identification and correction of clock asymmetry in the ers–1 and ers–2 radar altimeters
TRANSCRIPT
IDENTIFICATION AND CORRECTION OF CLOCK ASYMMETRY IN THE ERS–1 AND ERS–2 RADAR ALTIMETERS
Richard Francis(1), Annalisa Martini(2), Mònica Roca(1), Pierre Féménias(2), Sandra Mingot(1)
(1)ESA/ESTECKeplerlaan 1,
2200AG Noordwijk,The Netherlands
e-mail:[email protected] [email protected] [email protected]
(2)ESA/ESRINvia Galileo Galilei,I-00044 Frascati,
Italye-mail:
[email protected] [email protected]
INTRODUCTION
During the Commissioning Phase of ERS–2 a relative calibration exercise was performed in which ERS–1 and ERS–2altimeter data, obtained with a one-day separation in time, were compared. Many groups noticed a step change in the rel-ative bias of 3–4 cm which occurred on 1 August 1995, and closer examination of the data-set revealed others. An exampleof the relative bias estimate, revealing a number of such step-changes, or jumps, is shown in Fig. 1.
This event of 1 August 1995 was associated with a change in parameter Look-Up Tables in the ERS–1 Fast Delivery proc-essor, although this eventually turned out to be pure coincidence. The true reason for the jumps in bias is a series ofchanges in the on-board hardware, each one associated with the instrument being switched off, as we will demonstrate.Fortunately the instruments on both ERS–1 and ERS–2 had been operated in a special characterisation sequence at fre-quent intervals since launch, and this sequence contained sufficient raw measurement data to enable corrections to becomputed. A first version of the corrections was implemented in 1995 and regular updates have been made available, on-line, since then. More recently an extended modelling of the internal hardware and software operations has enabled us toimprove the accuracy of this correction.
Rela
tive
bia
s [c
m]
–10
–5
0
5
10
Time of ERS–2 pass [days of 1995]
KirunaGatineau
MaspalomasPrince Albert
130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310
[after R. Scharroo]
Fig. 1: Relative bias between ERS–1 and ERS–2, using collinear tracks, from QLOPR data. The ground station which generated thedata for each track is identified and the heavy and light lines indicates the mean and standard deviation for each day,respectively. Several “bias jumps” are visible, including the event of 1 August 1995, at day 213.
INTERNAL CALIBRATION AND RELATED MEASUREMENTS
The altimeter is able to generate several types of signals useful for calibration and characterisation by disabling the track-ing loops and directly controlling the range window instead. These include:
• measurement of system noise by setting the range window at some intermediate point between the satellite andthe Earth, where there is no echo;
• setting the range window at the effective location of the transmitter, in order to use the transmit signal to generatea Point Target Response (PTR);
• slowly sweeping the range window past the effective transmitter location to characterise the entire range window;a technique called Scanning Point Target Response (SPTR).
The first of these is not relevant in this context so we will not describe it. The other two types of signal are similar, andwe describe them here. The PTR is the basis of the internal calibration (called open loop calibration, OLC), and measure-ments using the SPTR technique enable the bias jumps to be corrected. The internal calibration makes a relative calibrationof time-dependent variations in the instrument measurements, caused by thermal variations around the orbit and ageingeffects. It is not is not an end-to-end absolute calibration.
Both of these signal generation techniques are based on similar hardware and software functions so the PTR, or OLC, isdescribed first, in some detail. The minor differences associated with the SPTR are then described.
Point Target Response
The essential elements of the PTR signal generation are shown in Fig. 2, the block diagram of the altimeter. The transmit-ted chirp signal (TX) is delayed by the 25 µs delay line, which is longer than the 20 µs duration of the signal, and afteramplification in the transmitter is directly injected into the receiver chain through the calibration coupler. After a suitabledelay a receive trigger (called RX trigger) is used to generate the deramping chirp which is routed to the deramp mixer.There the two chirps are mixed to generate a tone whose frequency depends on the time delay between the arrival of thechirps, as follows:
(1)
where ∆f is the frequency difference with respect to the Intermediate Frequency (IF), ∆t is the time delay, B is the chirpbandwidth and T the chirp duration (20 µs). The IF, which is introduced for technological reasons, is provided by the
Error Signal
FFT andPower
Averaging
Alpha-Beta Tracker
ADC
ADC
I
Q
fine2nd LocalOscillator
Deramp LocalOscillator
PhaseShift
DerampMixer
Front-EndElectronics
ChirpGenerator
f
t
f
t
f
t
TX trigger
RX trigger (coarse)DelayLine
Amplifier
Fig. 2: Block diagram of internal calibration. The transmit chirp is delayed and passed via the calibration coupler (shownwithin the box representing the chain of circulators in the Front-End Electronics) to the deramp mixer. Here thesignal is mixed with the second chirp signal from the chirp generator, triggered by the RX trigger. The resultingsignal is a constant-frequency tone which appears in the FFT as a point target.
∆f ∆tBT---=
different centre frequency of the deramping chirp compared to the TX chirp, which is produced by mixing it with a LocalOscillator signal. The time delay quantum is the period of the 80 MHz clock which is used for the RX trigger timing; i.e.12.5 ns. The chirp bandwidths are 82.5 MHz and 330 MHz for ice and ocean respectively. Thus the smallest real frequencyinterval between tones is 51.56 kHz and 206.25 kHz respectively.
In the frequency domain the resulting signature of the transmit pulse, the Point Target Response, is a sinc function (i.e.). The hardware digital signal processor samples the video signal and computes the 64 spectral components by a
FFT, using decimation in time. A squared modulus extraction of the complex components is performed and these powervalues are averaged over 50 consecutive pulses. Before the FFT, a signal weighting is applied to account for the finitewindow width: a Hamming window is used, as defined in eqn (2), where α = 0.54.
(2)
This weighting broadens the spectral lines and reduces the sidelobe amplitude; in theory to –43dB, with a 30% main lobewidening. The effect of this is to convert the sinc function of the point target response into a gaussian function.
The PTR for both ERS–1 and ERS–2, is shown in Fig. 3. In order to show this gaussian signal, however, we generatedthis example by the SPTR technique, described in the next section, since the PTR would only have generated 3 or 4 points,at the FFT samples.
Scanning Point Target Response
The PTR technique, which is used for the OLC, only generates measurements at the FFT samples, only 3 or 4 of whichare non-zero. The data telemetered are fully processed on-board so the input data are not available, otherwise zero-paddingcould be used to generate the full PTR signal.
