale factors. In a second step (c) the temperature biases were
applied to loggers T2, T3, and T4; and finally (d) the temperature
variations at loggers T2, T3, and T4 were scaled to fit those at logger T1.
In this example the raw data (a) of logger T1 were
shifted (b) relative to loggers T2, T3, or T4 until best correlation
was reached. Then (c) temperature biases were applied at logger T1
to fit either T2, T3, or T4 before finally, (d) the temperature
variations at logger T1 were scaled to fit loggers T2, T3, or T4.
Epoch-wise maxima of correlation (left panels) and corresponding
time shifts (right panels) for the three pairs of loggers T1 : T2 (top
panels), T1 : T3 (middle panels), and T1 : T4 (bottom panels); for
smoothing the centered moving mean of five samples was computed.
Epoch-wise biases (left panels) and scale factors (right panels) for
the three pairs of loggers T1 : T2 (top panels), T1 : T3 (middle panels),
and T1 : T4 (bottom panels); for smoothing the centered moving mean of five
samples was computed.
Epoch-wise standard deviations of biases (left panels) and scale
factors (right panels) for the three pairs of loggers T1 : T2 (top panels),
T1 : T3 (middle panels), and T1 : T4 (bottom panels).
Significantly determined parameters; correlation length is 101 samples (a) or 51 samples (b).
Correlation analysis
Two parameters have to be chosen carefully when actually correlating
the temperature data. First we have to define the number n of
samples we want to use for correlation. We inherently assume that the
airflow speed is constant for the time period covered by the n
samples. It is therefore desirable to choose n as small as possible
if we are interested in the temporal variability of the airflow speed
in the cave. On the other hand the part of the time series under
consideration has to be long enough to show a unique temperature
pattern for correlation. Due to the smoothness of the observed
temperatures they will resemble a linear trend during short stretches
of time. Cross-correlating two straight lines will produce constant
correlation coefficients of 1, and no distinct maximum will be
distinguishable.
To find an adequate n it is helpful to actually take a look at the
correlation function of example data observed in Schellenberger
Eishöhle. We analyzed temperatures observed by four different
loggers during periods of large temperature variations on 30 January
(Fig. ) or small temperature variations on 29 January
(Fig. ). The temperatures at logger T1 were taken as
a reference, while the temperatures recorded by loggers T2, T3, and T4 were
cross-correlated with the temperatures at logger T1 using different
numbers of samples. During periods with large temperature
variations, only a small number of samples is needed to produce
distinctive maxima in the correlation function (Fig. ,
bottom panels). Actually for our example epoch, correlation maxima are more
distinctive the fewer samples are used.
During periods of little temperature variations on the other
hand, no distinction of a maximum of correlation is possible, if too few
samples n are considered for cross-correlation
(Fig. , middle and bottom panels) and the determined
time shifts become meaningless. Generally we may assume
that a time span of 1 day (corresponding roughly to a correlation
length n of 51 samples in Figs.
and ) will most probably suffice in most cases to get
a clear correlation peak due to the day/night cycle in outside
temperature. Shorter time spans may suffice during periods of
pronounced weather patterns. Fine tuning of n will be worthwhile,
whenever time resolution of the determined airflow speeds is in the
center of interest.
The second parameter we have to choose is the maximum number of
samples we shift time series Y against time series X.
From a computational cost point of view, it is desirable to keep this number
small. Moreover, periodic temperature patterns like the day/night cycle
will lead to secondary maxima in the correlation function, if we shift
one time series by a full period of the cycle (i.e., 1 day). A rough
idea of the expected airflow speeds is helpful to adjust this
parameter. If the air is expected to move within 10 min from
logger A to logger B, it is in principle not necessary to shift
the time series at logger B by more than 10 min to catch the
maximum in the correlation function. In our examples we used
time windows of ±2d to also show the
variability of the correlation coefficient related to the
applied time shift.
It has to be stressed that the sampling rate of the temperature
measurements limits the time resolution of the correlation
analysis. The time shift of maximum correlation will always be an
integer multiple of the sampling rate, and its uncertainty corresponds
to half the sampling rate. Even if the smooth nature of
the temperature measurements suggests increasing the sampling rate by
interpolation, this will not introduce new information for the
correlation analysis. On the other hand, it does not disturb the
analysis according to our experience (not shown).
