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Switching Events in a Subglacial Drainage System : a Detection Algorithm and Observational Analysis Yeo, Kevin Mirng En 2016-04

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Switching Events in a SubglacialDrainage System: a DetectionAlgorithm and ObservationalAnalysisbyKevin Mirng En YeoA Thesis submitted in partial fulfillmentof the requirements for the degree ofBachelor of Science (Honours)inThe Faculty of Science(Geophysics)The University of British Columbia(Vancouver)April 2016c©Kevin Mirng En Yeo, 2016AbstractThe rate of glacier mass loss depends on the sliding velocity of the glacier. Fora temperate glacier, during the summer when the rate of surface melting is high, thesubglacial drainage system becomes active and affects the sliding velocity in a nontrivialmanner. An important component of the subglacial drainage system is the networkof cavities on the glacier bed. Connected cavities can be disconnected either partiallyor completely from each other and the rest of the drainage system. Connection anddisconnection between two cavities are collectively referred to as switching events. Thisparticular phenomenon has not been included in existing subglacial drainage models. Inorder to facilitate a physical understanding of how hydraulic connections at the glacierbed are made, and ultimately to allow further model development, this study aims toformulate a semi-automated algorithm to detect all switching events from an array ofboreholes with water pressure time series. Observational analysis of the output suggestsfour different possible mechanisms that could be considered in order to explain theseswitching events.iiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Field Site and Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . 53 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.1 Developing an algorithm to automatically find all switching events in twopressure time series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.1.1 Challenges in identifying switching events from pressure measure-ments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.1.2 A three-criterion test . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Finding actual switching times . . . . . . . . . . . . . . . . . . . . . . . . 153.2.1 Applying a moving interval over the two time series . . . . . . . . 183.3 Post-processing the results . . . . . . . . . . . . . . . . . . . . . . . . . . 184 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.1 Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.2 Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28iiiA Relating the connection tolerance to the switch-finding tolerance . . 31ivList of Tables3.1 Parameters used in finding all switching events . . . . . . . . . . . . . . . 18vList of Figures1.1 An example of two pressure time series showing switching events . . . . . 32.1 Map showing the location of South Glacier . . . . . . . . . . . . . . . . . 62.2 Map showing the boreholes drilled in the study site . . . . . . . . . . . . 73.1 Schematic of a well-connected pair of boreholes in a subglacial distributeddrainage system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Schematic of all the overlapping intervals within the time period in whichboth time series have data . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3 Schematic of the search interval for case 1 and case 2 . . . . . . . . . . . 174.1 A raw output of the algorithm in Section 3.1–3.2 . . . . . . . . . . . . . . 214.2 Post-processed results plotted on 1.1 . . . . . . . . . . . . . . . . . . . . 214.3 A second example of the post-processed results . . . . . . . . . . . . . . . 224.4 A third example of the post-processed results . . . . . . . . . . . . . . . 224.5 A fourth example of the post-processed results . . . . . . . . . . . . . . . 23viAcknowledgementsMy heartfelt gratitude to my supervisor, Christian Schoof, for his never-ending sup-port and guidance during my undergraduate years, particularly in my final year. Thecompletion of this thesis, let alone my experience of the true joy and frustration of re-search, would not have been possible without him.I would also like to extend my appreciation to my dearest friend, Arvin Boutik, whohas put up with my verbal abuse during my times of distress like a true friend. I canonly hope that our friendship will grow and last.Research can be a self-consuming process that deprives one from his loved ones. Myinfrequent yet brief conversations with my family over the past year may have distortedthe fact that I was very busy. To my dad, my mum and my sister, I am deeply apologeticif my actions were somehow misconstrued.Lastly, a big thank you to Camilo Rada for providing the data and maps of the studyglacier, and Marianne Haseloff for providing the UBC thesis LATEX class files.vii1 | IntroductionFor glaciers that have temperatures close to freezing throughout the thickness ofthe ice, which are called warm-based glaciers, surface melt can reach the ice-bedrockinterface through openings in the ice known as crevasses and moulins, and subsequentlydrain through a variety of conduit types to and along the glacier bed (Fountain & Walder,1998). Once it reaches the base of the glacier, surface melt can act as a lubricant thatreduces friction between ice and bedrock, facilitating sliding of the ice.A glacier experiences mass loss when the rate of melting in the ablation zone exceedsthe rate of ice formation in the accumulation zone. An increase in ice velocity will resultin a larger portion of the glacier in the ablation zone and a smaller portion of the glacierin the accumulation zone, consequently accelerating mass loss. There have been observedincrease in ice velocity relative to background winter values during the summer at valleyglaciers (Iken & Bindschadler, 1986; Sugiyama & Gudmundsson, 2004) and the marginalareas of the Greenland ice-sheet (Zwally et al., 2002; Bartholomew et al., 2010), andthis increase in velocity is attributed to the discharge of higher amounts of surface meltgenerated during the summer as compared to winter through the subglacial drainagesystem.The relationship between surface melt and basal sliding velocity is not trivial; moresurface melt does not necessarily lead to faster ice flow (Bartholomew et al., 2010).Whereas surface melt dictates the water discharge along the bed, effective pressure (de-fined as the difference between overburden and water pressure in the subglacial drainagesystem) controls ice-bed friction and therefore sliding velocity. If the effective pressureis lowered, the normal force applied by the glacier on the underlying bedrock is reduced,and therefore friction decreases and sliding velocity increases (Iken & Bindschadler, 1986;Schoof, 2005). In cases where water pressure is