Summary: A settlement limit is an angle with the length left out. The karst monitoring criteria are settlement amounts, settlement rates and crack widths, and none of them carries a base, while the damage criteria they stand in for, one in 500 for design and one in 300 for the onset of cracking, are ratios that do. In cover collapse the missing base is the trough half width, which the published collapse range ties to the cover thickness at about a third to a half of it, so the same twenty millimetres is six times the design criterion in three metres of cover and six tenths of it in thirty. The rock then decides whether the monitoring object is the water, the rock or the cavern, the precursor is several times wider than the hole, and the deepest point of a sag is the one place that does not rotate. A wireless tilt channel is a change detector here, not a compliance instrument.
1. The Settlement Limit Nobody Prints a Length For
Every settlement number in the geotechnical documents is a stand in for something else. The quantity that damages a building is not a settlement, it is a rotation, and the rotation is a settlement divided by the distance between the two points that settled differently. The documents carry both quantities, and they carry them in an odd arrangement: the ratio is stated as a ratio, and the settlement is stated as a bare number in millimetres with no distance anywhere near it.
The bare number has a genealogy. Terzaghi recorded in 1938 that ordinary buildings tolerate about 20 mm of differential settlement without cracking, and that figure passed into European and American practice and is still what a client means when asking for a settlement limit. It is meaningless on its own. Twenty millimetres over three metres is 6.67 millimetres per metre and over thirty metres is 0.67, a factor of ten, and the building responds to which of the two it is, not to the twenty.
Skempton and MacDonald put the ratio on a footing in 1956 with a study of 98 buildings, 40 of which showed damage. Cracking of load-bearing or panel walls becomes likely above a distortion of one in 300, structural damage becomes probable above one in 150, and one in 500 was recommended as a design criterion with a margin against cracking. For load-bearing brick walls, one in 1000 was to be avoided. Those three numbers convert to 3.33, 6.67 and 2.00 millimetres per metre, and to 0.191, 0.382 and 0.115 degrees, and they are what every settlement figure in the field is trying to approximate.
Eurocode 7 carries both kinds in Annex H and the pairing is instructive. Total settlements up to 50 mm are described as often acceptable for normal structures on isolated foundations, and the maximum relative rotation is given as one in 500 in a sagging mode and half of that, one in 1000, in a hogging mode. The code does not explain the settlement figure and it does explain the rotation, and the factor of two between sagging and hogging is itself an admission that the geometry rather than the magnitude is the subject.
The customary totals are the same story one step further removed. Twenty five millimetres on sand and 40 to 65 on clay are the numbers in circulation, and the reason given for them is that the differential settlement is difficult to predict, so the total is limited on the assumption that the differential is a fraction of it. That assumption is stated openly in the literature, and it is a ratio between two settlements, not a length. A review of settlement criteria prepared for satellite radar risk assessment says it in one sentence: the limit on total settlement is in essence a control on differential settlement. The same review records the arithmetic that produced the sand figure, that differential settlement runs at about three quarters of the total, so 20 mm of differential became 25 mm of total. Two proxies deep, and the base was already lost at the first step.
There is exactly one place in this literature where a settlement and a rotation appear side by side as a pair, and it is a first-stage screening rule rather than a design rule: a rotation of one in 500 together with a settlement of 10 mm, either of which sends a building to a second, more detailed assessment. Even there the 10 mm has no base attached.
Now the karst documents. The Chinese standard for karst ground collapse monitoring, T/CAGHP 075-2020, splits the object into three layers, the karst structure below, the groundwater as the trigger layer, and the surface deformation as the expression, and on the surface it asks for settlement, ground cracks and the appearance of new collapse pits, with the settlement rate as the index that matters. Published summaries of the same standard describe a layout of three rings and one line: a dense core around existing pits and thin cover, a transition ring, a sparse background ring for comparison, and a profile along the groundwater flow direction and the karst trend. Nothing in that description attaches a length to the settlement either, and the parallel standard for mined-out ground, T/CAGHP 078-2020, is a separate document for a separate mechanism, which is the first useful boundary in the field.