Bias and scale
The time shifts determined by the correlation analysis are inserted
into Eq. () to compute the partial derivatives
with respect to the scale factors. In a consecutive step, biases and
scale factors of our simple model can be determined for each pair of
data loggers. We perform the least-squares adjustment for the example epoch of
Fig. , applying the time shifts determined using 51 samples
(Fig. , middle row).
Figure shows the fit of observations of loggers T2, T3, and T4
to observations of logger T1,
Fig. , the inverse fit of T1 to data of either T2, T3, or T4.
Both definitions are valid in principle.
The parameters determined for the example epoch are listed in the
legends of Figs. and .
To compare them, the signs of the time
shift and bias of either Figs. or have to be
changed and the corresponding scale has to be inverted.
Note that bias and scale factor were determined together and are only
evaluated separately for Figs. and .
Between loggers T1 and T2 the air is warmed by 0.45 ∘C (or
0.47 ∘C); between T1 to T3 it is warmed by 1.05 ∘C, and between T1 and T4 by
1.12 ∘C (or 1.19 ∘C). This warming goes hand in hand with
an attenuation of temperature variations by a factor of 0.77 (or 1/1.22) between
T1 and T2, by a factor of 0.36 (or 1/2.75) between T1 and T3, and by a factor of
0.31 (or 1/3.49) between T1 and T4. We therefore assume
that the inflowing cold air passes T1 and T2 on its way to the
deepest reaches of the cave at T3 and that T4 records the outflowing warmed air
(see Sect. for detailed discussion).
For the distance of approximately 65 m from T1 to T3, passing T2
half way, we get a time shift of 0 min.
This means that it took the air less than half the sampling rate, i.e., 15 min
(corresponding to an air speed greater than 4 mmin-1) and that the
sampling rate of 30 min is too infrequent to determine the airflow speed along
this way for the example epoch.
For the distance of approximately 180 m from T1 to T4, assuming
that the air passes T1, descends via Wasserstelle (T2), and rises via Mörkdom, we
get a time shift of 270 min. Considering only the way from T3 to T4 this results
in an air speed at the order of 0.5 mmin-1.
The warming of the air along its way through
the cave and the attenuation of temperature variations agree well with the
assumptions that underlie the model design (see Sect. ).
The slightly different results in Figs. and
are due to the fact that the reference epochs differ by the determined time
shifts (depending on which logger is kept fixed as reference).
The very much comparable results prove that the method is robust and that
the parameters are stable for the period under investigation
(the temporal variability of the parameters is studied in Sect. ).
The validity of our model is further confirmed by the optically good fit
achieved for the example data (Figs. d
and d);
measures for the quality of the model fit are introduced in Sect. .
Temporal variability
In Sect. it was mentioned that the airflow speed is
supposed to be constant during the time period considered for
correlation. In this section we will estimate airflow speeds (time
shifts) for the whole period of about 6 days (see Fig. )
to check if this requirement is
met. To do so we repeat the analysis performed in Sect.
for an example epoch for all epochs of the period shown in Fig. .
We use either 51 or 101 samples for correlation. We also try the
effect of smoothing (by a centered moving mean of five samples) to filter out
short-term variations of unknown origin visible in Fig. .
The determined time shifts and the
corresponding maxima of correlation are displayed in
Fig. . The latter may serve to assess the reliability of
the time shifts. Comparably small correlation coefficients indicate
questionable results.
Only between loggers T1 and T2 the correlation, at least of the
smoothed temperature data, is high during the whole period analyzed
and the determined airflow speed is quite constant. As already
mentioned, the sampling rate of 30 min is too coarse to really resolve
it; the time shift varies between 0 and -30 min, indicating a true
value between both limits. The negative time shift, which is at first glance puzzling, may hint at the placement of logger T1 too high above the ground.
The cold air entering the cave moves along the floor of the passage
below T1 and reaches T2, before it is recorded by T1 (see discussion in Sect. ).
The results of the correlation analysis between loggers T1 and T3
indicate that the airflow speed in fact is not constant. The larger
time shifts determined for the beginning of the time period correspond
to higher temperatures and consequently a less pronounced
gravitational airflow. Near the end of the period the temperatures
rise so much that the air movement stops, the open period of the ice
cave is interrupted, and our model is no longer valid. Consequently
the correlation analysis fails.
The somewhat different values determined from the analysis of either
51 or 101 samples indicate that the slow airflow at the beginning of
the period affects the results for a longer time if 101 samples are
considered for correlation. In general the correlation of a larger
number of samples leads to smoother results.
