The mixing height is a key parameter for many applications that relate surface–atmosphere exchange fluxes to atmospheric mixing ratios, e.g., in atmospheric transport modeling of pollutants. The mixing height can be estimated with various methods: profile measurements from radiosondes as well as remote sensing (e.g., optical backscatter measurements). For quantitative applications, it is important to estimate not only the mixing height itself but also the uncertainty associated with this estimate. However, classical error propagation typically fails on mixing height estimates that use thresholds in vertical profiles of some measured or measurement-derived quantity. Therefore, we propose a method to estimate the uncertainty of an estimation of the mixing height. The uncertainty we calculate is related not to the physics of the boundary layer (e.g., entrainment zone thickness) but to the quality of the analyzed signals. The method relies on the concept of statistical confidence and on the knowledge of the measurement errors. It can also be applied to problems outside atmospheric mixing height retrievals where properties have to be assigned to a specific position, e.g., the location of a local extreme.

In good scientific practice, uncertainties or errors must be provided for all physical quantities which are measured or estimated. Unfortunately, for a wide class of estimations it is not straightforward to apply standard error propagation on the result. This is the case for many applications where thresholds have to be identified in noisy signals. The aim of this work is to provide a rigorous way to estimate uncertainties for this class of operations. This is the general case of the localization of a local property. Examples of local properties for a signal are maximum and minimum values. A more general example can be seen as the property to have a certain value or threshold. This is also the case for the location of mixing height (MH), which can be defined by local properties of the data used for its estimation.

The top of the mixed layer or MH is the thickness of the layer adjacent to
the ground where any pollutants or constituent emitted within it or entrained
from above will be vertically mixed by convection or mechanical turbulence on
a reasonably short timescale. This timescale is about 1 h or less
according to

The mixed layer is a sublayer of the planetary boundary layer (PBL), which is
the atmospheric layer that is closest to the ground. In the PBL, several
processes control exchange of energy, water and pollutants between the
surface and the free atmosphere. The structure of the PBL is variable as
detailed by

The knowledge of MH has been considered fundamental for modeling dispersion
of pollution since

Idealized profile of potential temperature

This paper introduces a rigorous method to derive uncertainties in the localization of a property, with a special focus on mixing height retrievals as an example. This allows for more quantitative assessments of the quality of retrievals and can provide useful information especially when comparing different observation-based retrieval methods with one another or with mixing heights diagnosed in weather prediction models. Note that this proposed method does not address uncertainties related to assumptions about the boundary layer physics, or related to its spatial and temporal variability.

In Sect.

Several methods for detecting MH are reported in the literature, depending on
meteorological conditions and instrumentation used

The driving idea is that warmer air in contact with the ground reaches an
altitude where a capping inversion is located. For practical use in
convective conditions – when the impact from wind shear can be neglected –
the MH is located at the altitude

We chose this methodology to estimate MH because it uses a smaller number of
environmental profiles than the bulk Richardson number method. In our
numerical examples, we do not consider humidity, so we will focus on
potential temperature

Real examples of vertical profiles of virtual potential temperature are presented in Sect.

We found convenient the use of an analytical function to describe the
potential temperature profile, because in this way we can control the more
relevant aspects of the profile, which are the excess temperature at the
ground and the uniformity of potential temperature within the mixed layer.
The use of an analytical function helps also to study effects of spatial
resolution and smoothing. The profile

Looking at Fig.

Under steady conditions, we would get an estimated MH and a properly estimated uncertainty by repeating the measurements many times. However, in the real word, the conditions are typically not steady and the measurements cannot be repeated often enough (if at all) to obtain a statistically consistent set of estimates. Therefore, a methodology is needed that retrieves the localization error from a single profile. Our methodology requires the knowledge of the errors of the measured profiles, so that it is possible to propagate it onto the signals we want to analyze. The error propagation on potential temperature and on the bulk Richardson number profiles are provided in the Appendix.

The meteorological quantities observed by radiosondes are pressure,
temperature, relative humidity, wind speed, and wind direction. The data for
the practical examples used in this work are part of the data set of
radiosonde data of the Lindenberg Meteorological Observatory in Germany
(WMO station 10393). The data are collected regularly every 6

Vaisala RS92-SGP

Results of a Monte Carlo for the parcel method as detailed in algorithms presented in Sect.

In practice, the method that is used more widely to produce estimates of MH
is the so-called bulk Richardson number method. In
Sect.

After the choice of a methodology to detect MH on meteorological profiles, we
have many options for implementing it as an algorithm. Again, the parcel
method defines the MH at the altitude where the virtual potential temperature
equals

From an abstract point of view – not related to the actual meteorological concept – the core of the method is detecting the location where a certain threshold value is reached. This is a very common task in signal analysis, commonly called threshold detection. To implement a threshold detection, one must consider different properties of the signal. The signal noise is the main source of erroneous and multiple detections, especially for non-monotonic signals.