The separate standard for mined-out ground is more than a filing convention. In a mine the void has a known plan position, a known depth and a known height, taken from the extraction record, the surface trough follows the influence function of the seam geometry, and the subsidence is therefore predictable before it happens and measurable against a design. A karst void is unknown in plan, unknown in size and still growing, and the surface survey is the only description of it that exists. The two mechanisms share a word and a sensor and almost nothing else, and a monitoring plan transferred from one to the other carries across the wrong assumption, which is that the geometry is known.
The claim of this article is narrow and it is arithmetic. In karst the missing length is not a choice, it is a soil property. The published collapse geometry ties the plan dimension of the depression to the thickness of the cover, so the same settlement number means things that differ by more than a factor of ten depending on how much soil sits over the void.
2. The Same Twenty Millimetres Is Six Times the Criterion
The surface over a growing void is neither a cone nor a dish. Profiles measured across cover-collapse features are bells: a rounded centre, an inflection where the curvature changes sign, and a tail that decays outward. Describe it by its half width, the radius at which the sag has fallen to about sixty one percent of its centre value, which is also where the slope is steepest. The steepest slope of such a bell is exactly 0.607 times the centre sag divided by the half width, because that is where the derivative of the profile peaks. That single relation carries the whole argument of this section.
| Centre sag | Angle over 3 m | Angle over 10 m | Angle over 30 m |
| 10 mm | 0.191 degrees | 0.057 degrees | 0.019 degrees |
| 20 mm | 0.382 degrees | 0.115 degrees | 0.038 degrees |
| 50 mm | 0.955 degrees | 0.286 degrees | 0.096 degrees |
Read the middle row. Twenty millimetres over three metres is 6.67 millimetres per metre and 0.382 degrees, which is three and a third times the design criterion of one in 500 and twice the one in 300 at which cracking is expected. The same twenty millimetres over thirty metres is 0.67 millimetres per metre and 0.038 degrees, a third of the design criterion. One settlement figure, a factor of ten in the quantity that actually causes damage, and no indication in the figure itself of which case is being specified.
The break-even is easy to state and worth remembering: twenty millimetres lands exactly on one in 500 when the half width is ten metres. Any narrower depression makes the same figure permissive, any wider one makes it conservative.
None of this would matter much in a building, where the base is the bay spacing and the structural engineer knows it. It matters in karst because nobody chooses the base. A review of cover-collapse case data reports that the collapse range is often approximately equal to the soil cover thickness, or two thirds of it, and that thin cover produces a narrow collapse while thick cover produces a wide one. Take the half width as a third to a half of the cover, which is what that relation implies once the depression is wider than the hole, and the geometry becomes a function of the depth of residual soil over the rockhead.
| Cover thickness | Half width taken as a third | Angle for a 20 mm sag | Times the one in 500 criterion |
| 3 m | 1.0 m | 0.695 degrees | 6.07 |
| 6 m | 2.0 m | 0.348 degrees | 3.03 |
| 12 m | 4.0 m | 0.174 degrees | 1.52 |
| 30 m | 10.0 m | 0.070 degrees | 0.61 |
Three metres of cover makes twenty millimetres six times the design criterion. Twelve metres puts it at one and a half times. Thirty metres puts it at six tenths. Across the half widths in the table, one metre to ten metres, the same number moves by a factor of ten, and the direction of the error is not constant: in thin cover the settlement figure is permissive, in thick cover it is conservative, and a single figure cannot be both. The covers that karst sites actually have, five to fifteen metres of residual soil over a dissolved rockhead, sit in the permissive half. That is the finding of this article, and it is not a criticism of the standards, which never claimed to publish a length: it is a statement that a settlement specification without a base has an unstated assumption inside it, and in karst that assumption is wrong by a factor of three to six.