In case of the analysis of loggers T1 and T4 we get very variable
results for the time shifts as well as for the value of maximum
correlation. A closer look at the correlation function at single
epochs would reveal that side maxima distort the analysis, leading to
jumps in the determined time shifts. A reduction of the search window
would probably help to remove some of these artifacts. The results
achieved for 51 or 101 samples agree best during the middle of the
period, where the spell of warm weather leads to a distinct
temperature pattern that facilitates the correlation analysis.
The smoothing of the data generally improves correlation by
reduction of uncorrelated noise, but
does not significantly alter the determined time shifts.
After applying the determined time shifts to the time series of
temperature observations at loggers T2, T3, and T4, optimal biases and
scale factors were estimated for each epoch. The results are
summarized in Fig. and show a strong dependency on
the temperature of the cold inflowing air. Colder inflowing air
goes hand in hand with larger temperature
gradients that lead to a faster inflow of the cold air. This
results in less pronounced attenuation of temperature variations, i.e.,
larger scale factors, because the time for energy exchange with the
cave (air, ice, rock) is reduced. The short spell of warm weather on 30 January
immediately leads to an increased attenuation, i.e., smaller scale factors.
The biases increase with the steepness of the temperature gradients.
Again, the parameters were fitted either from 51 temperature samples or
from 101 samples. Because the fit is optimal to all samples used, an
averaging takes place and the results obtained from more samples look
considerably smoother. A smoothing (moving mean) of
the temperature time series prior to the estimation of biases and
scales helped to derive scales between T1 and T2, where short-term variations of unknown origin superimpose the
temperature variability induced by outside temperature variation (Fig. b).
Validation of the model
The time shifts derived from the correlation analysis could most
easily be validated by actual airflow measurements. However, we do not have
airflow measurements available and so we depend on internal validation
methods that do not rely on external data. The presented tests allow
the plausibility of our model to be validated.
The determined correlation coefficients validate the general
applicability of the linear model assumed. In our analysis of data
collected in Schellenberger Eishöhle, correlation was generally high
(>0.9 for most of the time analyzed) and we can safely assume the
linear model to be valid.
The quality of the bias and scale parameters determined by a least-squares adjustment can be assessed by their formal errors. The overall
quality of the model is characterized by the post-fit error of the
modeled temperatures when compared to the ones actually observed.
The post-fit error σ of the modeled temperatures
is easily computed from the sum of squares of the residuals:
ν2=∑n(TB, observed-TB, modeled)2σ=ν2n-u,
with n the number of observations used to fit the model (in our
examples so far chosen to be equal to the number n of samples used
for the correlation analysis) and u the number of unknown parameters
estimated. The time shift is determined independently of bias and
scale factor; nevertheless we chose u=3.
The formal errors of bias σb and scale factor
σs are taken from the covariance matrix of the least-squares
adjustment:
K=σ2⋅σb2σbsσsbσs2=σ2⋅ATPA-1.K is a symmetric matrix; covariances σbs and
σsb are identical. Keep in mind that in
Sect. we chose P to be the identity matrix. The
formal errors are scaled by the post-fit error σ.
Note that from the covariance matrix one can also compute the
correlation coefficient between the bias and the scale factor:
rbs=σbsσb⋅σs.
This has not been evaluated in this study. In the case of the data
analyzed from Schellenberger Eishöhle the correlation between bias
and scale turned out to be small and could also be neglected (corresponding to
a separate estimation of both parameters).
The formal errors of bias and scale factor, scaled with the post-fit
error of the model, are shown in Fig. for the time
period analyzed in Sect. . The temperature biases are
rather well defined; the scale factors profit from a smoothing of the
data. Rising errors to the end of the period correspond to the rising
outside temperatures that finally lead to ceasing air movements in the
cave and an interruption of the open period.
As long as the time shifts are computed independently by cross-correlation
we cannot define their error bounds correspondingly to bias and scale.
However, in any case the accuracy of the determined time shifts is limited by
the sampling rate of the temperature observations to half the sampling
interval (in our case, this corresponds to error bounds of plus/minus 15 min).
Note that time shifts determined to be zero are not meaningless; they just
show that the air took less time than half the sampling period (i.e., 15 min)
from one logger to another.