high enough, uplifting of glacier may evenoccur. The key to understanding the relationship between surface melt and ice velocityis therefore to understand how melt supply affects water pressure at the bed.Subglacial drainage systems consist of two qualitatively distinct components, known1as channelized and distributed systems. The channelized system consists of isolated flowthrough channels known as Ro¨thlisberger channels (R-channels) which have semi-circularcross sections in models (Ro¨thlisberger, 1972). The two competing processes that keepR-channels open are widening by melting walls caused by heat dissipation from waterflow and narrowing caused by the inward creep of the surrounding ice motion (Schoof,2010). Distributed system comprises of linked subglacial cavities and possibly subglacialwater film (Iken & Bindschadler, 1986; Walder, 1986; Fountain & Walder, 1998; Schoof,2005). Cavities are formed when ice is forced upward by protrusions at the bedrock. Thisprocess causes a gap to be formed in the lee of the protrusion. The size of the cavity isdetermined by the opening rate due to sliding and the closing rate due to the ice creepclosure (Schoof, 2010).A channelized system drains basal water more effectively than the distributed system.In the less efficient distributed system, the conduits’ limited water-carrying capacity leadsto impeded flow and thus a higher basal water pressure, which results in a higher slidingvelocity. In contrast, a well channelized system can lead to more efficient drainage andthus a lower basal water pressure, which results in a lower sliding velocity (Hewitt, 2013;Schoof et al., 2014). Hence, the structure of the subglacial drainage in part determinesof the sliding velocity of a glacier.The subglacial drainage system evolves spatially and temporally on a seasonal basisand is observed to be more active during summer (Hubbard et al., 1995; Murray &Clarke, 1995; Fudge et al., 2005; Harper et al., 2005; Schoof et al., 2014). Diurnal cyclesin surface melt input during summer lead to diurnal cycles in basal water pressure,though not throughout the glacier. For the parts of the glacier that exhibit prominentdiurnal variations in water pressure, boreholes that are closely-spaced (approximately≤30 m apart) can have strongly correlated water pressures, suggesting that they areconnected to each other. This hydraulic connectivity, however, is not permanent andcan change with time: two boreholes showing highly correlated water pressure recordover a multi-day period can switch abruptly to exhibiting no correlation over a period oftime and subsequently switch back to being connected again (Figure 1.1). This switchingbehaviour is a result of distributed part of the drainage system alternating back and forth2Figure 1.1: Basal water pressure time series from two boreholes drilled at South Glacier,Yukon Territory, Canada. 1st January 2014 was set as serial date 0 in this plot. Duringwhich both water pressure time series exhibit diurnal variations between day 203 and 206and between day 216 and 219, there are multiple switching events where they switchedfrom being connected to disconnected (approximately day 204.2, 216.5, 217.5 and 218.2)or vice versa (approximately day 205.0, 216.3, 217.0 and 218.0).from being unconnected to being connected.For the purpose of this study, I characterize the ice-bedrock interface as consistingof hydraulically connected and unconnected parts. Only the connected part is likely toexhibit highly correlated water pressure signals and shows diurnal oscillations in waterpressure. The unconnected part has uncorrelated water pressure signals. As most bore-hole water pressure time series never show any diurnal pressure variation, the majorityof the bed remains unconnected most of the time. That, and the fact that the subglacialdrainage system covers only part of the bed, suggests that the connected part only coversa small part of the bed. The connected part links to the surface (and exhibits diurnal3oscillations) whereas the unconnected part remains isolated. There may be slow waterflow within the unconnected part (for instance through till) but it is likely that waterfluxes are very small compared to the connected part.Extensive work has been done to model the subglacial drainage systems (e.g., Schoof(2005, 2010); Hewitt (2011, 2013); Werder et al. (2013)). The aim of this work is tobetter understand the relationship between surface melt and ice flow velocity. Almostall models in use are elaborate diffusion models, similar to models for groundwater flowthrough a poroelastic medium, but with a smoothly evolving diffusivity. However, thepresence of switching events suggest that water flow is not always possible through thecavities or film that make up the distributed system. Existing models assume that watercan flow as long as there is a finite amount of water stored in the distributed system andconsequently do not account for the existence of the unconnected part of the glacier bed.The purpose of this thesis is to investigate the physical process that determines switch-ing behaviour in the expectation that this will guide future improvements to existingdrainage models. A better described subglacial drainage model enables us to better un-derstand how surface melt affects the sliding velocity of glacier, which in turn gives us abetter prediction of how fast glaciers are melting.42 | Field Site and InstrumentationThe study site is an unnamed valley glacier known informally as ”South Glacier”,located in the Donjek Range, southwest Yukon Territory, Canada, at 60◦49′ N, 139◦8′ W(Figure 2.1). South Glacier is a polythermal glacier (De Paoli & Flowers, 2009; Wilson,2012) and is known to have last surged (a short-lived event where a glacier advancessubstantially) in 1986 (Johnson & Kasper, 1992). Polythermal glaciers are ice massesthat have both temperate and cold ice. Temperate ice is ice at the freezing point andcoexists with liquid water whereas cold ice is below the freezing temperature without thecoexistence of liquid water. The study glacier is approximately 5 km long and has anarea of 4.4 km2, spanning an elevation range of 1960 to 2930 m above sea level (a.s.l.). Itsequilibrium line altitude is approximately at 2550 m a.s.l. (Wheler et al., 2014). It has amean slope of 13o. The studied area is the ablation zone of the glacier (slightly south ofthe equilibrium line) with ice thickness ranging from about 60 to 100 m. There are twomajor surface streams within the study area and numerous crevasses on the glacier. Icevelocities in the study area vary seasonally in the range 10 to 30 m yr−1. South Glacieris described in greater detail in De Paoli & Flowers (2009), Wheler (2009), Schoof et al.