In contrast to this limitation of the PTR, the SPTR is able to sample the full shape of the point target response. This is doneby slowly sweeping the range window through the transmit signal, from end to end, commanded as a very small rate term.This small commanded height rate is realised in the altimeter as a combination of coarse and fine height words. The coarseword selects different RX trigger values while the fine word selects different fine shifts of the FFT. By unwrapping these
xsin x⁄
W H n( )α 1 α–( )–
2πnN
---------- cos for 0 n N 1–≤ ≤
0 elsewhere
=
1000
800
600
400
200
0
Pow
er
6456484032241680Filter Number
1000
800
600
400
200
0
Pow
er
6456484032241680Filter Number
Fig. 3: Point target response measured in flight on ERS–1 (upper panel) and ERS–2 (lower panel) in June 1999. In each case the RXtrigger value is 2496. The slightly different position of the signal is due to differences in the delay line length.
fine shifts and relocating the measurements back to the position of the RX coarse trigger the detailed shape, and relativeposition, of the transmit signal at each position of the range window can be measured.
We describe this in considerable detail in Annex 1: Details of the SPTR Algorithm, when we describe all the characteris-tics of the instrument which are modelled in the new algorithm.
Open Loop Calibration
In generating the PTR used for the open loop calibration (OLC) a significant interval must be allowed between the TXand deramping chirps, so that both arrive at the deramping mixer at approximately the same time, due to the presence ofthe delay line. The actual value chosen for internal calibration is about 31 µs, corresponding to 2496 cycles of the 80 MHzclock. This is illustrated in Fig. 4. The PTR waveform is used to calculate the internal time delay, τC, as follows:
τC = (Nf - 32) µ + κ (3)
where Nf is the derived centre of the PTR, µ is the conversion factor from filter units to time and κ is the delay time to therange window position (2496×12.5 ns). The internal time delay computed in this way, is subtracted from all altimeter data.
We present an representative time series of the daily means of internal calibration results from ERS–1 in Fig. 5. ERS–2shows similar behaviour. The internal calibration value has a drift component as well as a series of abrupt changes. It wasevident early in the life of ERS–1 that these step changes were occurring at times when the instrument was switched off.
The delay line is the major contributor to the internal delay. It is fabricated as a Surface Acoustic Wave device in whichthe substrate is a crystal of lithium niobate (LiNbO3) of dimensions 8 × 1 cm; it is 1mm thick. Mechanical expansion caus-es it to be temperature sensitive. Therefore it is maintained in a temperature controlled environment in which the round-orbit temperature variations are less than 0.05°C. The step changes when the instrument was switched off were attributedto slight variations in the repeatability of this controlled temperature (of order 0.1°C).
The abrupt changes in the real internal path length have a magnitude of several centimetres; but these are corrected by theinternal calibration and are not directly responsible for the bias jumps.
However, we will show that the calculation of the internal time delay, τC, is the true source of the bias jumps, because:
• clock asymmetry causes systematic errors in κ;• distortions in the PTR shape causes systematic error in determining Nf;• changes in the chirp slope cause systematic errors in µ
and that these errors can change if the instrument is switched off, particularly if it is allowed to cool down significantly.It is these discrete changes in the errors in determining the internal delay which is the source of the bias jumps.
TX
τc
Internal delay (calibration length)
32
κ = 2496 × 12.5 ns
1 2 24962495
2494
Nf - 32
τc = κ + µ × (Nf - 32)
3 4 52497
Nf
Fig. 4: The calibration of internal time delay is performed by placing the range window at a position, κ,where the PTR will appear, and estimating the position of the PTR in the window, Nf. The twoterms are added, converting the units in the range window (FFT filters units) to range by a factor,µ, which is the chirp slope.
CLOCK ASYMMETRY
The operation of the altimeter is dependent on a high-speed digital counter which performs the radar timing. During pre-launch testing we found that part of this electronics had a systematic asymmetry which introduced a small repeatable tim-ing error. The magnitude of this was ameliorated by modifying the circuitry, but the effect did remain. This effect hasbecome known as clock asymmetry, which causes systematic errors in κ, as described above. In this section we describethe asymmetry and how it is measured.
Generation of Internal Timing Signals
The altimeter timing is referenced to an ultra-stable internal frequency source, or clock. The fundamental frequency ofthis oscillator is 5 MHz, but a number of other phase-locked frequencies are generated from it, including a signal at80 MHz which is used as the time reference in the digital electronics.
The 80 MHz clock frequency has a period of 12.5 ns, which is the “tick” of the clock. The trigger signal which generatesthe chirp at the time the echo is expected to return, called the RX Trigger, is synchronous with this tick. The number of80 MHz cycles between the transmit pulse and the RX trigger is the coarse part of the height word, with a resolution of12.5 ns. The fine part of the height word, with a further 8 bits of resolution, is used to generate a virtual time shift of theecho by frequency shifting the fourier transform of the echo later in the processing. The count of 80 MHz cycles in theRX coarse word is shown in Fig. 6.
Unfortunately, for technological reasons, the 80 MHz clock could not easily be used in the digital circuitry in the timingsection of the altimeter. Although (in the late 1980’s, when the design was made) some components were available whichcould operate at that speed they were very power consuming, and many important components were not available. There-fore the simple concept of counting 80 MHz clock cycles had to be modified in the design, and an approach was developedin which the majority of the circuitry operates at the significantly lower speed of 20 MHz while still retaining the timeresolution of the 80 MHz clock. This was done by synchronising the phase of four different 20 MHz signals to the 80 MHzclock and selecting one of them. The important elements of this design are shown in Fig. 7.
A 20 MHz signal is made by frequency division of the 80 MHz clock; this is not shown. The resulting 20 MHz signal isapplied to the input of a 4 cell shift register which is clocked by the 80 MHz signal. At each clock cycle the state of the
4680.50
4680.45
4680.40
4680.35
4680.301600155015001450140013501300
Ope
n Lo
op C
alib
rati
on [m
]
Days since Launch [d]
Fig. 5: Internal calibration results from ERS–1 during 1995, presented as daily mean values. The minimumand maximum values during each day are shown as lighter points. Times when the instrument wasswitched off are indicated by vertical marks.
input (i.e. the 20 MHz signal) is read into the first cell and the existing values all ripple down the register. An output istaken from each of the 4 cells by reading the state of the cell at the clock transition onto an output line. It is clear fromFig. 7 that the result of this is to generate 4 new 20 MHz signals with phase shifts between them: every 80 MHz transitionhas an associated 20 MHz transition, which may be selected by the switch (which is actually a multiplexer device).
The RX coarse trigger value is a 16 bit word with a resolution of 12.5 ns. The 14 most significant bits (MSB) of this valueare loaded into a counter while the 2 least significant bits (LSB) are applied to the switch to perform the selection of thewanted phase. This phase of the 20 MHz signal is then used for the whole duration of the count to decrement the countercontaining the rest of the RX coarse value. When this reaches zero a pulse is sent to the RF electronics to generate thederamp chirp.