3. Two Criteria That Move in Opposite Directions With the Cover
The stability of the cover has been measured rather than argued. Centrifuge models of weakly cemented sand over a cavity in limestone found a sharp transition governed by the ratio of cover to cavity diameter: at 0.25 or less, the plug of soil that fell into the cavity penetrated the full thickness of the layer and left an open hole at the surface, and at 0.31 or more the plug stopped partway and left a stable arch. Analysis of a cohesive soil at 25 kPa puts the threshold at a cover of 2.5 cavity diameters, and cemented sand at 43 kPa can lose its void all the way to the surface with a cover of a quarter of the diameter.
| Basis | Geometry at failure | Outcome |
| Centrifuge, weakly cemented sand | cover to cavity 0.25 or less | plug penetrates, open hole |
| Centrifuge, weakly cemented sand | cover to cavity 0.31 or more | stable arch, no hole |
| Analysis, cohesion 25 kPa | cover below 2.5 cavity diameters | void enlarges upward |
| Centrifuge, cohesion 43 kPa | cover below a quarter of the diameter | void reaches the surface |
| Field observation | cavity 0.2 m, cover above 15 m | collapse, water-driven |
| Case data review | cavity to cover 3 to 5 | stable, wider when it fails |
Read the same tests the other way up and the ratio of cavity to cover becomes the indicator. At or above 1.3 the cover is thin and highly unstable, with the failure surface reaching the ground easily and a narrow collapse above it. In the range 3 to 5 the cover is stable, arching carries the load to the sides, and the surface waits, but the collapse that eventually arrives is wider. The last row of the table is the one that matters for a monitoring plan: the criterion that says a site is safe is also the criterion that says the eventual hole will be large.
So the two consequences run in opposite directions from the same variable. Thin cover fails easily, fails often, and makes a small hole with a steep rim. Thick cover resists for a long time and then makes a wide hole with a gentle rim. The collapse range scales with the cover, and the rim angle goes as the sag divided by a half width that also scales with the cover, so a wide gentle bowl and a narrow steep one can carry the same centre sag and call for completely different responses.
A third handle comes from the volume relation. The volume of the settlement trough is proportional to the volume of the cavity, divided by one plus an expansion coefficient, so the integral of the surface depression is the only measurement of the void that the surface offers. That changes what a survey of a karst depression should report. The peak settlement is a poor descriptor, because two sites with the same 20 mm peak can differ by a factor of a hundred in the volume of ground that has gone missing, while the volume of the depression is a direct proxy for the volume of the void. A levelling run that reports only its maximum has thrown away the number that connects the surface to the cavity.
That is the practical trap, and it is a budget trap rather than a technical one. The instrument density that thin cover demands is the highest of any cover thickness, because the depression is small and the rim is steep, and the lead time that thin cover offers is the shortest, because there is little soil to ravel before the arch reaches the surface. The two requirements are anti-correlated with the money: the spend goes to the sites where it buys the least warning, and the sites where a sparse array would be adequate are the ones with years of notice.
4. The Rock Decides Whether You Are Watching a Trigger or a Generator
The dissolution rates are not a detail, they are the reason the same word covers two different hazards. For limestone the natural denudation measured across many terrains is 1 to 3 cm per 1,000 years, which is 0.01 to 0.03 mm per year, and a measurement in a Zagros basin put it at 0.236 cm per 100 years. Lowering a rock surface by one metre therefore takes 33,000 to 100,000 years, and by 100 mm takes 3,300 to 10,000 years. Nothing that happens during the design life of a structure dissolves a cavity of consequence in limestone.
In limestone the void is therefore inherited. It is older than the asset, older than the drainage scheme and usually older than the settlement, and the hazard is not generation but triggering. Field reviews of induced sinkholes converge on the same list of triggers and the same first cause: a water table lowered below the soil-rock contact, which removes buoyant support and raises the pore pressure gradient and its amplitude of variation; a concentrated discharge from a new drain, a leaking main or a change in surface grading; a wetting front arriving from above through unsaturated soil; and a load added over a void. Every item on that list is a change to water or to load, and none of them is a change to the rock.
