In this section we describe how quantisation error, ∆Q, arises when a Mode-S EHS transponder processes a time-varying analogue waveform, C (t), into a digital waveform, Zm(t). This process is referred to as analogue-to-digital conversion (ADC), where the
waveform is an electrical voltage. We suggest that it is the effect of this processing that gives rise to the oscillations observed in the Mode-S EHS observations shown in figures 3-6 (page 49) and 3-7 (page 50). A detailed discussion on ADC is beyond the scope of this project (see Begueret et al. (2008) or Pelgrom (2017, Ch. 8) for a description). However, a part of the process is described.
There are three steps for the Mode-S EHS processing:
(a) The first step is the conversion from an analogue waveform, C (t), to a sampled waveform S (tk),
S (tk) = C (t) Ik for sample k = 1, 2, 3, ...., (3.11)
where ts is the time between samples. The term Ik is an indicator function given by,
Ik=
(
0 for t − kts6= 0
1 for t − kts= 0.
(3.12) (b) The sampled waveform is passed through to a flash ADC where it is converted
to a binary number. The flash ADC is an electronic circuit composed of a series of state comparators. The output of a state comparator is a binary value. For further details on the ‘flash’ ADC and state comparators see Horowitz & Hill (2015, Ch. 13) or Pelgrom(2017, Ch. 8) .
Mathematically, the ADC binary encoding of the sample S (tk) is in terms of the
ADCs quantisation step size, , which is given by (Pelgrom 2017, p. 92),
= |Smax|
2n , (3.13)
where |Smax| is the maximum amplitude of the continuous-time waveform, |C(t)|, and
n is the number of binary bits used by the ADC. The ADC process causes the sample S(tk) to be rounded-up or rounded-down by up to ±0.5. (Note that if |Smax| <
|C(t)| then this results in distortion of the digital waveform called clipping, i.e., when |C(t)| > |Smax| then S (tk) = |Smax|. However, ADCs are usually designed to avoid
this problem.) The ADC converts the sample S (tk) to binary form Zn(tk) (Pelgrom
2017, p. 92), Zn(tk) = n X p=0 bp2p, (3.14)
where bp is an indicator that represents the state of the ADC at the binary bit position
p,
bp=
(
1 ADC state ‘on’
0 ADC state ‘off.’ (3.15)
(c) The final step in the Mode-S EHS processing is truncation, which reduces the number of bits from n to m. This step uses the first m higher-order terms from Zn(tk)
so the truncated value is,
Zm(tk) = n X p=n−m bp2p, (3.16) where m < n.
The quantisation error is then the difference between the analogue input value and the digitised output value (Widrow et al. 1996, Bennett 1948),
∆Q = C(t) − Zm(t), (3.17)
where Zm(t) = Zm(tk) for kts < t < (k + 1)ts,i.e., Zm(t) remains constant until the
next sample is taken (this step is called ‘sample and hold’).
distribution on the interval [a, b] (Widrow et al. 1996),
F (∆Q) = ( 1
b−a a < ∆Q < b
0 elsewhere (3.18)
the mean and variance of its distribution are given by Walpole et al. (2011, Ch 6) as, E(∆Q) = b + a 2 (3.19) and V ar (∆Q) = (b − a) 2 12 (3.20)
respectively. When a = −12 and b = 12 then the mean E(∆Q) = 0 and the standard deviation of quantisation error is given by,
σ =pV ar (∆Q) = s 2 12 = √ 12. (3.21)
We hypothesize that it is the quantisation effect from ADC used by the Mode-S EHS processor that causes the step changes observed in the Mach number and true airspeed, as shown in figures 3-6 (page 49) and 3-7 (page 50). Furthermore, we suggest that this variability can be expressed as a contribution to observation errorusing the standard deviation of quantisation error.
3.10
Summary
In this chapter we have described how NATS use a network of SSRs to actively inter- rogate individual aircraft to requestMode-S EHSreports of the aircraft’s state vector. We have used a sample of Mode-S EHS data from the NATS archive to construct ver- tical profiles of temperature and wind speed. We create these profiles by aggregating the Mode-S EHS reports into altitude bins within a defined region around London Heathrow airport. We have used these profiles to identify the occurrence of meteoro- logical features, the temperature inversion and the low-level jet. We suggest that these may provide information that may be useful for operational meteorological forecasting and for numerical weather prediction. However, we note that quality of the obser- vations requires further investigation. We suggest that quantisation error is a major contributor to the estimation of error in the aggregated observations and which affects the quality, and therefore, the utility of the resulting derived observations.
the studies in the rest of this thesis. In chapter 4 we address our first thesis question
aboutthe accuracy of the measurements for horizontal wind and ambient temperature.
We use thestandard deviationof quantisation error to develop and validate error models for the Mode-S EHS observations. We use reports collected by the Met Office Mode-S EHS receiver network described in chapter 5 to address our second thesis question in chapter 6, what atmospheric phenomena within the boundary layer can be observed from using observations derived from these routine messages. For this question we construct vertical profiles for ambient temperature, and we show that quantisation is still a dominant source of error. In chapter 7 we address the third thesis question on the benefit that assimilation of these high frequency observations brings to the Met Office UKV numerical weather predictions. We suggest that the quantisation error remains a significant source for the observation error used in data assimilation.
Chapter 4
Comparison of Aircraft Derived
Observations with in situ
Research Aircraft Measurements.
We acknowledge that sections 4.2 to 4.9 inclusive are extracts from the paper published by Mirza et al. (2016).4.1
Introduction
This chapter investigates the errors of Mode-S EHS derived observations resulting from the limited precision of the transmitted aircraft state vector. We do this using in situ recordings made with research-grade high precision instruments aboard the Facility for Atmospheric Airborne Measurements (FAAM BAe-146) (Smith & Gratton 2004). Unfortunately the FAAM aircraft does not have a Mode-S transponder so we emulate the processing of the aircraft’s state vector, using the in situ research observations, so as to generate Mode-S EHS type reports. We then derive the temperature and wind and compare with the actual in situ observations. We also derive novel error models based on a consideration of Mode-S EHS processing and validate these models using the in situ observations.
This chapter is organized as follows: in section 4.2 we describe the FAAM aircraft, the data parameters available from its avionics systems and how we emulate Mode-S EHS reports using data recorded by the FAAM BAe-146. Section 4.3 describes the instruments used to obtain the in situ observations and details the six case studies to be examined. We evaluate one case study in detail and describe the methodology used for the analysis. Section 4.4 defines our notation and metrics for our quality assessment
of the derived temperature, section 4.5, and horizontal wind, sections 4.6 and 4.7. We derive and apply our error models for the Mode-S EHS processing. Section 4.8 presents the results for all of the cases studied, which suggest that theobservation error standard
deviation for temperature increases from 2 K at 10,000 m to 4.5 K near the surface,
an order of magnitude greater than AMDAR’s 0.4 K (Painting 2003). For the cases studied,the observation error standard deviation for horizontal wind is up to 0.5 ms−1 compared with AMDAR’s 2 to 3 ms−1 (Painting 2003). These results are shown to be due to the reduced precision of the aircraft state vector that results from Mode-S
EHS processing. Section 4.9 summarizes the findings of this work and concludes that
horizontal wind derived from Mode-S EHS observations may have practical applications for high-resolution NWP, while derived temperatures may be aggregated from multiple aircraft to provide useful information. However, this requires further investigation to assess how to minimize their errors. Finally, in light of the findings in thischapter, we note that direct reports of higher precision data would be preferable.