(2014) and Flowers et al. (2014).To obtain the basal water pressure time series for this study, a total of 318 bore-holes were melted through the ice to the glacier bed using hot water drilling, and weresubsequently instrumented with pressure transducers (Figure 2.2). Schoof et al. (2014)provide additional technical detail on the instrumentation and drilling equipment used.Barksdale 422H2-06A pressure transducers were used from 2008 onwards and Honeywell19C200PG5K pressure transducers were used for some of the boreholes starting in 2013.Additionally, eight digital transducers built at UBC were deployed in 2014 and 2015.All pressure transducers were cast in epoxy to ensure waterproofing. From 2013 onward,most pressure transducers installed have attached 1/4′′ brass ray piston style snubbers toprevent damage to the transducer from extreme pressure pulses (Kavanaugh & Clarke,2000). The pressure transducers were calibrated in water-filled boreholes prior to 2015,5Figure 2.1: The red highlighted region on the map, labelled SG, is South Glacier andthe box indicates the study area. Additional properties of South Glacier are given in thetext box.while a custom made pressure vessel was used in 2015. Generally, the pressure transduc-ers conformed to factory calibration (within a few percent), except for a change in offset(zero pressure reading) that appeared to be specific to the length of signal cable used. Thepressure transducers were installed about 20 cm above the bottom of the boreholes, andwere attached to either Campbell Scientific CR10X or CR1000 data loggers via 24-gaugecopper signal cables. The data loggers were set to record data continuously with loggingintervals of 1 minute for CR1000s and 2 minutes for CR10Xs during July and August,changing to 20 minutes from September to June. The glacier bed was assumed to havereached when drilling is notably slowed or prohibited, and water samples taken from thebottom of the boreholes were significantly turbid. In cases of clear water samples beingreturned, a borehole camera was used to check for bed contact from 2012 onwards. Thefull set of water pressure records collected between 2008 (the initiation of the project)and 2015 were used for this study.6Figure 2.2: Each empty circle represents a borehole drilled at the study site. The blacklines represent contour levels with 20 m spacing. The thickness of the ice is representedby the colour based on the colour bar on the left of the figure.73 | Methodology3.1 Developing an algorithm to automatically findall switching events in two pressure time seriesTwo boreholes are hydraulically well-connected when there is at least one path (ori-fice) between them that allows water to flow through without significant resistance (Fig-ure 3.1). The hydraulic potential φ is defined as the liquid pressure measured above a ref-erence elevation. Mathematically, the hydraulic potential can be expressed as φ = pw+pelwhere pw is the water pressure exerted on the surrounding container due to the container’slimited storage capacity and pel is the water pressure due to the height of the water col-umn above the reference elevation. A difference in φ between two locations, termed thehydraulic gradient ∇φ, drives water to flow down the hydraulic gradient. The flux of thedischarge q is given byq ∝ −κ(|∇φ|β−2∇φ)where κ is the permeability and β(> 1) is a constant. When there is a good connectionbetween two boreholes, κ is large. To have fluxes that are not excessive between thetwo boreholes will then require |∇φ| to be small. Hence, a well-connected pair of bore-holes demands that in addition to having at least a connecting conduit, φ needs to beapproximately the same in both boreholes. As creep closure driven by the pressure dif-ference between ice and water occurs, orifices (smaller cavities connecting larger cavities)may be shut down and two previously connected boreholes can become disconnected inwater path. Even though some parts of the ice-bedrock contact region are made up ofbasal sediments may allow water flow through, the flow may be insignificant. Figure 3.1illustrates such a possible scenario.In this study, I limit myself to looking for well-connected boreholes. I would like toidentify pairs of boreholes which exhibit the same hydraulic potential for some lengthof time (typically of length that is a significant fraction of a day). Switching events8Figure 3.1: A top-down view depicting a possible morphology of the subglacial distributeddrainage system. The left panel shows a scenario in which two boreholes are hydraulicallywell-connected whereas the right panel shows how a change in the subglacial drainagestructure causes the two previously connected boreholes to become disconnected.correspond to times at which a pair of boreholes changes from exhibiting the same to adifferent hydraulic potential and vice versa. I will refer to those events as a switch-offand switch-on event respectively. There are 280 pressure time series from the study site;the other 38 time series were discarded as their outputs were considered unreliable. Assuch, there are 39060 pairs of time series in which to look for possible switching events.An algorithm to automate the finding of all the switching events is needed to cataloguethese events.3.1.1 Challenges in identifying switching events from pressuremeasurementsThe pressure transducers used in this study are force collector types. Changing waterpressures deform a diaphragm within the transducer and cause the resistance of a straingauge bonded to the diaphragm to vary. The strain gauge is part of a Wheatstone bridgecircuit that allows the ratio of an applied (“excitation”) to a measured (“differential”)voltage to be used to determine the resistance of the strain gauge. The transducers havea linear relationship between that voltage ratio and the applied pressure. The voltagemeasurements can either be made at a data logger on the ice surface as is the case forthe analogue transducers with which the majority of our data were collected, or inside a9digital transducer which then transfers its measurements to the logger in binary format.Converting voltage ratio to pressure requires the coefficients in the linear relation-ship between the two to be known. Factory calibration provides such values based ondiaphragm and strain gauge design and the circuitry of the Wheatstone bridge. Theaddition of signal cable affects that circuitry for analogue transducers, requiring recal-ibration. Moreover, the calibration is also expected to degrade over time. The maincauses of degradation are water leaking into the circuitry and extreme pressure pulses.Pressure transducers with snubbers attached are protected to some degree from the lattereffect.Partial short-circuiting can occur when water leaks into the transducers or into thecables connecting the transducers to the data loggers, affecting the calibration, whileexcessive pressure pulses can damage the diaphragm with the