This is also why the cover ratio criterion has to be used with care. Openings 0.2 m across have produced collapses through more than 15 m of cover, which no bearing failure criterion predicts, because the mechanism is transport and not support. Soil ravels into the opening, the void migrates upward toward the seepage source, the seepage path shortens, and the process accelerates. Tank experiments showed collapse reaching the surface regardless of the thickness of the soil as long as the fracture was not blocked by the fallen material, and the process stopping each time the fracture filled and restarting when it was cleared. A bearing criterion bounds the dry case. The water case is a conveyor.
Gypsum changes the clock by two orders of magnitude. Measured in the same Zagros basin, the dissolution rate was 23 cm per 100 years against 0.236 for the limestone, a factor of 97. Under the steep hydraulic gradients beneath a dam, the modelling puts the widening at 0.1 cm per year in limestone and at least 1 cm per year in gypsum, with leakage becoming excessive within 25 years in the first case and 5 in the second. In flowing water the rate leaves the geological range entirely: gypsum balls suspended in Spanish irrigation canals lost 0.393 m of surface per year at a flow of 0.35 m per second and 0.853 m per year at 0.99 m per second, and a three metre block of gypsum that fell from a cliff at Ripon Parks into a river dissolved completely in about 18 months.
| Material and condition | Rate | Time to open 100 mm |
| Limestone, natural denudation, lower figure | 0.01 mm per year | 10,000 years |
| Limestone, natural denudation, upper figure | 0.03 mm per year | 3,300 years |
| Limestone, measured in a Zagros basin | 0.024 mm per year | 4,200 years |
| Gypsum, measured in the same basin | 2.3 mm per year | 43 years |
| Gypsum, beneath a dam gradient | up to 10 mm per year | 10 years |
| Gypsum, flowing water at 0.35 m per second | 393 mm per year | 3 months |
| Gypsum, flowing water at 0.99 m per second | 853 mm per year | 6 weeks |
The right hand column is the one that decides a monitoring plan. One hundred millimetres of opening takes ten thousand years in limestone and forty three years in gypsum, and the gypsum sinkhole of 376 cubic metres reported from the same basin took 71 years to form. That is the order of a design life rather than of a geological epoch, and it is the figure that separates gypsum from limestone in any risk register. Salt completes the series: relative solubility runs 1 to 100 to 10,000 for limestone, gypsum and halite, salt caves develop to accessible dimensions in a few hundred years against more than ten thousand in limestone, and a cavern collapse in a salt dome in 1954 left a hole 270 m across.
So the rock decides what the channel is for. In limestone, watch the water, and treat the tilt array as a secondary confirmation that a trigger has begun to move something. In gypsum, watch the rock, because generation and triggering have collapsed into a single question and the rate shows up in the record as a trend rather than a step. In salt, watch the cavern, because the object that fails is a mapped or a man-made one and its geometry is known in advance. Three different monitoring objects for one word, and the rock type belongs in the first sentence of the plan rather than in the geology appendix.
5. Three Clocks, and the Alarm Belongs to the Slowest One
The published precursor times run from a few weeks to more than two years, and the range is the most useful thing about them. Along the Dead Sea shore, radar series show precursory subsidence before the collapse of most sinkholes, over 2 to 5 months, at about 3 mm per month accelerating to 2 cm per month before three consecutive events inside one month. Four documented events give precursory times of 32 to 48 days, 66 days, 88 days and 92 days, with a linear relation reported between the depth of the salt layer and the precursor time and an exponential one between hole diameter and precursor time. The 16 day repeat of the radar is the floor of the method, and anything faster is invisible to it.
At Bayou Corne in Louisiana the surface deformed at least a month before the collapse of 3 August 2012, and the deformation was mostly horizontal, up to 260 mm toward where the hole would later form, spread over an area 500 m by 500 m against an initial hole of about one hectare. The cavern was more than 900 m below the surface and had been mined closer to the edge of a salt dome than the design assumed. The direction is the part that matters for a ground array: horizontal movement on flat ground produces almost no angle change at the surface, and an instrument that measures rotation is nearly blind to it.