Finally the significance of the estimated parameters may be calculated,
assuming that their errors are normally distributed (their variances
are χ2-distributed). This test tells us if the parameter in
question is indispensable to improve the model. We expect that during
the closed phase, only the biases are significant parameters of the
model (corresponding to time shifts of 0 and scales of 1 that do not
contribute to the modeled temperatures), while during the open phase
all three parameters are rated as significant.
The test of significance will not tell us if
the estimated values represent the physical quantities the parameters
were intended to model. A parameter that absorbs systematic
noise will be rated as significant, even if the determined numerical
values may not be interpreted in a meaningful way.
To test the significance of the
parameter in question, two different models are compared: one
including all parameters (full model), the other one including all but
the parameter in question (reduced model).
The reduced model to test the significance of Δt reads
TBΔt(t)=s⋅(TA(t)-T‾A)+b∗;
the reduced model to test the significance of s reads
TBs(t)=TA(t-Δt)+b∗;
and finally, the reduced model to test the significance of b∗
reads
TBb∗(t)=s⋅(TA(t-Δt)-T‾A)+T‾A.
Note that it is not correct to determine the parameters of the full
model once and subsequently insert them into the reduced models.
Instead, the parameters of each of the reduced models have to be
determined in a separate estimation procedure to also take into
account the correlations between the different parameters. As
mentioned before, the correlations may be neglected here for the test
of bias and scale, which can be determined quite independently, but
for the significance test of the time shift, both parameters of the
reduced model (Eq. ) have to be re-estimated with a time
shift of Δt=0.
We perform an F test (e.g., ) computing the ratio
Φ=νr2-νf2/(rr-rf)νf2/rf,
where νr2 and νf2 are the sum of squares of the observed
temperatures after substraction of the modeled ones (see
Eq. ); subscript f refers to the full model,
subscript r to the reduced model. rr and rf are the
corresponding degrees of freedom n-u of the two models, the number
of unknowns ur=uf-1 of the reduced model being smaller than that
of the full model uf, and therefore rr=rf+1.
Φ is F-distributed, its probability density function
Fnm(Φ), with n=rf and m=rr-rf, is a measure for
the probability that the additional parameter in the full model could
have been estimated in the same way from normally distributed random
numbers. We evaluate the associated cumulative distribution function
and reject all parameters for which it is smaller than 0.99
(corresponding to a 99 % confidence level). The remaining biases,
scales, and time shifts are marked in Fig. a for
a correlation length of 101 samples, and in Fig. b for
a correlation length of 51 samples. Bias and
scale turn out to significantly improve the model for most of the time.
The results look different for the time shift, which is only rated as significant for short periods of time. Comparing Fig.
with Fig. we realize that the time shift is always rated
insignificant when it is estimated to be zero. This is reasonable because
a time shift of zero corresponds to not estimating the time shift at
all. As stated above, the estimates of zero for the time shift are
artifacts caused by the coarse sampling rate. With an increased
sampling rate, it can be expected that the time shifts are rated as significant whenever the scale factors that benefit from the
high temperature resolution of the loggers indicate air movements
in the cave. The message of Fig. therefore is that
for the time period analyzed, all three parameters – time shift (as soon as it is larger than half the sampling rate),
bias, and scale – are indispensable for our model. The test
probably will become more interesting when the model is refined
to include effects like insolation of the entrance hall that
most probably affects the loggers in Angermayerhalle and should
be measurable, at least during the closed phase of the cave.
Discussion of results
The calcFLOW method proved very helpful to better understand the
test data observed in Schellenberger Eishöhle.
As described in , the two loggers T1 and T4 in Angermayerhalle
show very different behavior (Fig. a) that could not yet be explained. Our
analysis revealed a significant time shift (Fig. b) as
well as a pronounced positive temperature bias
(Fig. c) of T4 relative to T1, as well as a pronounced
attenuation of the temperature variations
(Fig. d) recorded by T4. We therefore assume that logger T1
records the cold inflowing air, while T4 records the relatively warmer
air flowing out of the cave. We further assume that the
cold inflowing air passes by logger T2 to the deepest point in
Fuggerhalle, where logger T3 is positioned. Temperature
biases are positive and
increase with the distance the air has traveled into the cave (as long
as we assume that T4 records the outflowing air), as predicted by our
model. The scaling factors are smaller than 1 (attenuation of signal)
and are inversely proportional to the distance the air has traveled.