strain gauge (Kavanaugh& Clarke, 2000). Provided there is no damage to the diaphragm, the digital transducersshould be less prone to calibration degradation as long as binary information can berelayed to the data logger and there is no short circuit inside the transducers, theircalibrations remain reliable. Conversely, analogue transducers should be more prone tocalibration degradation since the ratiometric conversion occurs within the data loggers.The applied pressure on the diaphragm is not φ but rather pw. Define the linearrelationship between the applied pressure and the ratiometric voltage output aspw,1 = a1V˜1 + b1 (3.1)pw,2 = a2V˜2 + b2 (3.2)where V˜1 and V˜2 are the voltage ratios measured using the two transducers. The hydraulicpotential at any two boreholes, φ1 and φ2, are given byφ1 = pw,1 + pel,1 (3.3)φ2 = pw,2 + pel,2 (3.4)10When two boreholes are well-connected, using (3.3) and (3.4),φ1 = φ2pw,1 + pel,1 = pw,2 + pel,2pw,1 = pw,2 + (pel,2 − pel,1) (3.5)A linear relationship between pw,1 and pw,2 is therefore expected for well-connected bore-holes. Combining (3.1) and (3.2) with (3.5),V˜1 =(a2a1)V˜2 + [(b2 − b1) + (pel,2 − pel,1)]= mV˜2 + c (3.6)where m = a2a1is the multiplier and c = (b2− b1) + (pel,2− pel,1) is the offset. It is obviousfrom (3.6) that c is not just a calibration offset but also includes pel, which cannotbe known. m on the other hand is purely a calibration constant. Thus, for perfectlycalibrated instruments, m is expected to be unity. A significant difference from 1 istherefore likely to indicate a significant damage to the instrument(s) (making the outputquestionable), or a lack of good connection between two boreholes. However, a linearrelationship between pw,1 and pw,2 can potentially exist even when the two boreholesare not well-connected. This happens when both pw,1 and pw,2 are ”featureless”. Forinstance, if water pressure in both boreholes varies linearly with time (as is likely to bethe case in an approximate sense over short time intervals when the pressures are varyingsmoothly), then a good linear fit between the two time series will result, even though itneed not reflect a good connection.3.1.2 A three-criterion testA series of three tests can be designed to identify intervals in which two boreholesare well-connected. Consider two water pressure time series, P1(t) and P2(t). Assumethat the two boreholes are well-connected. Within an interval Ti where {Ti : tk ∈ Ti, k =111, 2, . . . , Ni}, performing linear regression on P1(t) and P2(t), t ∈ Ti givesP˜1(t) = miP2(t) + ci when t ∈ Ti (3.7)where P˜1 is the predicted response for P2. The root-mean-squared error RMSEi isRMSEi =√√√√√∑tk∈Ti(P1(tk)− P˜1(tk))2Ni=√√√√√∑tk∈Ti(P1(tk)−miP2(tk)− ci)2Niwhere Ni is the number of data points in Ti.A linear regression minimizes the RMSE. An approximate linear fit requires a smallRMSE but the RMSE cannot be too small such that it is less than the random error.A natural way of determining the quality of the linear fit is to compare the RMSE withthe natural variability in one of the pressure signals, defined as a standard deviation.However, as the two time series are typically compared over short intervals, the stan-dard deviation is calculated over a longer period of time to define the required naturalvariability. In other words, a first test, Test 1RMSEiσ(P1)≤ toltest 1 (3.8)is used to determine whether there is a good linear fit, whereσ(P1, T¯i) =√√√√√∑tk∈T¯i(Pi(tk)− P¯1)2N¯iis the standard deviation and toltest 1 is a tolerance. T¯i is a representative interval (typ-ically larger than Ti) such that Ti ⊂ T¯i and N¯i is the number of data points in T¯i. Thelength of the interval Ti needs to be sufficiently short such that it is on the same scale asthe expected length of time two boreholes stay well-connected. Otherwise, Test 1 may12fail to identify an interval in which two boreholes are well-connected.In an interval of well-connectedness (labelled Ti), if factory calibrations are reliable,mi from (3.7) should effectively be unity. To account for calibration errors, assumingthat they are not large and remain unaltered within Ti, mi is allowed to differ from unityonly within a given tolerance. That is, we impose a second test of the form1− tol′test 2 ≤ mi ≤ 1 + toltest 2 (3.9)where tol′test 2 and toltest 2 are tolerances. The reason for having different lower and upperlimits on mi is that, if the roles of P1 and P2 were reversed, we would want the test in(3.9) to yield the same result as it does with P1 and P2 taking their original roles. Thisimplies a relationship between tol′test 2 and toltest 2.If the roles of P1 and P2 are reversed in (3.7) and P˜2 was calculated instead, combining(3.1), (3.2) and (3.5) yieldsV˜2 =(a1a2)V˜1 + [(b1 − b2) + (pel,1 − pel,2)]= mV˜1 + cwhere m is now a1a2instead of a2a1as in (3.6). If both definitions of mi are to satisfy (3.9),then this requires1− toltest 2 ≤ a2a1≤ 1 + toltest 2 and 1− toltest 2 ≤ a1a2≤ 1 + toltest 2so that implies11 + toltest 2≤ a2a1≤ 11− toltest 2It is then plain to see that 1− toltest 2 = 11+toltest 2 . The second test, Test 2, is thus11 + toltest 2≤ mi ≤ 1 + toltest 2 (3.10)There is one further complication. Suppose both time series are approximately linear13in time in the interval Ti. This is always likely to happen over very short intervals ifpressure varies smoothly (as follows from a Taylor expansion in time) If both time seriesare linear in time, it follows that one time series is a linear function of the other, with-out this necessarily indicating a good connection. We call intervals where this happens“featureless”. To identify such intervals, define the detrended time series asPdetrended(t) = P (t)− (αt+ β)where the linear fit coefficients α and β are chosen such that the sum of squared error∑t(P (t)− (αt+ β))2is minimized. A time series in an interval Ti is deemed featureless if the standard deviationof the detrended time series σ(Pdetrended(Ti)) is less than a prescribed fraction of thestandard deviation of the time series σ(P (Ti)).Whether the two boreholes are well-connected during a “featureless” interval or nothas to be determined from the behaviour in the last interval. In the case where both timeseries are featureless, the boreholes are considered to be well-connected in the currentinterval Ti if and only if the following holds:a) The boreholes are well-connected in the previous interval Ti−1b) The relative difference between the multiplier from the last and current interval,mi−1 and mi, is less than a given tolerance.c) The difference between ci−1 and ci is less than a prescribed fraction of the scale forpressure variation σ(P (Ti)).14Figure 3.2: A schematic showing all the overlapping intervals within the time period inwhich both time series have data, in which an interval T is represented by a rectangularbox. The two unlabelled boxes after