As an algorithm for applying the parcel method, we decided to use the location
of the last data point (starting from the bottom) that is still smaller
than

From the more physical point of view, the parcel method can be implemented as
the simple parcel method introduced by

Referring to the synthetic profile of Fig.

We must point out that the parcel method as it is implemented can be
considered just an algorithm for threshold detection in a signal. So all the
considerations that we made could be applied to other methods, for
example the bulk Richardson number explained in Sect.

So far, we have used a simple Monte Carlo (MC) simulation to illustrate the impact of measurement noise on the error in the retrieved MH. However, for application to large data sets this is too expensive to perform, and a more analytical method is needed.

In a continuous signal, a property can be defined as local when it occurs in an arbitrarily small neighborhood of points. However, real signals are not continuous but rather discrete data series of ordered points. For such discrete data series, the neighborhood concept must be adapted since it is not possible to consider arbitrarily small neighborhoods. Instead, a neighborhood would be a set of contiguous points. It contains a reference data point and some other points in its vicinity.

Two measurements can be considered equivalent when their difference is
smaller than their errors. The degree of equivalence is commonly called
confidence. Confidence is rigorously defined in several textbooks. It is
used to verify a hypothesis or, in other words, to see if an estimated value
agrees with a theoretical expectation. The most general case is presented in
Eq. (

A local property on an ordered data series can be shared between data points. This is due to the fact that data have errors, which has the consequence that different data values at different points can be differentiated from each other only within a certain degree of confidence. This sharing of properties by contiguous data points is the key to defining a rigorous concept of localization error.

To give an example, a data point located at 400

The formal description of the method requires the introduction of some
symbols. An ordered data series

When a local property in a signal can be defined, there are two choices to define its location: the local property can be located exactly at a data point or between two data points. The second possibility will not be discussed. Instead, for simplicity, we assume that the localization is located at the first data point that defines the interval where the property is detected.

The general assumption of the method is that the measurement errors are known, and they are normally distributed and uncorrelated. We focus on data points that have neighbors on both sides – not the end points of a series.

The method relies on one main idea: (a) the results of an algorithm are expected to fall in a neighborhood of the true location, and (b) this neighborhood can be seen as a set of data points that have similar values within the errors of the measurements. The similarity of values is measured with the quantity commonly called confidence.

The confidence

Welch's t test and other similar tests are typically used to evaluate
hypotheses. In this particular case, we try to verify the null hypothesis
that two estimations

All plots are obtained starting from the synthetic profile

For a normal distribution, confidence intervals are typically defined as
a distance in units of the SD

optimal confidence:

good confidence:

acceptable confidence:

bad confidence:

Given a series of locations

This reflects the idea that the neighborhood

To estimate a local property in an ordered data series

By merging the concept of the discrete neighborhood expressed in
Eq. (

To refer to confidence neighborhoods, we use the following notation:

We take a monotonic series of locations

Referring to the test function presented in Fig.

In Fig.

This definition of confidence neighborhood is more than a mathematical
abstraction. From the physical point of view it reflects the idea of
probability to obtain an estimation of a local property starting from
a signal which has its own uncertainties. Qualitatively, some properties of
the distributions of results can also be inferred. In particular, the
skewness or asymmetry of the distributions is captured by differences between the
leftward

This diagram shows the difference between confidence neighborhood (above) and strict confidence neighborhood (below).
The lines connecting the points represent where the relation of confidence Eq. (

Smoothing the data with a window of three points produces a profile whose
error is

When an algorithm defines a location

Comparing the confidence neighborhoods and the strict confidence neighborhoods in Fig.

Despite the definition of confidence neighborhood, the strict confidence neighborhood is not always unique. This can be understood by examining the process that is used to estimate it.

We start from the first three points

When checking for couples of points that agree with all the other points, but
not with each other, a choice must be made and one of the two points must be
rejected. Rejecting a point means that the confidence neighborhood stops
growing in that direction. For the other direction, one can still add points as
long as the confidence relation remains satisfied. A practical way to
determine which one of two conflicting data points should be kept is to
select the one that has a better confidence with

The width of the confidence neighborhood is a measure for the quality of
a localization. In particular, the introduced left width

In Fig.

One way to measure localization error would be to use the SD of the output
distribution of a Monte Carlo. However, the Monte Carlo distribution overlaps
well with the confidence neighborhoods for different values of

Left panel: the strict confidence neighborhoods

As a measure for uncertainty we used the square root of the second moment about the mode. For normal distributions this is also called SD. So we can calculate the square root of the second-order moment about the mode of the distribution of the results.

The second-order moment expresses clearly the uncertainty of the localization. However, its calculation requires performing a Monte Carlo experiment. Moreover, it depends strongly on the algorithm used. If the data have reasonably small errors and the algorithms provide a useful estimate of the target quantity, the results will have good confidence with respect to the true target quantity.