Imperfections in the Circuit
Very late in the test program of ERS–1 we found that the four phases of the 20 MHz clock were not generated synchro-nously with the 80 MHz clock, and had systematic deviations of the leading edge by almost one nanosecond (equivalentto about 15 cm of range!). This was caused, partially, by the internal design of the shift register chip which did not provideequal delays between its cells. There was also a contribution from slight variations in the threshold switching voltage dur-ing the rising edge of the digital signals which was compounded by the effects of electrical loading in this high-speedcircuitry. The nature of the problem is schematically illustrated in Fig. 8.
Fig. 6: The RX Coarse value is an integral number of 80 MHz clock periods.
f
t
f
t
TX RX
RX coarse
Shift Register
20 MHz input
80 MHz Clock
RX Coarse
Counter
RX Trigger Output
Switch
MSB LSB
0
1
0
1
20 MHz Clock Phase
1
1
0
0
1
0
1
0
0
0
1
1
0
1
0
1
Fig. 7: Generation of 20 MHz signals to give an effective resolution equal to a 80 MHz signal, by using a shiftregister as a phase shifter. The appropriate phase is then selected by a switch controlled by the two leastsignificant bits of the height word.
Clearly the circuit did provide RX coarse trigger signals with an interval of 12.5 ns on average, as otherwise the entirecircuit would become unsynchronised, which did not happen. Measurements in the laboratory showed that the effect wasstable and repeatable and had a modulo 4 behaviour, as we would expect from its generation mechanism.
In early 1999 Alenia Spazio provided us with components from the same batch which had been used to manufacture theERS–1 and ERS–2 instruments. A series of tests were performed in the Component Division at ESTEC [1] which con-firmed the existence and stability of the relative offsets between the different shift register ouputs. The offsets are in therange 350 to 750 ns, and these on-ground component measurements compare extremely well with our in-flight system-level measurements, as we will present in Fig. 11, later.
Under intense schedule pressure the contractor, Selenia Spazio, found and implemented a technical solution in which theselected 20 MHz phase was re-synchronised with the 80 MHz clock by a flip-flop, before being used in the counter. Thistechnique was able to reduce the magnitude of the errors, although they still remained at a significant level. It should benoted that the contractor (now called Alenia Aerospazio) has introduced further improvements for the RA–2 which willfly on EnviSat, and this will have negligible errors due to this cause.
Although this concept of re-synchronisation appears simple in principle it was rather delicate in practice. To illustrate thiswe show in Fig. 9 traces of one of the 20 MHz signals and the 80 MHz clock as measured during the development of thecircuit modification. The real signal characteristics are very different from the clean digital signals drawn schematicallyin Fig. 8. These differences are due to the high frequency of the signals – the circuit tracks are behaving as transmissionlines and the loading on them is critical to the conditioning of the signals. In fact much of the effort in developing themodification was in optimising this loading.
All the measurements we made of this behaviour, during the development of the circuit modification and the ERS–1 Com-missioning Phase, indicated that it was stable. We also knew that it would not be possible to produce an undistorted PTRby the SPTR technique unless the circuitry was stable — each PTR requires about 3000 consecutive RX trigger pulseswith the same value.
Despite these indications, however, we will show that this assumption of stability was wrong. Abrupt changes in the pat-tern of asymmetry do occur in certain circumstances which were not experienced prior to launch or very often in the lifeof ERS–1. These events occur when the instrument is switched off and allowed to cool down significantly, as we willdiscuss later.
Shift Register
20 MHz input
80 MHz Clock
RX Coarse
Counter
RX Trigger Output
Switch
MSB LSB
0
1
0
1
Re-Sync
Fig. 8: Imperfections in the 4 clock phases and the method of re-synchronisation. Each phase from the shiftregister is delayed with respect to the master clock, by varying amounts. The values shown arerepresentative of the real delays, magnified by a factor of 4 compared to the clock period.
Measurement of Clock Asymmetry
The clock asymmetry may be characterised by processing the SPTR measurements, as the point target response sweepsthrough the filterbank, and determining the centre of each of the point target responses in the filterbank. Each PTR is as-sociated with a distinct value of RX trigger so variations in their timing will appear as uneven spacing in the filterbank.
An example is shown in Fig. 10. Each of the PTR’s in the filter bank is generated by a successive value of the RX trigger.The separation between them should be 12.5 ns, or 4.125 filters. To characterise the asymmetry the actual spacing is meas-ured and the relative position of individual RX trigger values is determined. In Fig. 11 we show the differences of theindividual measured PTR’s from the linear regression. this figure also shows the results of the component tests, displayedas equivalent PTR positions. The modulo-4 behaviour of the instrument is visible, but masked by the effect of the re-syn-chronisation circuit describe earlier and by inaccuracies in the recovery of the true position of the PTR’s.
The development of the algorithm to faithfully generate the PTR’s and recover their positions has formed a major part ofthe development of our new approach. The details are too complex to be included in the main body of this text and weprovide them in Annex 1: Details of the SPTR Algorithm.
This analysis provides only the relative offsets of the RX trigger pulses; however we know that the frequencies are derivedfrom a stable 80 MHz oscillator so the offsets must have zero mean. If we had sufficient samples the regression linethrough the measured positions would provide this mean, and the offset of the pulse 2496 from this regression line couldbe absolutely determined.
Fig. 9: Traces recorded during the development of the circuit re-synchronisation. The upper signal isone of the 20 MHz phases and the lower is the 80 MHz clock.
12.5 ns
1200
1000
800
600
400
200
0
PTR
Powe
r
706050403020100-10Filter Number
Fig. 10: Point Target Response of the ERS–2 altimeter measured in June 1999 by the SPTR technique. A distinct PTR is generated foreach RX trigger value ranging from 2489 at the right to 2506 at the left. The pulse corresponding to RX trigger 2496 ishighlighted. The transfer function of the anti-alias filter has modified the power of the signals from each RX trigger.
Referring back to Fig. 4 it is evident that the true value of κ is 2496×12.5 ns plus the offset of the pulse 2496 from its “true”position. Thus the offset of the pulse 2496 from the regression line provides the error in κ, and thus a correction.
Due to the limited width of the filterbank only about 15 recovered RX trigger positions (ie PTR positions) are availableand this reduces the confidence in the determination of the regression. We performed several experiments in fitting theregression line but have found that the best and most stable results are obtained by using an unweighted fit to the full en-semble of available measurements.
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2Filt
er P
osit
ion
(wit
h re
spec
t to
nom
inal
slo
pe)
25042502250024982496249424922490RX Coarse Value
Fig. 11: For each of the PTR’s shown in Fig. 10 the position in the filterbank has been determined. This plot shows, as greydots, the residuals between these extracted positions and the ideal values considering the nominal separation of 4.125filters, for the nominal chirp slope of 330 MHz / 20 µs. The overall gradient of the residuals is due to the deviation ofthe actual chirp slope from the theoretical value. The component-level test results (converted to equivalent residuals)are shown by open circles, revealing the modulo-4 behaviour as indicated by the dotted blue lines.