the interval T3 represent all the intervals betweenT3 and Tmax overlap. The subsequent interval of an interval is the current interval shiftedforward in time by a moving interval ∆tmove.The last test, Test 3, is thus(a) σ(P1,detrended(t) > toltest 3,std × σ(P1(t) or (3.11a)(b) σ(P1,detrended(t) ≤ toltest 3,std × σ(P1(t),the two boreholes are well-connected in Ti−1,∣∣∣∣mi −mi−1mi−1∣∣∣∣ ≤ toltest 3,m and ∣∣∣∣ci − ci−1σ(P (t))∣∣∣∣ ≤ toltest 3,c (3.11b)for all t ∈ Ti. toltest 3,std, toltest 3,m and toltest 3,c are tolerances, and either (3.11a) or(3.11b) needs to be satisfied.Test 2 is only executed if Test 1 is passed. Similarly, Test 3 is only executed if Test2 is passed. As such, for Test 3, it is only necessary to check whether P1 is featurelessor not, as P2 will have the same characteristic. In Ti, if (3.8), (3.10), and one of (3.11a)and (3.11b) are passed, then the two boreholes are well-connected. Otherwise, they areconsidered not to be well-connected.3.2 Finding actual switching timesThe time period in which both time series contain data is divided into overlappingintervals T1, T2, T3, . . . , Tmaxoverlap. For an interval Ti−1, the next overlapping intervalTi is Ti−1 shifted forward in time by a moving interval ∆tmove. Figure 3.2 depicts theschematic of this. Whenever the three criterion test indicates that two boreholes change15from being well-connected in an interval Ti−1 to being not well-connected in the nextinterval Ti or vice versa, a search needs to be done to find all the possible switchingevents within the combined intervals of Ti−1 and Ti. Applying the three criterion teston each of such intervals increases the likelihood of capturing short-lived well-connectedperiods. If the intervals are not overlapped, an interval containing a section of a short-lived well-connected period may be misconstrued as a not well-connected period, causinga search not to be done when needed.To identify these switching events, we define the absolute difference between P1(tj)and P˜1(tj) asej =∣∣∣P1(tj)− P˜1(tj)∣∣∣= |P1(tj)− (mAP2(tj) + cA)|where mA and cA are the regression coefficients from the well-connected interval TAidentified by the three criterion test of section 3.1.2. This can be Ti or Ti−1. In twopressure time series, the well-connected times tconnect are defined as{tj : ej < tolconnection} (3.12)where tolconnection is a tolerance for well-connectedness. A detailed description of how theconnection tolerance tolconnection relates to the switch-finding tolerance tolfind is given inAppendix A. tunconnect are simply times tj for which ej > tolconnection. Mathematically,the set of switch-on events is defined as{tk : ej < tolerror for j = k, k + 1, . . . , k +D andej > tolconnection for j = k − 1, k − 2, . . . , k −D − 1} (3.13)whereas a switch off is defined as analogously by reversing the inequality signs in (3.13).D is a minimum duration for well-connectedness or a lack of well-connectedness that isimposed to prevent outliers from generating switching events.The case of the three criterion test indicating a switch from not-well-connectedness to16Figure 3.3: A schematic showing the search interval for case 1 and case 2, in which aninterval T is represented by a rectangular box. For case 1, the three criterion test iden-tified Ti−1 and Ti as being not well-connected and well-connected respectively. The twounlabelled boxes between the intervals Ti−K+1 and Ti−1 represent all the well-connectedintervals. For case 2, Ti−K and Ti are not well-connected intervals and Ti−K+1 and Ti arewell-connected intervals.well-connectedness will be referred to below as “case 1” whereas “case 2” is the opposite.The time interval search interval is simply the union of Ti−1 and Ti in case 1. For case 2,the search interval needs to be extended back further. This is because the last switch-onevent prior to Ti−1 was in general found using different regression coefficients from thosedetermined in Ti−1. There is also a possibility that short-lived switches occurred betweenthe last search interval and the interval Ti−1. In order to identify mutually consistentswitching events (a switch-on must be followed by a switch-off), it is necessary to extendthe search back to the last not well-connected interval in the last search interval (then case1). This leaves us with a choice of which regression coefficients to use however. Case 2 mayterminate a sequence of multiple time intervals TB in which the three criterion test waspassed. I pick the regression coefficients from the interval with the smallest correspondingnormalized root-mean-squared error RMSEA/(toltest 1 × σ(P1)). Therefore, the searchinterval of case 2 will consist of a not well-connected interval, followed by a series ofconnected intervals, and a terminal not well-connected interval. Figure 3.3 illustrates thesearch interval for case 1 and case 2.17Descriptions Parameters Values Unitsinterval Ti 4 hoursmoving interval ∆tmove 2 hoursrepresentative interval T¯i 6 daysminimum duration D 60Test 1 tolerance toltest 1 0.2Test 2 tolerance toltest 2 0.4Test 3 standard deviation tolerance toltest 3,std 0.05Test 3 multiplier tolerance toltest 3,m 0.1Test 3 offset tolerance toltest 3,c 0.1switch-finding tolerance tolfind 0.9Table 3.1: Parameters used in finding all switching events3.2.1 Applying a moving interval over the two time seriesThe following steps are implemented to find all the switching events in two pressuretime series within the period in which both time series have data points.Step 1 The two time series P1 and P2 are interpolated onto common time stamps witha sampling interval of 2 minutes.Step 2 The period of time in which time series contain data is divided into overlappingintervals Ti. In each Ti, P1 and P2 are tested whether to be well-connected usingthe three criterion test.Step 3 When there is a change in connectivity, a search for the switching events isperformed as described in the previous subsection.Table 3.1 shows the value of the parameters used in the algorithm when comparingall possible pairs of time series.3.3 Post-processing the resultsThe algorithm described in section 3.1–3.2 is not robust to producing artificial switch-ing events as will be shown in Chapter 4, figure 4.1, of which I will explain the reasonin chapter 5. A way to post-process the output from the algorithm is to apply a set18of regression coefficients to the time series P2 that yield a set of best fit versions of P2,among which the best is deduced visually to be the version that produces the most num-ber of switching events accurately. This leaves us with a choice of the set of regressioncoefficients to be used. Naturally, we want to use the set of best regression coefficientsavailable within a specified period in which both time series P1 and P2 have data. Thequality of the regression coefficients m and c should not only