2.1 Background

Any modification of instrumentation might introduce a systematic change to the measurement time series. This change is typically assumed to be a constant difference (Δ) as a first-order approximation resulting from differences in the individual instrument biases, i.e. their systematic deviations from the true value. As the true value of the quantity being measured is unknown in practice, it is not possible to estimate each instrument's individual bias. It is possible, however, to estimate the difference $\mathrm{\Delta }={\text{Bias}}_A-{\text{Bias}}_B$ in biases Bias_A and Bias_B of instruments A and B. If temporally and spatially coincident measurements are made using instrument A and B (i.e. dual flights), this difference can be easily obtained: consider some quantity of interest, e.g. air temperature (T), measured with instrument A and instrument B at the same location and time t. The bias of each instrument is the difference between the expectation value of the instrument's measurement and the unknown true value T_t:

where T_t,A and T_t,B are the temperatures at time t measured with instrument A and B, respectively. The difference in the instrument bias is therefore

Consider now that T_t,B differs from T_t,A only by a constant offset Δ, i.e.

which is independent of the true value and thus the measurement time t. Under this assumption, an estimate for the stationary difference in biases can be obtained from N dual measurements according to

with $\widehat{\mathrm{\Delta }}$ denoting an estimate of the constant offset Δ. This equation applies even if the true value T_t is changing with time as it depends only on anomalies $T_{t,A/B}-T_t$. Under suitable conditions, the uncertainty (expressed in terms of standard deviation, SD) of this estimate decreases with $\sqrt{N}$ and depends on the persistence (i.e. autocorrelation) of the time series (Wilks, 2011).

Figure 2Example time series for interlaced measurements of instrument A (red dots) and instrument B (green dots). Horizontal lines are the means of the measurements using instrument A (red) and instrument B (green). Smooth dashed lines (red for instrument A, green for instrument B) are spline estimates with the differences being an estimate for the differences in the instrument biases.

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2.2 A statistical model for interlaced measurements

As dual measurements using both instrument types require additional resources and therefore inherent additional costs, estimating a systematic difference between the instruments using interlaced measurements, i.e. using instrument A on odd days $t\in \mathit{\{}\mathrm{1},\mathrm{3},\mathrm{5},\mathrm{\dots }\mathit{\}}$ and instrument B on even days $t\in \mathit{\{}\mathrm{2},\mathrm{4},\mathrm{6},\mathrm{\dots }\mathit{\}}$, is explored in this study. Using this approach, at every time t only one measurement from one instrument is available, and hence Eq. (4) is not applicable.

The underlying assumption for the approach outlined here to work is that the quantity of interest fluctuates around a smooth climatological signal (i.e. a seasonal cycle) and the fluctuations show a certain degree of persistence at the weather timescale; e.g. the fluctuations show a day to day dependence. For a typical difference in the biases between radiosondes this persistence (i.e. autocorrelation) is key to the idea of estimating a bias from interlaced measurements. The difference in the biases tested here is smaller than the day to day fluctuations themselves as it carries information from the measurement A to the measurement B.

In the following, a simplified model for air temperature time series complying with the above-mentioned assumptions is constructed. The true (unobserved) time series is represented by a smooth seasonal cycle with an autoregressive process of first order (AR[1], e.g. Box and Jenkins, 1976; Wilks, 2011) added to the time series; i.e.

with $d_t\in [\mathrm{1},\mathrm{\dots },\mathrm{365}]$ giving the day in the year for date t, where a is the autocorrelation coefficient which describes the degree of persistence in the time series at the weather timescale, e.g. the fluctuations show a day to day dependence, and ${\mathit{\eta }}_t\sim \mathcal{N}(\mathrm{0},{\mathit{\sigma }}^{\mathrm{2}})$ is the driving noise of the AR[1] process selected randomly from a Gaussian distribution. The latter is taken to be Gaussian white noise with zero mean and variance σ². This is a well-established model for the persistence of e.g. daily air temperatures (e.g. Wilks, 2011).