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
Devia
tion
from
Nom
inal
Chi
rp S
lope
[filt
ers]
25042502250024982496249424922490
RX coarse value
y
2 Oct 97 5 Oct 97 15 Oct 97 18 Oct 97 21 Oct 97
25 Oct 97Anomal 28 Jan 98 7 Feb 98 10 Feb 98 19 Feb 98
10 Mar 98 14 Mar 98
Fig. 12: Residuals of the PTR positions for a series of SPTR measurements, before and after an anomaly. The positionsare unmarked, but joined by solid lines. The marked points are the respective average positions of each RXtrigger value for each set. The regression lines, which are proportional to the chirp slope are also shown.
The stability of this behaviour in the absence of a switch-off of the instrument is striking; in Fig. 12 we show the recoveredpositions of the PTR’s associated with the successive RX trigger positions for 6 SPTR measurement campaigns spreadover a month before an anomaly, during which the instrument was switched off and allowed to cool down, and 6 sets ofresults spread over more than a month after the event. The internal consistency of the sets of results, as well as the differ-ences between them, is a clear indication of the abruptness of the internal change of the instrument which occurs at ananomaly.
ERRORS IN THE OLC POSITION DUE TO POINT TARGET RESPONSE DISTORTION
Operational Processing of Open Loop Calibration
We described above, in eqn (3), the means by which the internal delay, τC, is computed from the OLC measurement. Ineqn (3) the derived centre of the PTR, Nf, plays an important role. In this section we will examine the errors in this deter-mination due to the imperfect shape of the PTR.
In operational processing the evaluation of the PTR position in the filter bank is made by means of the three-point fit al-gorithm, which assumes that the PTR is a perfect gaussian function. The three-point fit algorithm fits an exact gaussianshape to three points, providing f0, the centre of the fitted gaussian, in filter units. The first step is to find the 3 consecutivefilters with the maximum signal. Let the values be V
1, V
2 and V
3, in filters f
1, f
2 and f
3. Then,
z1
= loge V
1z2
= loge V
2z3
= loge V
3
The centre of the three-point gaussian fit to the OLC is then obtained by the following equation:
(4)
An example of measurements around five orbits, from ERS–1, is shown in Fig. 13. The around-orbit variations, of order1 cm peak-to-peak, are corrected by the results of the three-point-fit in the Fast Delivery and off-line processing.
We have developed an empirical fit to this behaviour which, although unused in the present work, is described in Annex 2:Round-Orbit Behaviour of OLC.
f 0
z3 f 22 f 1
2–( ) z2 f 12 f 3
2–( ) z1 f 32 f 2
2–( )+ +
2 z2 f 1 z3 f 1– z1 f 2– z3 f 2 z1 f 3 z2 f 3–+ +( )-------------------------------------------------------------------------------------------------------=
0.60
0.55
0.50
0.45
0.40
Ope
n Lo
op C
alib
rati
on [n
s]
0.900.850.800.750.700.650.600.55Fractional Part of Day [d]
OLC value Model Function
1 cm
Fig. 13: Internal calibration results around five orbits (left and bottom scale). These data are from ERS–1 on30 August 1992; ERS–2 shows similar behaviour. The variation around the orbit is caused bythermal effects, primarily in the delay line. The solid line shows a fitted function, described inAnnex 2: Round-Orbit Behaviour of OLC, based on sinusoid decomposition.
Distortions in the Point Target Response
The three-point-fit algorithm, described above, would accurately find the centre of the PTR, Nf, if the PTR signal weregaussian. This would require that the point-target response, prior to the hamming weighting in the on-board FFT was atrue sinc function.
Prior to launch the PTR response at hardware level was measured; the test point is unmarked in Fig. 2, being at the outputsof the Chirp Up-Converter and Amplifier (CUCA), which is not included in the figure. However it provided a measure-ment equivalent to a measurement at the Deramping Mixer. The results of this measurement are shown in Fig. 14.
The sidelobes of the measured PTR are significantly higher than the theoretical sinc response, also shown in Fig. 14. Thereare also no true nulls visible in the spectrum. These observations indicate the presence of spurious signals, the most likelysource of which is ripples in the phase and amplitude of the chirp. Indeed such ripples were known to be present.
The spurious signals have an effect on the PTR, both during the SPTR and the OLC measurements. This can be clearlyseen in Fig. 15 where the point target response, represented as points determined by the SPTR technique, are shown on alog scale. They clearly deviate from a fitted gaussian function, but they can be well fitted by a family of three gaussians.The additional two gaussians, one on each side and with reduced magnitude compared to the central gaussian, fit the twodominant spurious signals which have also been converted from the sinc function by the Hamming weighting. The addi-tional spurii have, in principle, a fixed offset and relative power compared to the main lobe. With some deviations(attributed to imperfect corrections for the instrument transfer function) this is observed for all the RX trigger values sam-pled by the SPTR technique. However there are clear differences between ERS–1 and ERS–2.
The initial algorithm, used since 1995 to determine the clock asymmetry, handled the distortion of the point target re-sponse caused by these spurii by representing the PTR positions by the centre of gravity (COG) of the PTR shapes asdetermined by the SPTR technique. In our current work the main lobe and spurii are independently determined as a familyof gaussians.
We present typical examples of the fitted functions for both ERS–1 and ERS–2 in Fig. 16. Due to the different character-istic spurii in the two instruments the COG systematically appears on opposite sides of the main lobe centre in each case.
0-100-200-250 -150 -50 50 100 150 200 250Frequency [kHz]
-25
-30
-35
-40
-45
-50
-55
-60
-65
-70
-75
Powe
r [dB
m]
Fig. 14: Point Target Response from the ERS–2 A-chain, measured at the outputs of the Chirp Up-Converter and Amplifier. The dashed line shows the theoretical sinc function
This will immediately imply a shift in the relative bias of the two altimeters when the new SPTR algorithm, which is basedon the main lobe centre, is used. This shift amounts to about 3 cm.
Effect of PTR Distortions on OLC Results
We have shown that the PTR is not a pure gaussian signal, yet the OLC Three-Point Fit algorithm, used operationally sincethe launch of ERS–1 assumes that it is. Furthermore the algorithm is critically dependent on just three measurements andso it can be influenced by hardware effects such as quantisation in the signal processor. We present, in Fig. 17, a typicalexample of the comparison between the reconstituted OLC gaussian fit and the actual pulse shape measured simultane-ously by the SPTR technique. This example shows a reasonable agreement; an earlier study of the effect [2] identified thatdeviations of up to 8 cm can occur; the histogram inset into Fig. 17 summarises these results.
0.1
1
10
100
1000
Powe
r
41403938373635Filter Number
SPTR Measurements 3-Gaussian Fit Single Gaussian Fit
Fig. 15: Detailed Point Target Response from ERS–2, recorded in the filter bank following Hamming weighting andFFT, presented on a log scale. The recovered SPTR measurements are shown in grey, a best-fit single gaussianis shown in blue and the result of fitting a family of 3 gaussians is shown in black. The additional two gaussiansare caused by spurious signals in the PTR.
