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

Ice cave research in its historical dimension has a long history in
Europe

Location of Schellenberger Eishöhle at the foot of the east face of Untersberg.
The mountain is viewed from the East, the length of the edges is 11 km
(orthophotos: ^{©}2003/2004, Salzburg AG and DI Wenger-Oehn,
digital elevation model: Bundesamt für Eich- und Vermessungswesen in Wien).
The map inlay shows the location of Untersberg in Germany.

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.

As described in Sect.

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 express the temperature modeled for logger B as a function of the
temperature measured by logger A:

To set up the design matrix

To determine the third unknown parameter

The correlation between cave and outside temperatures to our knowledge
was first studied by

Pearson's correlation coefficient between two linearly correlated time
series

To illustrate the methods introduced in Sect.

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

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

Raw data

In this example the raw data

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

Two parameters have to be chosen carefully when actually correlating
the temperature data. First we have to define the number

To find an adequate

The second parameter we have to choose is the maximum number of
samples we shift time series

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).

The time shifts determined by the correlation analysis are inserted
into Eq. (

Figure

Between loggers T1 and T2 the air is warmed by 0.45

The slightly different results in Figs.

In Sect.

The determined time shifts and the
corresponding maxima of correlation are displayed in
Fig.

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.

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
(

The post-fit error

The formal errors of bias

The formal errors of bias and scale factor, scaled with the post-fit
error of the model, are shown in Fig.

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

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

We perform an

The calcFLOW method proved very helpful to better understand the
test data observed in Schellenberger Eishöhle.
As described in

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

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

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.

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.

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.

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