incorporate the associatednormalized root-mean-squared error in (3.8) from the linear regression model, but alsothe duration in which these regression coefficients are maintained (defined as the timefrom the earliest switch-on to the last switch-off found using those regression coefficients).This translates mathematically intoξ =tmaintainRMSEnormwhere ξ is the quality of the regression coefficients, tmaintain is the duration for whichthe regression coefficients was maintained and RMSEnorm is the normalized root-mean-squared error. The higher the value of ξ, the better the quality of the regression coeffi-cients. The regression coefficients are then ranked according to their associated qualityand the first n best fit regression coefficients are selected. The best fit of P2 is thenPˆ2 = mA,bestP2 + cA,bestwhere mA,best and cA,best are the best fit regression coefficients from the set of valuesavailable.194 | ResultsThe algorithm detected 2844 pairs of time series that have switching events of whichthe majority are false positives. The following figures show a subset of all the switchingevents found. These are intended in part to show the output the algorithm produces(including the false positives mentioned above). The main aim is to illustrate differenttypes of ‘real’ switching events detected. In each figure, a pair of unscaled water pressuretime series P1 and P2 are plotted as red crosses and blue circles, respectively. Each timeseries is labelled by a unique identifier of the pressure transducer used in collecting thepressure data, composed of the year of installation and a serial number. The ‘best fit’version of P2, Pˆ2 = mA,bestP2 + cA,best is plotted as green circles. Switch-on times areplotted as black lines whereas switch-off times are plotted as magenta lines. In figure 4.1,I plot switching events based on the algorithm in sections 3.1–3.2, with no post-processingapplied, to indicate the raw output of the algorithm and to illustrate the degree of likelyslow drift in regression (and therefore calibration) coefficients. The remainder of thefigures show the post-processed output as described in section 3.3. With the exceptionof figure 4.2, each panel 2 shows the continuation of the time series in panel 1, whichincludes a brief overlap in the last few days in panel 1. In figures 4.2–4.5, I have removedthe false switching events. I have also added switching events in figures 4.3 and 4.5 thatthe post-processing algorithm fails to detect.20Figure 4.1: The top panel shows all the switching events found in a pair of time serieswithout implementing the post-processing as described in Section 3.3. Panels 2 and 3show the regression coefficients m and c needed to make the two time series agree witheach other during each period of well-connectedness.Figure 4.2: This is the same two time series in 1.1 with the post-processing algorithmapplied. There is a data gap after day 205 and before day 214 due to the data loggermalfunctioning.21Figure 4.3: A second example of the post-processed result.Figure 4.4: A third example of the post-processed result.22Figure 4.5: A fourth example of the post-processed result.235 | Discussion5.1 PhenomenaFigure 4.1 shows two time series that, judged purely visually, appear to indicate avery good connection (see also Schoof et al. (2014)). Prior to day 217, water pressurein both boreholes undergoes the same pattern of diurnal oscillations with near-identicalamplitudes and similar high-frequency features (which are not apparent when plotted atthe scale shown). Later in the time series, the diurnal oscillations are absent but theslow evolution of the pressure as well as short-lived pressure variations are common toboth time series. Despite this, the algorithm detects a multitude of switching events.There are two reasons for this. The first is that the low amplitude of pressure variationsσ(P ) sets an artificially tight threshold for detecting well-connectedness. The secondreason is that the regression coefficients m and c needed to bring the two time series intoagreement actually change over time (panel 2 and 3), as might be expected due to thereasons described in subsection 3.1.1.Figure 4.2 presents the switching events shown previously in figure 1.1, but nowprocessed using the algorithm (including post-processing). S14P32 (red) stays connectedto the subglacial drainage system in both panels as indicated by the diurnal pressureoscillation. The switching events are very clear, with S14P36 (blue) switching off withno abrupt change in pressure. Actual transition into well-connectedness, except the firstof the two that occurred on day 216, may actually precede the switch-on detected bythe post-processing. In these switch-on events, the pressure at S14P36 rises rapidlybut initially lags that at S14P32, with the lag varying from 40 minutes (day 217) to120 minutes (day 204). Typically, S14P32 connects when pressure at S14P36 is risingor at its peak, day 216 being an exception, and disconnects when pressure at S14P36is dropping. If actual well-connectedness precede the switch-on events detected by thealgorithm, there is no apparent critical pressure at which the connections are established.Pressures at disconnection are mostly lower than that of connection.24In figure 4.3, the disconnection between S15P83 (red) and S14P18 (blue) that occursearly in the season (days 176 to 211) is not a complete disconnection. While S15P83 is nolonger well-connected to S14P18, it shows signs of weak diurnal oscillation as comparedto the stronger diurnal oscillations in S14P18, suggesting that S15P83 is still connectedto the rest of the active drainage system in the early season. The switching events ondays 182, 185 and 190 occur without any abrupt changes in the rate of change of pressurewith respect to time. These switching events also occur at very similar pressures. Thissuggests multiple physical conduits connecting the two boreholes, some of which openand close at the switching times detected. This could potentially be the explanation asto why there are no sudden changes in pressure during times of disconnection.A more interesting observation in figure 4.3 is that after the average pressure in bothboreholes drops from day 193 onwards. Between day 193 and 210, both boreholes showdiurnal pressure oscillations. These are at times more pronounced in one borehole or theother, suggesting that the strength of connection of either with the rest of the drainagesystem varies in time. During this period, there are sporadic switching events beforeS15P83 disconnects from the drainage system after day 210. A good connection by thetwo boreholes to the drainage system and with each other, identified as a switch-on, isformed on day 214. This is similar to that on day 216 in figure 4.2. Several switchingevents ensue, two of which were not picked up by