The sampling rate of
30 min proved to be too coarse to determine the airflow speed from T1 to T3
for most of the time analyzed. The estimate of 0 min means that the air took
less than 15 min for the distance of approximately
65 m between T1 and T3, corresponding to a speed of more than 4 mmin-1
(agreeing well with air speeds of gravitational flow of 6 mmin-1 reported by
). Negative values for the time shift between
T1 and T2 may indicate a position of logger T1 too high above the floor so that the cold air
that flows along the floor of the passage passes T1 without being noticed and reaches T2 before it
is recorded at T1. This suspicion was confirmed by in situ inspection of logger T1.
While T2 shows distinctive variations of rather short duration (and
unknown origin) that clearly correspond to the temperature variations
recorded by T1, the same variations are very much attenuated at T3 and not
at all visible any more at T4. This
may be explained by the distance the air traveled inside the cave and by the attenuation
of the temperature variations due to energy exchange with stagnant cave air,
ice, and rock. Moreover, Fuggerhalle acts as a dead end where the cold air that enters via
Wasserstelle and probably also via Mörkdom
is thoroughly mixed with the stagnant air. The assumption that
Fuggerhalle is probably warmed by dynamic ventilation from deeper reaches of
the cave could not be confirmed. The temperature biases and scaling factors determined
for T3 fit our model very well. We conclude that Fuggerhalle is warmer than
Angermayerhalle or Wasserstelle just because it is farther from the entrance.
From T3 at the furthest end of Fuggerhalle the warm air takes
a significant amount of time before it reaches T4 on its way out of the
cave. For this remaining distance of 115 m, a time shift of
270 min was determined for our example epoch (Sect. ),
corresponding to an air speed of 0.5 mmin-1.
Not much more signal attenuation or warming takes place along this
path. Unfortunately, in the time period analyzed, no logger was
positioned in the second passage (Mörkdom) connecting
Angermayerhalle and Fuggerhalle, so it cannot be clarified if one of
the passages acts as the primary way down for the cold air while the
other channels the warm air back to the surface. The determined air speeds
have to be considered as mean speeds for the way the air traveled between
loggers;
they will of course vary depending on the cross section of the passage. The
different speeds determined for the inflowing cold air and the replaced warm air
may also be explained by the cross section of the passage occupied by the
corresponding air flow. Independent of all the factors that complicate
interpretation, we can state that the results appear to be realistic.
The resolution of the correlation analysis is drastically limited
by the coarse sampling rate of the loggers and the missing synchronization.
This fact does not reduce the applicability or validity of our model, but it
limits the interpretation of the results. Nevertheless we were able to
characterize the general patterns of air movement and their slow
temporal variations.
The analysis of the temporal variability of the determined parameters (Sect. )
confirmed the basic principles on which the model is based. Low outside
temperatures correspond to steep temperature gradients that result in
small time shifts (high air speeds). The energy exchange with the cave environment
is limited by the short time the cold air stays in the cave, and the attenuation factors
are closer to 1 when outside temperatures are low. The biases correspond to the temperature gradients
and are larger during spells of cold weather.
But the analysis of the temporal variability also revealed problems in the correlation analysis.
The cross-correlation of loggers T1 and T4 exhibits an unrealistic variability, including a number
of jumps. These clearly are artifacts that are caused by side maxima of the correlation analysis,
stressing the need to limit the search window to a sensible width, which depends on the cave,
the placement of the loggers, and the distance between loggers, and can only be refined after some
tentative analysis.
Generally it can be stated that
times of poor correlation correspond to periods of little temperature variations. Long
correlation lengths may help but also reduce the time resolution of the determined
time shifts due to averaging over the number of samples used for the correlation analysis.
A rise of the outside temperatures above the cave temperature will lead to ceasing
air flow and an interruption in the open phase of the cave. In this case the correlation analysis fails.
The determination of biases is robust, while the determination of scaling factors is
only limited by the signal-to-noise ratio of the observations. The time series of
T1 and T2 show a number of short-term variations superimposing the long-term variations
of the outside temperature. They cannot be explained by our
simple model and hinder the estimation of scale factors for logger pair T1/T2.