0.1
1
10
100
1000
Powe
r [FF
T un
its]
43424140393837
Filter Number
COGMain Lobe
Centre
0.1
1
10
100
1000
Powe
r [FF
T unit
s]
41403938373635
Filter Number
COGMain LobeCentre
Fig. 16: Typical fitted point target response functions for ERS–1 (left) and ERS–2 (right).Due to the different nature of thespurious signals in these two instruments the Centre of Gravity of the function is systematically to the right of the centreof the main lobe for ERS–1 and to the left for ERS–2, as indicated schematically.
CHIRP SLOPE VARIATIONS
The final element in the computation of the internal time delay, τC, is µ, the scale factor between FFT filter numbers andtime delay. This parameter is essentially the chirp slope. We have seen, for example in Fig. 12, that the regression of thePTR positions in the filter bank enables us to determine this coefficient µ.
The chirp generator is fabricated as a Surface Acoustic Wave (SAW) dispersive delay line, and, like the delay line, it istemperature sensitive, and so we may expect that it may change its characteristics following a switch off. A typical set ofvalues of the ERS–1 chirp slope, derived from the SPTR regressions, and showing this effect, is shown in Fig. 18.
Thus the SPTR regression analysis enables us to directly measure the coefficient µ.
Powe
r [F
FT u
nits
]
1000
800
600
400
200
0403938373635
Filter Number
SPTR Measurements OLC OLC 3-Point Fit
50
40
30
20
10
0O
ccur
ence
s
876543210OLC offset [cm]
Fig. 17: Reconstruction of the gaussian fit by the OLC Three-Point Fit algorithm, based on the measuredpoints shown. This OLC measurement was made within 30 s of the SPTR measurement of the samepulse 2496, which is also shown. The inset histogram shows the deviation between the OLC andSPTR position for a sample of 240 ERS–2 SPTR analyses.
52500480460440420400380360340320300280260240220200180160140120-4.17
-4.16
-4.15
-4.14
-4.13
Date [days of 1995]
Chirp
Slo
pe [f
ilter
s/cl
ock
cycl
e]
Fig. 18: Values of the ERS–1 chirp slope, determined from the SPTR regression. Times of instrument anomalies are indicated byvertical lines.
CHARACTERISATION OF BIAS JUMPS
Events which cause the instrument to be switched off, typically due to internal software failures (or, rarely, other events)are called anomalies. Anomalies were already known to cause jumps in the delay line length, as we described earlier.
We have processed SPTR data from ERS–2 and ERS–1 during the tandem phase, taking account of the three terms wehave described, to produce corrections for the errors introduced by the application of the simplified three-point-fit OLCalgorithm used in the operational processing. These correction values are shown in Fig. 19. The relationship betweenchanges in the correction value and instrument anomalies is readily apparent. It is also evident that ERS–1 suffers jumpsof smaller magnitude than ERS–2 and its anomalies are less likely to cause jumps.
The bias jumps are almost always associated with switch-offs, but the converse is not true. During the early investigationwe found a weak correlation between the magnitude of the bias jump and the duration of the switch-off. We speculatedthat this was due to a temperature effect. The normal operation temperature of the signal processor sub-assembly (SPSA)is about 35°C internally, while the temperature of the panel on which it is mounted (which is an external wall of the sat-ellite) is about 5°C for ERS–2 and around 8°C for ERS–1. The warmer temperature for ERS–1 is mostly due to thedegradation of thermal control surfaces in the space environment. During an anomaly the temperature of the electronicscools down – the longer the switch-off, the closer the internal temperature will become to the panel temperature. Further-more, the panel itself cools down without power dissipation in the (now switched off) electronics.
Three temperature measurement points are available inside the SPSA but monitoring of these is unavailable during aswitch-off. Immediately after switch-on the measurements become available, polled on a 16 s cycle by the main spacecraftcomputer. We have used the three values immediately following the switch-on to perform a regression back to the timeof switch-on, in order to determine the minimum temperature reached during each event. Furthermore, the minimum ofthe three samples available was taken. The relationship between the minimum temperature reached and the magnitude ofthe associated bias jump is shown in Fig. 20, which includes both ERS–1 and ERS–2.
We did not have correction values available before and after each anomaly, so the number of cases shown in Fig. 20 islimited. Nevertheless there is an indication that jumps are associated with lower minimum temperatures during a switch-off. There is no clear threshold although the differing behaviour above and below 5°C is noticeable.
Following Fig. 20, the apparent immunity of ERS–1 to bias jumps, noted above, could be explained by the higher paneltemperature of ERS–1, which would imply that it is less likely to fall to low temperatures during an anomaly. None of theERS–1 minimum temperatures are lower than 3°C.
The tests made at component level described in [1] included an attempted to detect such temperature dependence. How-ever the test was performed with a low temperature applied during the switch-off for only 5 minutes, instead of the severalhours experienced in-orbit. No change in the clock-asymmetry was observed, except for a transient effect during thewarm-up phase. Due to the unrepresentative nature of this test we have made no conclusion from it.
520500480460440420400380360340320300280260240220200180160140120-2
8
0
Time of SPTR Measurement [days of 1995]
Bias
Cor
rect
ion
[cm
]
4
2
-2
-4
6
4
2
0
ERS-1
ERS-2
Fig. 19: Bias correction values derived by the methods described in this paper, for ERS–1 and ERS–2. Interruptions to the instrumentoperations, typically caused by instrument anomalies are indicated by vertical bars.
BIAS CORRECTION RESULTS
We have established that the jumps in bias in each of the two altimeters is associated with anomalies. We now make theassumption, supported by the observations which we have, that there are no jumps in-between anomalies. This assumptionhas enabled us, from the combination of the known times of anomalies and all the available SPTR measurements, to gen-erate correction values during the sample period. For the intervals where no SPTR data are available we have estimatedthe appropriate bias value by extrapolation of the observed behaviour of the instrument during anomalies, taking accountof the relationships to temperature described above.
In order to perform a first validation of our results we have used a data set prepared by CLS [3] which contains an estimateof the relative sea surface height between ERS–1 and ERS–2 for each day during the tandem phase. This data set, whichis similar to the quick-look data-set we presented in Fig. 1, is shown in Fig. 21. It contains all corrections except the so-called SPTR correction, which is the common name for the correction term we made available on-line following the initialinvestigation.
We present, in Fig. 22, the results obtained when this data set is corrected by the currently available SPTR correction.
-4
-2
0
2
4
Bias
Cha
ge [c
m]
20151050-5Minimum SPSA Temperature [deg C]
ERS–1ERS–2
Fig. 20: Relationship between the minimum temperature in the SPSA and the associatedbias jump, for both ERS–1 and ERS–2.