the algorithm (the switch-on andswitch-off on day 222). During the times S14P18 is disconnected from S15P83, S15P83remains connected to the rest of the drainage system. The pressure at S15P83 tends toincrease sharply just before connection with S14P18. As also shown in figure 4.2, thepreviously disconnected borehole (S14P83) initially lags the pressure in the connectedborehole (S15P83) before the two become well-connected to each other near the peak inthe pressure spike. The pressures at which connections are established appear to varyfrom event to event.Overall, the example in 4.3 can be divided into two distinct periods, the first (earlierin the season) suggesting that there may be multiple physical connections that, eitherdirectly or indirectly through the rest of the drainage system, link the two boreholes, andonly some of these connections are opened or closed during switching events. The second25period is distinguished by the abrupt transition into a series of switching events whichtypically have a pair of switch-on and switch-off associated with a pressure spike. Fur-thermore, the actual switch-ons may well be earlier than the switch-on events detected,with varying lags as seen in figure 4.2.In figure 4.4, S14P18, now red but previously plotted in blue in figure 4.3, is nowpaired up with S14P61 (blue) instead. An abrupt connection is made by both boreholeson day 214. The same switching event is also detected at the same time in figure 4.3, andlater followed by a series of switching events. This suggests a larger-scale reorganizationof the drainage system, not just a single connection being created. During my searchthrough the data output by the algorithm, this event showed up in many more pairs ofboreholes. The event itself is preceded by several days during which there is little if anyevidence for diurnal pressure spikes. A plausible explanation is that there was a lack ofsurface melt for several days prior to day 214, and the resumption of water supply to thesystem led to a reactivation of the drainange system.Figure 4.5 shows a pair of borehole records (S14P11 and S14P25) from a different partof the glacier in 2014. The behaviour during the first few days (day 193 to 197) appearssimilar to the first period described in figure 4.3 exhibiting switch-on and switch-off eventswith no abrupt change in pressure. However, the subsequent connections (days 208 and216) are established while both pressures are increasing and are not followed by a seriesof switching events. Note that the pair of switching events on day 216 was not detectedby the algorithm, but added manually. This behaviour appears qualitatively differentfrom that seen in the other figures, such as figure 4.3. Borehole S14P11 disconnects fora long period of time but then sporadically reconnects much later in the summer seasonat times when water pressure in S14P25 rises above to match that in S14P11.5.2 PhysicsBelow, I will describe four plausible mechanism by which the switching events seenin the data might be cause. This list of mechanisms is not intended to be exhaustivebut provides a basis for future analysis of the data. The simplest mechanism is that26the openings in the drainage system are “elastic”, in which the conduit size is simply afunction of the effective pressure. If this is the case, we would expect opening and closingto happen at a single critical effective pressure (or water pressure). This is evidently notconsistent with the results in 4.3 as the pressure at which connections and disconnectionsare made are typically not the same, with the disconnection pressures being lower thanat connection, at least later in the season.A “viscous” mechanism would be on in which conduit size evolves in time, with therate of change of conduit size being a function of effective pressure and conduit size. Thisis the usual model for dynamically evolving cavities (Schoof, 2010). Such a mechanismcould explain the time lags in pressure rise of connecting cavities via a diffusive process,but requires testing through a modelling exercise. This can potentially build on existingmodels for using time evolving cavities (e.g. Werder et al. (2013)).Crack formation may possibly connect two cavities as well. This idea is similar tothe “elastic” conduit except the conduit does not pre-exist prior to connection and theconnection need not establish on the glacier bed. This can potentially explain the abruptconnection observed in figure 4.2 to 4.4. Hydraulic fracturing could potentially causethe cracks to form due to an excessive amount of surface melt entering the subglacialdrainage system. Such a mechanism could conceivably explain the opening of connectionsassociated with sharp pressure spikes, especially after a longer period of apparent lack ofdrainage activity (such as day 214 in figures 4.3 and 4.4). Hydrofracturing is associatedwith moving water opening a crack for itself, and should lead to time delays in theopening of connections across a larger domain (Tsai & Rice, 2010). Given that manyconnections were formed on day 214 in 2015, it may be possible to test for this by lookingat the sequence of switching events relative to borehole location.A last possibility is an overtopping of a connection between two basal crevasses atsome height above the bed connecting two cavities. This could explain the observationmade in figure 4.5 where the two boreholes connect when the water pressure in theborehole that is generally at a lower pressure and connected to an active drainage system,reaches the water pressure in the adjoining, generally disconnected borehole. However,this phenomenon is not consistent with observations made in the other figures.27BibliographyBartholomew, I., Nienow, P., Mair, D., Hubbard, A., King, M.A., & Sole,A. 2010 Seasonal evolution of subglacial drainage and acceleration in a Greenlandoutlet glacier. Nature Geoscience 3 (6), 408–411.De Paoli, L. & Flowers, G.E. 2009 Dynamics of a small surge-type glacier inves-tigated using one-dimensional geophysical inversion. Journal of Glaciology 55, 1101–1112.Flowers, G.E., Copland, L. & Schoof, C.G. 2014 Contemporary glacier processesand global change: Recent observations from Kaskawulsh Glacier and the DonjekRange, St. Elias Mountains. Arctic 67 (5), 22–34.Fountain, A.G. & Walder, J.S. 1998 Water flow through temperate glaciers. Reviewsof Geophysics 36 (3), 299–328.Fudge, T.J., Harper, J.T., Humphrey, N.F. & Pfeffer, W.T. 2005 Diurnalwater-pressure fluctuations: Timing and pattern of termination below Bench Glacier,Alaska, USA. Annals of Glaciology 40 (1), 102–106.Harper, J.T., Humphrey, N.F., Pfeffer, W.T., Fudge, T. & O’Neel, S. 2005Evolution of subglacial water pressure along a glacier’s length. Annals of Glaciology40 (1), 31–36.Hewitt, I.J. 2011 Modelling distributed and channelized subglacial drainage: The spac-ing of