The confidence neighborhoods as from Eq. (

We found that Eq. (

This definition of

Probability density functions of five MCs performed 100 000 times each.
From left to right we created

For our final definition of localization error, we take
Eq. (

With

The strict confidence neighborhood

Probability density functions of six MCs performed 100 000 times each.
For this example we used a fixed spatial resolution of 0.1

In general, a confidence neighborhood can be asymmetric if the signal does
not depend on the location in a linear way. Especially if there is no change
in signal towards one side of

The localization error as defined in Eq. (

In the example shown in Fig.

Often a local property can be defined as a location where more than one
condition must be fulfilled. The localization might depend on different data
series defined in the same series of locations

The location vector

The results clearly show that the resolution has an impact
(Fig.

Note that in Sect.

From the example of the rightmost panel of Fig.

There are two main effects when running averages are used before an algorithm is applied:

the increased window size reduces the localization error,

the median of the results changes.

From Fig.

An interesting effect of increasing resolution can be seen in
Fig.

The estimation of MH comes with many dubious aspects.
The first problem is the choice of a method to detect MH. The second problem
is the choice of an algorithm to apply the method. It is well known

In this context, we do not want to evaluate the uncertainties that MH has due
to the choice of a method. This was done successfully by

The retrieved uncertainty can be used for several purposes, e.g., to compare
two different methods to see the degree of confidence by using
Eq. (

As a practical application, we used the described methods to retrieve the
uncertainties of MH from radiosonde data. Together with the already-introduced
parcel method, we applied two variants of the bulk Richardson number method as described by

The first definition

The second definition

The reference level as used by

To locate the MH, an appropriate critical or threshold value for Ri has to be selected.
The MH is located where this threshold value is reached.
A typical value for the threshold for the first method

The critical number for Ri

Examples of the profiles of Ri

Profiles used for estimating MH on 24 June 2010, 12:00

As introduced in Sect.

Using the radiosonde data, we calculated

The following steps were taken to describe the process that we used to estimate

propagate the errors using Eqs. (

use the algorithm presented in Sect.

estimate the

use Eq. (

The result of such an analysis are presented in Fig.

The methodologies for retrieving MH should be applied in proper
meteorological conditions. The use of a wrong methodology directly results in
a large localization error. This is clear from the time series of results in
Fig.

Profiles used for estimating MH on 3 June 2010, 06:00

Time series of MHs calculated with three methods: blue is the parcel method (PM), green the Ri

Potential temperature, Ri

Potential temperature, Ri

Another reason for high uncertainties is that the wind speed might not be
strong enough to justify the use of either Ri

We consider the points with small error bars in
Figs.

In the examples for bad localization (Figs.

The values of

In fact, a more symmetric error or confidence neighborhood can allow us to
use the localization error like an ordinary Gaussian error. However, when the
distribution of the results is not Gaussian, the retrieved localization
errors must be used with care. In particular, the

We defined a rigorous method for evaluating uncertainties in estimated
quantities where standard error propagation cannot be used. It is
particularly useful when a complex algorithm is necessary to find the
location of a certain property and a Monte Carlo approach is too expensive.
We call this uncertainty of a localization the localization error

Our work has proved to be a useful tool for qualitative analyses, and in
particular for filtering data by quality of the retrieval

Depending on the actual signal, the interval defined by the localization
error may be strongly asymmetric. In such a case, the expected distribution
of possible results is not Gaussian, and the distribution will have
non-negligible skewness. The correct use of the localization error then
requires considering the left and right contributions to

Our methodology was applied to compare different methods for retrieving mixing height from radiosonde data. This was not done to provide a better algorithm, nor to perform a general study on the best way to estimate mixing height. We rather provided a tool that can be used to better and more quantitatively compare different algorithms.

All the methodologies described in the literature provide values of MH
without a specific error estimate. Instead, the uncertainty was estimated on
the basis of the spatial and temporal variability of large data sets or by
comparing results of different methods. Our goal was to estimate a reasonable
uncertainty for one singular estimate of MH that depends only on the signals
used and their uncertainties. The uncertainties that we retrieve this way are
consistent with the climatological results of

Our method is not limited to mixing height retrieval or atmospheric science
at all. It can be applied to many problems where data points in a signal have
to be localized: for example to find minima or maxima, or values exceeding
a certain threshold. The localization error provided by our method can be
used for error propagation in almost the same way as SD. It opens the
possibility to check hypotheses by use of Welch's t test

This work is a contribution to the FP7 project ICOS–INWIRE, funded under grant agreement no. 313169. We are much obliged to Frank Beyrich of the Deutscher Wetterdienst (DWD), who provided us with the high-resolution radiosonde profiles. Edited by: L. Bianco The article processing charges for this open-access publication were covered by the Max Planck Society.