520500480460440420400380360340320300280260240220200180160140120-10
-5
0
5
10
Date [days of 1995]
SSH
diff
eren
ce [c
m]
ERS-1 Anomalies
ERS-2 Anomalies
Fig. 21: Daily values of the relative bias between ERS–1 and ERS–2, prepared by CLS. Times of anomalies are marked by verticallines, in the upper panel for ERS–1 and the lower panel for ERS–2.
It is evident that the original SPTR correction does not remove the step changes in bias which occur at anomalies, althoughthe magnitude of the steps is reduced.
We present, in Fig. 23, the equivalent results where our new correction values are used. There is a clear improvement inthe removal of steps associated with anomalies. It is also evident that there is a shift in the relative bias, of about 3 cm,which was foreseen in the section “Distortions in the Point Target Response” above.
This has to be regarded as an initial validation, as full validation requires the correction to be applied prior to generationof daily means. Such, more complete, validation exercises are underway at several institutes.
The operational processing of the full data-sets covering the ERS–1 and ERS–2 missions in their entirety has been com-pleted and are available at the following URL: http://pcswww.esrin.esa.it. The results are updated weekly.
At the time of the external calibration of ERS–1 (August and September 1991) the ERS–1 altimeter also suffered from theerrors in the application of OLC that we have described in this paper. The associated error in the OLC was therefore in-corporated into the range bias value which was determined by external calibration. However SPTR data from this periodare not available, as explained in [4], available from the same URL.
ERS-1 Anomalies
ERS-2 Anomalies
520500480460440420400380360340320300280260240220200180160140120-10
-5
0
5
10
Date [days of 1995]
SSH
diff
eren
ce [c
m]
Fig. 22: Uncorrected daily values of relative bias (as presented in Fig. 21) are shown by the grey line. The original SPTR correction,applied to the daily values, provides the results shown by the black line. This is only plotted where there are SPTRmeasurements within the inter-anomaly period. Spikes appear on anomaly days since the correction is applied on a daily basis,instead of taking account of applying the appropriate correction before averaging over each day. Furthermore the date in theSSH dataset is the ERS–1 date, while the ERS–2 collinear track is one day later, and this is not taken into account.
ERS-1 Anomalies
ERS-2 Anomalies
520500480460440420400380360340320300280260240220200180160140120-10
-5
0
5
10
Date [days of 1995]
SSH
diff
eren
ce [c
m]
Fig. 23: Uncorrected daily values of relative bias are shown by the grey line. The correction computed according to the methodsdescribed here, as applied to the daily values, gives the results shown by the black line. Again this is only plotted where thereare SPTR measurements within the inter-anomaly period and spikes appear on anomaly days as described above.
CONCLUSIONS
We have been able to demonstrate that the observed jumps in the relative bias of ERS–1 and ERS–2 are an instrumenteffect, with contributions from both instruments; ERS–2, however, tends to contribute more than ERS–1. We have iden-tified which specific element of the instrument is responsible, and demonstrated that it may be characterised in-orbit. Wehave been able to link the on-board changes to specific events, in which the relevant electronic unit is switched off, andhave characterised the circumstances in which such a switch-off is more likely to cause a bias jump.
A first-order correction scheme was implemented and has been available on-line since 1995. We have now developed andimplemented a full correction scheme, including simulation of detailed instrument operations. These data are also availa-ble on-line.
In the Ice Mode an exactly equivalent process occurs. Ice Mode was ignored in the earlier work as a correction of ordercentimetres was considered negligible in comparison to the reduction in precision of Ice mode measurements comparedto the Ocean Mode. However in recent years the Ice Mode data have been used in more demanding applications and thegeneration of the full set of corrections will be extended to include appropriate Ice Mode corrections.
The initial validation results based on tandem phase results are promising, and a full validation is under way.
REFERENCES
[1] Characteristics of MC10541L ECL Shift Register, Components Division Analysis Report MI0679, Noordwijk, 1999
[2] S. Mingot, Investigation of a Systematic Error in the Range Measurements of the ERS–2 Radar Altimeter, EWP 2070,Noordwijk, 1998
[3] J. Stum, F. Ogor, P-Y. LeTraon, J. Dorandeu, P. Gaspar, J. P. Dumont, An Inter-Calibration Study of Topex/Poseidon,ERS–1 and ERS–2 Altimetric Missions, CLS/DOS/NT/98.070, April 1998
[4] A. Martini and P. Féménias, The ERS SPTR2000 Altimetric Range Correction: Results and Validation, ERE-TN-ADQ-GSO-6001, Frascati, November 2000
ANNEX 1: DETAILS OF THE SPTR ALGORITHM
The new SPTR algorithm has two fundamental improvements over the initial algorithm:
• the modelling of the instrument hardware and software is highly accurate, so that the information in the instrument raw data may be converted into the data points representing the PTR (for example the grey data points in Fig. 15);
• the establishment of the true location of the PTR, by identifying the main lobe of the response independently of the spurious signals.
These are described in this Annex.
Instrument Modelling
Quantisation
The waveform sample is accumulated over 50 pulses. In the case of the SPTR, which is achieved by Preset Tracking, eachof these waveform samples is generated with a slightly different set of RXcoarse and RXfine, and this must be taken intoaccount to obtain the highest accuracy in the results. In the case of the OLC the 50 tones have the same values for RXcoarseand RXfine, but an amplitude weighting is applied to each tone according to a table stored in the SPSA. This is an artifactremaining from an earlier calibration concept which could not be disabled, being realised in hardware. The weighting fac-tors are applied as multiplication factors during the FFT processing.
The 50-pulse averaging is actually performed by dividing the squared modulus by 50 and then accumulating in a memory.This approach was taken to avoid overflow problems in view of the high level of speckle noise on individual waveforms.However it does lead to a quantisation effect in the waveforms with a consequent suppression of small signal levels. Nor-mally the results of this effect are concealed because the echo has a spectrum with random phases and statistically varyingamplitudes. Fifty such waveforms are averaged, and the quantisation errors can be treated as random quantities. Howeverthe SPTR and OLC signals the single tones have deterministic phase and amplitude, and thus systematic errors may occurwhich distort the synthetic waveform. The magnitude of the fractional error due to this truncation depends on the signallevel itself – at higher levels it is negligible. This is illustrated in Fig. 24.
Power levels in the PTR waveform reach about 1000 FFT units, but, based on Fig. 24, at power levels of less than about10 FFT units some significant distortions of the power level may be expected. This can be seen in Fig. 15 where, on theleft side, power falls off, compared to the fitted curve, below 10 FFT units. On the right side the power exceeds the fittedcurve below 10 FFT units; this is attributed to an unmodelled, low-level, spurious signal.