Smoothing helps to separate the long-term from the short-term variations
and stabilize the estimated scale factors. A better
solution surely would be to find the reason for the short-term temperature
variations and include corresponding parameters in the model; the forcing of air into
the entrance hall by outside winds would be a probable candidate, though difficult to model. As is the
case for the time shifts, a reduced number of samples used for the determination of bias and
scale factor leads to an improved time resolution, while an increased number
of samples stabilizes the estimation. As can be expected, the uncertainty of the fit
(i.e., the formal errors of bias and scale factor)
increases with the distance between loggers.
Conclusions
The objective of this paper is to present the principles and the
methodology of the calcFLOW method that was developed in order to be able to
use air temperature measurements in static ice caves to define the
airflow regime. The idea of calcFLOW is based on the fact that in many ice
caves in remote places, airflow measurements are difficult. However, in
every ice cave where cave climate related studies are conducted, at
least temperature measurements (air, rock, ice) are performed. Based
on this data we calculate three different parameters to
better characterize the processes that dominate the cave climate and to understand
the temperature differences observed between the measuring points: the airflow
speed, the change of the mean air temperature, and
the attenuation of the temperature variations dependent on the
location inside the cave.
The primary objective is to calculate airflow speeds inside
a static ice cave to define the airflow regime.
It is achieved by cross-correlating air temperature data of
different logger sites. The method was applied to temperatures
recorded in Schellenberger Eishöhle during the open period,
when air movement inside the cave is governed by gravitational flow.
The method of cross-correlation we use for the determination of time
shifts in general depends
on rather distinctive temperature variations to successfully correlate
the observations of different loggers. On the other hand, the airflow
speed is supposed to be relatively constant during the time span used
for correlation. These two requirements contradict each other and it
has to be shown by further studies to what extent the temporal
variability of the air movements inside the cave may be
resolved. Most probably the reliability of the analysis will benefit from
an increased sampling rate of the temperature observations. Regardless
of the complexity of the situation at our test site, we may state that
the presented method is well suited to uncover the complicated air
movements in the cave. The results of the analysis will help to
optimize the placement of the loggers. An increased number of loggers
positioned near the floor as well as near the ceiling of the passages
will allow the paths of the inflowing and outflowing
air to be distinguished with much better spatial resolution and reliability. Decreased sampling
intervals will enable the determination of the speed of the rather
fast inflowing cold air and generally improve the reliability of the
correlation analysis.
We have already tested calcFLOW with air temperature data from Fossil
Mountain Ice Cave (USA), but these results will be part of future
publications. What we can already state for the moment is that
calcFLOW is applicable to other ice caves, too. This is one major
reason for the publication of this pilot study and also a reason
for us to keep the model
as simple as possible. We want to present a basic tool for cave
climate studies which everyone can use for their specific
site.
To summarize the outcome of this study, we can state that
calcFLOW is useful in the following way:
to characterize the airflow regime inside a static ice cave;
to compute (interpolate) the temperatures between two
loggers with one simple model, based on only three determined parameters;
to indicate possible problems in the measuring setup (e.g., position and height of loggers); and
to indicate useful observation intervals.
In a next step we will address key problems of calcFLOW in a dedicated
simulation study with the objective to provide measures for the
signal content of the time series of temperature observations,
evaluated by the root-mean-square, and
for the quality of the cross-correlation. The latter will be based on the
shape of the peak of maximum correlation, exploiting characteristics
like its dominance and width. The simulation study will also provide a
test bed for cross-validation methods to assess the reliability of the
determined air speeds; and of course we also hope to
validate the calculated airflow speeds by
comparison to real-time airflow measurements.
Meanwhile the logger setup in Schellenberger Eishöhle has been revised.
With the expected results we hope to be able to further differentiate the specific
paths of the airflow and to tackle questions of energy exchange in the cave.
For this task, finally a much denser network of temperature loggers, which also
probe ice and rock temperatures, and a volume model of the cave and its
ice filling, will be indispensable. The evaluation of the temperature
observations has to be automatized, based on the criteria developed
in the simulation study.
Acknowledgements
This work is part of the Italian Project of Strategic Interest
NEXTDATA (PNR2011–2013) funded by the Italian National Research
Council (CNR). It is also part of a PhD project, “Ice deposit evolution
and cave climatology of ice caves”, at Ruhr University Bochum
(Germany). For the logistical support and the good cooperation, we
would like to thank the Verein für Höhlenkunde Schellenberg e.V.
Moreover, we would like to thank Martina Grudzielanek for the revision
of the mathematical part of the paper.
Edited by: M. van den Broeke
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