Pseudo-observations are now obtained from a realization of T_t (Eq. 5) with an instrument bias and random measurement noise added. Here, we aim for interlaced temperature measurements T_t,A and T_t,B from instruments A and B and thus add the instrument biases c_A and c_B, respectively, and independent Gaussian measurement uncertainties ${\mathit{ϵ}}_{t,A}\sim \mathcal{N}(\mathrm{0},{\mathit{\sigma }}_A^{\mathrm{2}})$ and ${\mathit{ϵ}}_{t,B}\sim \mathcal{N}(\mathrm{0},{\mathit{\sigma }}_B^{\mathrm{2}})$:

For simplicity, we assume equal variances ${\mathit{\sigma }}_A^{\mathrm{2}}$ = ${\mathit{\sigma }}_B^{\mathrm{2}}$ for the measurement uncertainties. The continuous series of combined interlaced measurements T_t,AB for $t\in \mathit{\{}\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{\dots }\mathit{\}}$ is therefore

with indicator function χ being 1 if t is a member of the set t_A or t_B and 0 otherwise. Figure 2 shows an example of such a synthetic time series of interlaced measurements. This example is based on a simulated temperature time series using a realization of an AR[1] process using an autocorrelation coefficient of a=0.5 in Eq. (6), similar to the autocorrelation coefficient of radiosonde measurements at 300 hPa above Lindenberg, Germany (see Sec. 2.4).

2.3 Estimating the difference in instrument biases

A direct approach to estimate the difference in instrument biases $\mathrm{\Delta }=c_A-c_B$ is an estimation using the differences in means ${\stackrel{\mathrm{‾}}{T}}_A$ and ${\stackrel{\mathrm{‾}}{T}}_B$ of instrument A and B, respectively, over a common time period t₁ to t₂; i.e.

with

being the arithmetic means for the individual instruments; N_A and N_B are the number of measurements made by instrument A and B, respectively, in the given time period. The uncertainty in this estimate of the difference in instrument biases decreases with increasing N_A and N_B but also depends on the persistence of the underlying time series: larger persistence leads to larger uncertainties when calculating arithmetic means (e.g. von Storch and Zwiers, 1999).

Here, we exploit the persistence and suggest an approach based on the estimation of a slowly varying signal common to both instruments. Imagine, for example, a smooth temperature time series in the absence of weather-induced noise. Measurements are then made of that signal using instrument A and this measurement series is represented by s(t) and an additional measurement noise ϵ_t. Analogously, measurements of the same slowly varying signal are made using instrument B and can be represented by the same s(t) but with the difference in instrument biases Δ and again measurement noise ϵ_t; i.e. $s\left(t\right)+\mathrm{\Delta }+{\mathit{ϵ}}_t$. A model for these interlaced measurements T_t,AB is constructed using the indicator function χ:

For t∈t_B, the indicator function χ(t∈t_B) returns 1 and we obtain a measurement with instrument B, i.e. ${\widehat{T}}_{t,B}=s\left(t\right)+\mathrm{\Delta }+{\mathit{ϵ}}_t$. For other time steps t∈t_A the indicator function returns 0 and we obtain a measurement of instrument A, i.e. ${\widehat{T}}_{t,A}=s\left(t\right)+{\mathit{ϵ}}_t$, excluding the difference in instrument bias Δ. The statistical model described in Eq. (12) belongs to the class of generalized additive models (GAMs; e.g. Chambers and Hastie, 1992), a fundamental class of regression models. GAMs extend generalized linear models (or linear regression) by additionally introducing to the classical linear components a smooth term s. This smooth term can be estimated using a smooth spline fit with its degrees of freedom (i.e. its flexibility of smoothness) determined by generalized cross validation (Wood, 2006). This functionality is implemented in the R package mgcv (Wood, 2006).

2.4 Simulation set-up

To investigate whether interlaced measurements diagnosed using the methodology described above can be used to estimate potential biases between instruments, we design a simulation study wherein an ensemble of synthetic upper-air temperature time series is generated using a stochastic process. For each member of the ensemble, interlaced measurements for two instruments are obtained by adding a systematic measurement uncertainty (i.e. bias) for each instrument plus some random measurement noise. As the instrument biases are known, their difference Δ is also known. The questions to be answered in this study are the following.

1. Can a combination of interlaced measurements, together with an adequate statistical model, be used to estimate the difference in instrument biases?