Filter Renumbering
During Ocean Tracking (but not during the OLC sub-mode) an artificial shift is applied to the tracking point of the wave-form, in order to reduce a potential error in the AGC calculation due to the action of the fine-shift mechanism in the FFT.The fine-shifting of the spectrum is done in one direction only (‘rightwards’ in the filter-bank), due to the design of theDSP. If the spectrum were correctly centred when the fine-shift is zero there would be no aliased energy in the ‘left’ partof the spectrum (assuming a rectangular anti-alias IF filter). As the fine-shift increased to maximum the aliased energywould grow in the ‘left’ part, to a maximum of 4.125 filters. In order to reduce the impact of this error the position of thespectrum in the zero-shift case is offset by 2 filters so that the fine-shift effectively becomes bi-directional about the midpoint. The filter-bank is thus centred on filter number 34.
The shift is achieved by a combination of renumbering the filters with an offset of 2, and subtracting an amount equivalentto 2 filters from the RX trigger value sent to the hardware. This is illustrated in Fig. 25.
10
8
6
4
2
0
Powe
r Lev
el [F
FT U
nits
]
1086420Input Power Level [FFT Units]
2.0
1.8
1.6
1.4
1.2
1.0
Fractional Error
Truncated Power True Power Fractional Error
Fig. 24: Effect on signal power of truncation during the waveform averaging. The power axesare given in terms of FFT Units, which are the telemetry units divided by 32.
tekton
The tracking point is at the centre of the leading edge and to maintain this at an offset from the zero frequency in the hard-ware it is necessary to apply the offset, equivalent to 2 filters, in the RX values just before sending to the hardware. Thusthe values of RXcoarse and RXfine which appear in the telemetry are different from the values sent to the hardware whenin Ocean Tracking mode.
In calculating the RX trigger values it will be necessary to implement this subtraction function, since it leads to a discrep-ancy between the telemetered RXcoarse and the value actually used, for about half the time.
Since the hardware value of RX trigger will be used then the hardware values for the FFT numbering also needs to beused, as shown in the third panel of Fig. 25. The renumbered values, in the fourth panel, would be used in combinationwith the telemetered RX trigger value, RXTM.
Sign Convention Adjustment
The resulting RXcoarse and RXfine values are further modified to account for the differing sign convention resulting fromthe hardware realisation of the coarse and fine shifting mechanisms. This is because an increment in the coarse value re-sults in a later generation of the LO chirp, and hence a shift of the filterbank to the right and the position of the echo to the
Frequency (MHz)-3.2 0 3.2
310 32 63 310
Input Filter Numbers
310 32 63
Re-ordered Filter Numbers
310 32 63
Re-numbered Filter Numbers
2 34 1
37 38
37 38
Open-LoopCalibration
RXtrigger = RXTM
Ocean Trackingand SPTR
RXtrigger = RXTM - 0.4848
0 31 32 633536
37 38
0.3
6 filters
OLCSPTR
Fig. 25: Renumbering of filters during Ocean Tracking. The first panel shows signals (an ocean echo, an OLC and a full PTR,labelled SPTR) as they appear at the input to the FFT. Note that the tracking point (zero frequency) is not at the mid-pointof the leading edge. The second panel shows the output from the squared modulus of the FFT, where the wanted spectrum isthe shaded part. In the OLC sub-mode a renumbering is performed as shown in the third panel to bring the wanted spectruminto the range 0…63. The fourth panel shows the additional renumbering (by 2) performed prior to evaluation of the errorsignals in the HTL window, centred on the mid-point of the leading edge. In order to maintain this point offset from the zerofrequency, as shown in the first panel, the RX trigger value sent to the hardware has 0.4848 (equivalent to 2 filters)subtracted.
left in the filter bank. Conversely, an increment in the fine value increases the fine shift which simply moves the echo tothe right.
This function also, generally, results in a change in the RXcoarse value sent to the hardware. Therefore it has to be repro-duced in the SPTR algorithm.
Conversion of RXfine to Actual Fine Shift
The value of RXfine resulting from the sign convention adjustment is not sent to the hardware directly, as it has to be con-verted into the appropriate value for the implementation of fine shifting in the hardware. This function is slightly confusingbecause the full range of RXfine, 0–255, has to encompass a shift of 4.125 filters in the FFT (corresponding to one RXcoarseincrement of 12.5 ns). This is accomplished by addressing the table OT_TAB_FINE_TUN[256] which is called (mislead-ingly) the shifting weights table in the SPSA. Here it will be referred to as the Fine Tuning table, to avoid confusion withthe Shifting Weights PROM in the hardware.
The output values from the table are in the range 0–255. The table is illustrated in Fig. 26.
The mapping used in this table uses 56 values of RXfine to represent one filter, with the last filter requiring 64 values as ithas to cover 1.125 filters. Therefore when the fine tuning word is interpreted as an offset in units of FFT filters, a singlefilter is equivalent to 64/1.125 units of fine-tuning. The discontinuities in the table have to be interpreted as integral num-bers of filters, as further described below.
In the case of the Ice Chirp, the same table is used but the input parameter, RXfine, is divided by 4 before the access to thetable. This is accomplished by two right shifts.
Inside the hardware signal processor the output from this fine tuning table is used during the FFT process to perform afrequency shift of the input signal, by multiplying each sample by the term , where ∆f is the desired (pos-itive) frequency shift (in units of filters).
256
224
192
160
128
96
64
32
0
Fine
Tun
ing
2562241921601289664320RX fine
filter 1 filter 2 filter 3 filter 4
Fig. 26: Visualisation of the fine-tuning table implemented in the SPSA software. The table mapsthe 256 possible values of the RXfine word to the fine tuning values applied in the hardware.These fine tuning values account for the fact that the 256 values (equivalent to one RXcoarsevalue, 12.5 ns) have to span 4.125 filters (of 3.03 ns each).
jexp 2π N⁄( )∆f
In order to reduce the number of operations this is combined with the Hamming windowing described above; both theHamming weight and the shifting weight are multiplicative terms applied to the successive samples of the input signal.Thus the nth ADC sample, x(n), is multiplied by the term:
(5)
The combination of the Hamming window and fine shift for each of the 64 I and Q samples from the ADC are stored ina PROM. This PROM contains a set of 64 Hamming weights for each value of fine shift. Since fine shift varies from 0–255 there are, in principle, 256 such tables, with 64 complex entries in each. However, only the first 64 of these tablescontain unique information; the remainder are synthesised by shifting the contents of the preceding 64 by one entry. Thestructure of this Shifting Weight PROM is shown in Fig. 27.
A representation of the contents of the PROM is shown in Fig. 28. This is “reversed” compared to Fig. 27; the Hammingweights for the samples 0…63 appear along the leading edge and are arranged in accordance with the contents of thePROM, such that the ordering is 32…63, 0…31. The table row corresponding to zero fine shift (at the leading edge) hasno imaginary part as it is the pure Hamming window. This can be seen by juxtaposing the left and right halves of the plot.