channels. Journal of Glaciology 57 (202), 302–314.Hewitt, I.J. 2013 Seasonal changes in ice sheet motion due to meltwater lubrication.Earth and Planetary Science Letters 371-372, 16–25.Hubbard, B.P., Sharp, M.J., Willis, I.C., Nielsen, M.K. & Smart, C.C. 1995Borehole water-level variations and the structure of the subglacial hydrological systemof Haut Glacier d’Arolla, Valais, Switzerland. Journal of Glaciology 41 (139), 572–583.28Iken, A. & Bindschadler, R.A. 1986 Combined measurements of subglacial wa-ter pressure and surface velocity of Findelengletscher, Switzerland: Conclusion aboutdrainage system and sliding mechanism. Journal of Glaciology 32 (110), 101–119.Johnson, P.G. & Kasper, J.N. 1992 The development of an ice-dammed lake: thecontemporary and older sedimentary record. Arctic and Alpine Research 24, 304–313.Kavanaugh, J. & Clarke, G.K.C. 2000 Evidence for extreme pressure pulses in thesubglacial water system. Journal of Glaciology 46 (153), 206–212.Murray, T. & Clarke, G.K.C. 1995 Black-box modeling of the subglacial water-system. Journal of Geophysical Research—Solid Earth 100 (B6), 10231–10245.Ro¨thlisberger, H. 1972 Water pressure in intra- and subglacial channels. Journal ofGlaciology 11, 177–203.Schoof, C.G. 2005 The effect of cavitation on glacier sliding. Proceedings of the RoyalSociety A: Mathematical, Physical and Engineering Science 461 (2055), 609–627.Schoof, C.G. 2010 Ice-sheet acceleration driven by melt supply variability. Nature468 (7325), 803–806.Schoof, C.G., Rada, C.A., Wilson, N.J., Flowers, G.E. & Haseloff, M.2014 Oscillatory subglacial drainage in the absence of surface melt. The Cryosphere 8,959–976.Sugiyama, S. & Gudmundsson, G.H. 2004 Short-term variations in glacier flow con-trolled by subglacial water pressure at Lauteraargletscher, Bernese Alps, Switzerland.Journal of Glaciology 50 (170), 353–362.Tsai, V.C. & Rice, J.R. 2010 A Model for Turbulent Hydraulic Fracture and Ap-plication to Crack Propagation at Glacier Beds. Journal of Geophysical Research115 (F03007), doi:10.1029/2009JF001474.Walder, J. 1986 Hydraulics of subglacial cavities. Journal of Glaciology 32, 439–445.29Werder, M.A., Hewitt, I.J., Schoof, C.G. & Flowers, G.E. 2013 Evolutionof subglacial water pressure along a glacier’s length. Journal of Geophysical Research:Earth Surface 118 (4), 2140–2158.Wheler, B.A. 2009 Glacier melt modelling in the Donjek Range, St. Elias Mountains,Yukon Territory. Master’s thesis, Simon Fraser University.Wheler, B.A., MacDougall, A.H., Flowers, G.E., Petersen, E.I., Whit-field, P.H. & Kohfeld, K.E. 2014 Effects of Temperature Forcing Provenanceand Extrapolation on the Performance of an Empirical Glacier-Melt Model. Arctic,Antarctic, and Alpine Research 46 (2), 379–393.Wilson, N. 2012 Characterization and interpretation of polythermal structure in twosubarctic glaciers. Master’s thesis, Simon Fraser University.Zwally, H.J., Abdalati, W., Herring, T., Larson, K., Saba, J. & Steffen, K.2002 Surface melt-induced acceleration of Greenland ice-sheet flow. Science 297 (5579),218–222.30A | Relating the connection toler-ance to the switch-finding tol-eranceDefine case 1 as a change from being well-connected to not well-connected and case2 as the opposite change. Let the not well-connected interval in the search intervalof case 1 and the first not well-connected interval in the search interval of case 2 beTB1 . Linear regression in (3.7) yields the fit coefficients mB1 and cB1 . The last notwell-connected interval interval in the search interval of case 2 is designated as TB2 andcorrespondingly has an associated mB2 and cB2 . A well-connected interval that gives thebest regression coefficients mA and cA as described in Section 3.2 is TA. In TB1 , if Test1 was failed, from (3.8), RMSEB1 > toltest 1 × σ(P1) = tol1,B1 . If Test 2 or 3 was failed,RMSEB1 ≤ tol1,B1 . Similarly, in TB2 , RMSEB2 > toltest 1 × σ(P1) = tol1,B2 if Test 1was failed and RMSEB2 ≤ tol1,B2 if Test 2 or 3 failed. If we demand that at least oneswitch-on must be found in case 1, then there needs to be at least one well-connectedinterval and not well-connected interval interval. The absolute minimum and maximumerror within the combined intervals B1∪A, minB1∪Aej and maxB1∪Aej must therefore satisfy thefollowingminB1∪Aej < tolconnection < maxB1∪Aej (A.1)where ej = |P1,j −mAP2,j − cA|, j is the jth data point in the search interval, andtolconnection is a range of values to be found. Likewise, a pair of switch-on and switch-off in case 2 demands that there are at least two well-connected intervals and one notwell-connected interval interval. The absolute minimum and maximum error within thecombined intervals B1 ∪ A ∪ B2, minB1∪A∪B2ej and maxB1∪A∪B2ej must therefore satisfy the fol-lowingminB1∪A∪B2ej < tolconnection < maxB1∪A∪B2ej (A.2)31In situations where Test 1 was failed in TB1 and TB2 , we have maxB1ej ≥ RMSE ′B1and maxB2ej ≥ RMSE ′B2 whereRMSE′B =√√√√√√NB∑j=1(P1,j −mAP2,j − cA)2NBBy the definition of a linear regression, we have RMSE′B1≥ RMSEB1 and RMSE′B2≥RMSEB2 . Since RMSEB1 > tol1,B1 and RMSEB2 > tol1,B2 , this givesmaxB1∪A∪B2ej > min(tol1,B1 , tol1,B2)Next,minAej ≤ RMSEA ≤ toltest 1 × σ(P1) = tol1,A⇒ minB1∪A∪B2ej ≤ tol1,AAssuming tol1,A ≤ min(tol1,B1 , tol1,B2), setting tolconnection = min(tol1,B1 , tol1,B2) will obey(A.2). However, in general, minAej  min(tol1,B1 , tol1,B2), so I set tolconnection = tolfind ×min(tol1,B1 , tol1,B2), where the switch-finding tolerance tolfind ∈ (0, 1]. For case 1, sinceTB2 is not (yet) available when the search is being done, setting tolconnection = tolfind ×tol1,B1 will satisfy (A.1).In situations where Test 2 or 3 was failed in TB1 and/or TB2 , since RMSEB1 ≤ tol1,B1and/or RMSEB2 ≤ tol1,B2 , tolconnection is no longer necessarily less than minB1∪A∪B2ej. Iftolconnection ≥ minB1∪A∪B2ej, then no switching event will be detected in the search intervalB1 ∪ A ∪ B2 even though the three-criterion test suggests that there are at least twoswitching events in that search interval. A similar argument can be made for the searchinterval B1 ∪ A. This usually happens when the two boreholes are slowly phasing frombeing well-connected to being not well-connected interval or vice versa. To deal with thisproblem, I do the followinga) In case 1, if no switch-on was found, then the left limit of the search interval is32extended by an interval T and the search for all switching events is reconducted.The process is repeated until a switch-on is found. However, the search intervalcan only be extended up until but not including the last switch-off found.b) In case 2, if there is at least one switching event found but the last switching eventis not a switch-off, TB2 is reset as being well-connected. This enables a switch-offto be found subsequently for the last switch-on found.33


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