2. If so, how effective is this estimation compared to an approach requiring dual measurements?

An analysis of the 300 hPa temperatures measured by radiosondes at Lindenberg, Germany forms the basis for this simulation study. After subtracting the seasonal cycle, the temperature anomalies show a variance of about ${\mathit{\sigma }}_{\text{anomalies}}^{\mathrm{2}}=\mathrm{10}{\mathrm{K}}^{\mathrm{2}}$ and can be adequately described with an AR[1] process as in Eq. (6) with a∼0.5. To provide a realistic synthetic time series for analysis, we use driving Gaussian white noise $\mathit{\eta }\sim \mathcal{N}(\mathrm{0},{\mathit{\sigma }}_a^{\mathrm{2}}$) with variance ${\mathit{\sigma }}_a^{\mathrm{2}}=(\mathrm{1}-a^{\mathrm{2}}){\mathit{\sigma }}_{\text{anomalies}}^{\mathrm{2}}$. This choice of ${\mathit{\sigma }}_a^{\mathrm{2}}$ ensures that the anomaly variance is fixed at ${\mathit{\sigma }}_{\text{anomalies}}^{\mathrm{2}}=\mathrm{10}{\mathrm{K}}^{\mathrm{2}}$ independent of the value of a. This is necessary as we vary the persistence parameter (i.e. the autocorrelation coefficient) $a\in (\mathrm{0},\mathrm{1})$ to study time series with different persistence but identical anomaly variance.

The synthetic temperature series is generated using Eq. (9) that includes a seasonal cycle and a realization of an AR[1] process. The instrument biases in Eq. (9) are prescribed at $c_A=-\mathrm{0.1}$ K and c_B=0.2 K and are added to the time series together with a measurement uncertainty being specified as Gaussian white noise $\mathit{ϵ}\sim \mathcal{N}(\mathrm{0},{\mathit{\sigma }}^{\mathrm{2}})$. The resulting two time series for instruments A and B are combined to (a) a synthetic time series of dual measurements and (b) an interlaced observational counterpart. The difference in instrument biases between the two time series is prescribed as $\mathrm{\Delta }=c_A-c_B=-\mathrm{0.1}-\mathrm{0.2}=-\mathrm{0.3}\mathrm{K}$. To investigate the influence of (i) persistence in the temperature series, (ii) measurement noise, and (iii) the number of measurements on our ability to estimate the difference in biases between two instruments, the following parameters are prescribed and controlled in our study:

leading to $\mathrm{6}\times \mathrm{7}=\mathrm{42}$ combinations, i.e. 42 synthetic time series to be analysed. The instrument noise is fixed at σ²∈ 0.1. To generate a synthetic time series for a given a, N, and σ, the following steps were taken.

1. Generate a time series of length N consisting of an annual cycle and a realization of an AR[1] process as described above.

2. Add an offset of −0.1 K (instrument bias of instrument A) and Gaussian noise with variance σ²=0.1 to produce a synthetic time series for instrument A.

3. Add an offset of 0.2 K (instrument bias of instrument B) and Gaussian noise with variance σ²=0.1 to produce a synthetic time series for instrument B.

4. Select measurements from A for odd days and from B for even days to generate an interlaced time series.

5. Repeat steps 1 to 4 many times (e.g. M=1000, where M denotes the number of repetitions) to generate 1000 synthetic time series to derive statistically robust estimates of $\widehat{\mathrm{\Delta }}$.

The difference in instrument biases is then estimated based on

1. the calculated mean values of N dual measurements (Eq. 10), i.e. N measurements for A and N measurements for B made simultaneously, and

2. results from the statistical model (Eq. 12) using the time series of N interlaced measurement, i.e. N∕2 measurements for A and N∕2 measurements for B.

Figure 3Box and whisker plots of bias estimates ($\widehat{\mathrm{\Delta }}$) against the number of interlaced flights N (50 flights means 25 flights of instrument A and 25 flights of instrument B) as derived from M=1000 simulations using an autocorrelation coefficient of a=0.5 (a), a=0.8 (b), and a=0.9 (c) and a measurement noise of σ²=0.1. The boxes show the inter-quartile range. The upper and lower whiskers represent the maximum (excluding outliers) and minimum (excluding outliers). Suspected outliers are shown as dots and are located outside the fences (“whiskers”) of the box plot (e.g. outside 1.5 times the inter-quartile range above the upper quartile and below the lower quartile). The true difference in biases $\mathrm{\Delta }=-\mathrm{0.3}\mathrm{K}$ is marked with a red line.

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Figure 4SD of $\widehat{\mathrm{\Delta }}$ against the number of flights N for different AR[1] coefficients a. The black solid line represents the reference experiment with dual flights of instruments A and B, i.e. 2 N measurements. To compare the results from the dual flights (black solid line) with the results obtained from interlaced flights, the number of dual flights has to be doubled. Note the logarithmic vertical scale.

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