The structure of the word used to address this PROM is shown in Fig. 29. The least significant bits, 20…25, are increment-ed to address the successive 64 samples from the ADC, while the 64 distinct tables containing the fine shift values areaddressed by the least significant 6 bits of the fine-tuning word from the software, provided in bits 26…211
The most significant 2 bits of the fine-tuning word will effectively produce a renumbering of the filters. This is achievedby the addition of these two bits to the least significant bits, 20…25, coming from the processor timing. This producessuccessive rotations in the eventual FFT.
Synchronisation
Raw data from the altimeter are contained in a data structure called a source packet. In Ocean and Ice Tracking modes,the measurement data are divided into 20 Auxiliary/Science data-blocks, each representing one 50-pulse average.
0.54 0.46–2πnN
---------- cos
j2πN------∆f
exp×
Fig. 27: Combination of Hamming weights and fine shift in the Shifting Weight PROM. The sixLSB’s of the fine shift word from the software are used to address the area containing the64 complex values which will be used to multiply each of the 64 (complex) samplesfrom the ADC. They are stored in the order shown for hardware implementation reasons.
Hamming weights for samples 0 … 63
0 132 33 63
fine shift
0123
63
30 31
During each data-block the FFT values are those accumulated during the previous 50 pulses, while the other values in thedata-block (such as RXcoarse and RXfine) are those in use during the current data-block. Specifically they are the valuesexisting between pulse 37 and 38. Thus for each set of parameters in a data-block, the corresponding FFT appears in thenext data-block. This is illustrated in Fig. 30. The re-alignment algorithm replaces, for every data-block i, the FFT valueswith those from data-block i + 1.
Determination of PTR Measurements
The FFT associated with the data-block is an average which has been generated during 50 pulses. It is necessary to findthe average value of the shift to be applied to the FFT samples in the data-block, ∆. This is determined by computing, foreach pulse in the data-block, in the range 0 … 49, the actual RXfine value, taking account of the system timing shown inFig. 30. Each RXfine value is then converting to the equivalent fine-tuning value by means of the Fine Tuning table. ThisFine Tuning value is converted to fractional filter units and the mean, ∆, over the 50 pulses is found.
The FFT values within the data-block are accumulated into the appropriate PTR. There is one PTR corresponding to eachvalue of RXcoarse. Thus if the RXcoarse value has been incremented a new PTR record is initiated. Otherwise the accumu-lation of the FFT data into the current PTR is performed, as follows.
Let:
Vi = power of FFT sample, for i = 0 … 63;
fi = FFT filter number, in the range 0 … 63.
Then, for all non-zero values of Vi,
(6)
(7)
0
0.25
0.5
0.75
1
-1
-0.5
0
0.5
1
3240
4856
08
1624
SampleNumber (n)
FineShift
(filters)
Real Imaginary
0
0.25
0.5
0.75
1
-1
-0.5
0
0.5
1
3240
4856
08
1624
SampleNumber (n)
FineShift
(filters)
Fig. 28: Visualisation of the contents of the Shifting Weights PROM. Due to the re-ordering of the sample numbers, 32…63,0…31, the characteristic shape of the Hamming window at the leading edge of the real part has its two halves reversed.
20
sample
Fig. 29: Structure of the address word used to access the Shifting Weight PROM.
25
numberfineshift
finetuning
213 211
V j V i=
f j f i ∆–=
where j is the cumulative index in the PTR.
Due to aliasing in the hardware it is possible, for PTR signals at the edges of the IF bandwidth, for the PTR power to appearat the wrong end of the FFT range. This is detected and the signal placed at the correct position, outside the range of theFFT. A correction for the IF transfer function (extracted from the set of about 60 closely spaced Ice Mode PTR’s) is per-formed over the full data set by means of a cubic spline interpolation.
Non-Linear PTR Fit
As we explained in the main text, hamming weighting is applied on-board during the FFT processing, so the n data pairscontaining corrected power Vn’ and position in the filter bank, fn, should follow a gaussian curve. Due to the spurious sig-nals the curve deviates significantly from a gaussian. However it may be accurately represented as the sum of threegaussians, representing the main lobe and two spurii at lower level, one on each side.
The quantisation effect described above causes a deviation from this function at low signal levels so the first editing cri-terion is only based on this quantisation level.
We fit this reduced data set by the Levenberg-Marquardt method. This method attempts to reduce the χ2 value of a fitbetween a set of N data points fn, Vn’ with individual standard deviations σn, and a nonlinear function; which in this caseis the sum of three gaussian functions defined in eqn (8).
Source Packet
0 1 2 17 18 19
1 20 36 37 38 49 0TX TriggerRX interval
RX Trigger and 1 20 36 37 38 49
Compute
999
New telemetryPacket start
Height Wordwrittento Telemetry
3534… 3933
Slot
for RX Trigger…
Radar Pulse Propagation
FFT Accumulation
1 20
FFT written toTM and availablefor computingnext RX values
Fig. 30: Timing of events within the RA data acquisition cycle. FFT values are accumulated during a data-block (50pulses) and transferred to the SPSA for the control of the next data-block. Other telemetry values are writtenduring slot 37 of the data-block, as they are used.
(8)
We extract the centre position, xSo, the power, ASo, and the standard deviation, σSo for the central lobe, the centre position,
UxSo, the power, UASo, and the standard deviation, UσSo for the upper spurious, and the centre position, LxSo, the power,
LASo, and the standard deviation, LσSo for the lower spurious.
The initial guess for the fitted parameters is derived from a survey of the data set, searching for the maximum value Vmaxand corresponding position fmax. These provide the initial guess for ASo, and xSo respectively. For σSo a constant empiricalvalue (0.85 filter units) is used. The initial values for the other parameters are derived from these by offsetting the centrea fixed number of filters above and below, and the power by fixed fractions. These offsets are empirically derived.
Following the removal of outliers, using an editing criterion based on the standard deviation of the residuals to the fit, wemake a second pass to the fit.
ANNEX 2: ROUND-ORBIT BEHAVIOUR OF OLC
In developing the algorithms described here we have fitted the following function to the OLC data:
(9)
where CS, AS, and φS are, respectively the offset (being the mean value of OLC), a scaling factor and a phase term, de-pending on the satellite (ERS–1 or ERS–2); an, and φn are the amplitude and phase coefficients of the sinusoiddecomposition; t is the orbit time since ascending node and T the orbital period. This function is seasonally dependent,due to the changing declination of the sun with respect to the ascending node, and this is expressed in the phase term φdefined by eqn (10):
(10)
where d is the day number in the year, d0 is a reference day, D is the number of days in the year and α and β are amplitudeand offset terms respectively.
However in the final algorithm this function is not used.
V n′ ASi e
f n x0–
2σ-----------------
–
AU Si e
f n xU 0–
2 σU--------------------
–
AL Si e
f n xL 0–
2 σL-------------------
–
+ +=
τC CS A+ S an n 2π tT--- φ φS+ +
φn+ sin
n 1=
4
∑=
φ α 2πd d0–
D--------------
sin β+=