At Road 90 on the western shore of the Dead Sea the sequence is documented to the day. Baseline subsidence below 0.2 mm per day through the first nine months of 2012, then a doubling to about 0.4, a formal alert on 14 October 2012, a bypass planned and built from the radar maps, the road closed on 26 January 2015 when cracks appeared, and the hole confirmed on 1 February 2015 at 5 m wide and 9 m deep, directly beneath reinforcement laid in 2002. Twenty seven months of notice. The quantity that fired was the acceleration rather than the rate, because 0.2 mm per day is 73 mm per year, an ordinary figure for a settling fill.
| Case | Lead time | What moved |
| Dead Sea shore, radar series | 32 to 48, 66, 88 and 92 days | 3 mm per month, then 2 cm per month |
| Bayou Corne, Louisiana, 2012 | at least one month | 260 mm horizontal over 500 m |
| Road 90, Dead Sea shore | 27 months from alert to closure | 0.2 rising to 0.4 mm per day |
| Guatemala City, 2007 | weeks of noise and odour | nothing instrumented, residents reported it |
| Chongqing, 2020, two collapses | alarm raised before both | groundwater level and air pressure steps |
Two of those rows are worth pausing on. In Guatemala City the warning existed and was not read: weeks of tremors, subterranean noises, sewage odour seeping into houses and insects leaving the ground, and three deaths when the surface went. In Chongqing in 2020 an air, space and ground monitoring system raised the alarm before two karst collapses, and the quantities it used were groundwater level and air pressure steps rather than ground angle. The successful instrumented case in the record is a water and gas case.
The Chinese technical requirement for karst collapse monitoring and warning, circulated on the national group standard platform, defines its four levels by time window rather than by magnitude: red for a probability of large scale occurrence within hours or days, orange within days or weeks, yellow within weeks or months, blue within a year. Its criteria are grouped as rainfall, deformation and pre-collapse anomaly. That is a clock hierarchy, and it is the most useful sentence in the document for an instrument selection, because it says plainly that no single instrument serves all four windows.
The rainfall criteria in that requirement deserve a note, because they make the alarm multivariate by design. Effective rainfall, accumulated rainfall and the relation between deformation and rainfall are listed beside the deformation rate, which means the standard expects a ground movement to be read against a water input rather than against a fixed threshold. That is the only practical way to separate a seasonal cycle from a developing failure: the same 0.05 degrees of tilt is routine after a wet month and alarming in a dry one, and a channel that does not carry the local rainfall series alongside the angle cannot make the distinction, however good its resolution.
A tilt channel sampled hourly and alarmed on level is a red window instrument in its sampling and a yellow window instrument in its zero stability, and the two do not have to agree. The band it can actually serve is the days to weeks one, which is the orange level: fast enough to see a sag developing, stable enough over the interval that the drift has not consumed the signal. Alarming a slow clock with a fast instrument produces a record that is mostly false alarms and one event that was missed because the threshold had been raised to stop the false ones.
6. The Precursor Is Five Times Wider Than the Hole
The Bayou Corne numbers give the ratio directly. The precursor covered 500 m by 500 m, 25 hectares, while the initial hole was about one hectare, an equivalent diameter of 113 m. That is 4.4 times the linear dimension and 25 times the area. The precursor is not a slightly larger version of the hole, it is a different size of object, and any array sized to the hole will watch the wrong footprint.
The arithmetic of coverage follows. A unit sees the slope within a radius of about twice the half width, beyond which the slope has fallen below any practical detection threshold. The area it watches is therefore a disc of radius 2i, and on a grid of spacing S the fraction of ground covered is pi times 4 i squared divided by S squared. Where that fraction exceeds one, the discs overlap and the grid is fully covered.
| Half width | Radius to be covered | Spacing for full cover | Coverage at a 20 m spacing |
| 2 m | 4 m | 4 m | 13 percent |
| 4 m | 8 m | 8 m | 50 percent |
| 8 m | 16 m | 16 m | 201 percent |
| 16 m | 32 m | 32 m | 804 percent |
At a half width of 2 m the disc is 4 m in radius and needs a 4 m grid; at a 20 m grid each unit watches 13 percent of its own catchment and 87 percent of the site is between sensors. At a half width of 8 m the same 20 m grid covers the ground twice over. Since the half width tracks the cover, a site with 3 m of soil needs a grid about six times denser than a site with 18 m, for the same nominal protection, and it is the 3 m site that gives the least notice.
That is the honest reason karst detection is not a sensor grid problem. The area to instrument grows as the square of the cover and the sensor count with it, while the lead time falls. Areal methods, radar interferometry, laser scanning, gravity and drilling, are the detection instruments in this field, and a ground array is a densification around a known feature or a watch on a mapped cavity. A plan that proposes a grid across an uncharacterised karst site is proposing a lottery with an expensive ticket.
The 25 to 1 area ratio also defines the exclusion zone, and the two uses conflict. The area that has to be cleared when an alarm sounds is the precursor area and not the hole, which at Bayou Corne meant 25 hectares rather than one, and that same 25 hectares is the area the array was supposed to have been watching. This is the structural reason the areal methods dominate karst work: they are the only ones whose coverage scales with the size of the phenomenon, while a point instrument’s coverage is fixed by its own spacing and shrinks in relative terms as the feature gets bigger.
The three ring and one line layout in the karst monitoring standard is exactly this logic expressed as geography: dense at the known pits and the thin cover, medium through the transition, sparse at the background, plus a profile along the groundwater flow and the karst trend. It tells you where the rings go. It does not tell you the spacing, and the spacing is the entire cost.
7. The Point of Maximum Settlement Reads Zero
The tangent at the centre of a bell is horizontal. A tilt unit standing at the deepest point of a sag therefore reads nothing, and the steepest ground in the whole depression is a ring at the half width radius, where the slope reaches the 0.607 figure. The instinct of anyone who has not differentiated the profile is to put the instrument where the ground has gone down most, and that is the one place in the feature where the ground has not rotated.
| Half width | 20 mm sag in mm per m | In degrees | Times the one in 500 criterion |
| 1 m | 12.13 | 0.695 | 6.07 |
| 2 m | 6.07 | 0.348 | 3.03 |
| 4 m | 3.03 | 0.174 | 1.52 |
| 10 m | 1.21 | 0.070 | 0.61 |
| 20 m | 0.61 | 0.035 | 0.30 |
The direction is radial everywhere, which gives the second half of the method. In a bowl the steepest descent points toward the centre at every radius, so the magnitude peaks in a ring and decays to zero at the centre and in the tail, while a rigid rotation of the whole area, from a slope, a settling fill or the block the units stand on moving as one, produces a uniform field: same magnitude and same direction everywhere. The two states differ in shape and not in magnitude, and no threshold on a single unit can separate them.
A difference between two units a known distance apart is therefore the cheapest useful quantity, and divided by the distance it is the rate at which the slope is changing. It cancels what the pair genuinely shares, which is the rigid rotation and the common temperature, and it does not cancel their two zero offsets, which add in quadrature, so a pair is still a comparator and not an absolute reference. What a pair cannot do is tell a bowl from a slope, because both give a steady change along the line.
The signature of a bowl is that the rate of change reverses. Inside the inflection radius the slope steepens as radius grows; outside it the slope relaxes. Seeing a reversal needs three units on a line. Locating the centre in plan needs a fourth, off that line. A pair is a comparator, three on a line is a shape test, and four in plan is a location, which is a more useful design rule than any minimum spacing and it comes out of the geometry rather than out of a catalogue.
The division of labour with the survey then follows exactly. The survey recovers the profile and therefore the half width, which is the length that converts a settlement into an angle, but it samples time coarsely. The array samples time continuously and has no length of its own. Neither substitutes for the other: the survey supplies the base and the array supplies the date.
8. What the Channel Can and Cannot Do Here
The same literature carries three rungs and they are not interchangeable: one in 1000 for load-bearing brick walls and for the hogging mode, one in 500 as the design criterion in sagging, one in 300 at which cracking is expected. Converted, those are 1.0, 2.0 and 3.33 millimetres per metre, and 0.057, 0.115 and 0.191 degrees.
| Datasheet term | In degrees | In mm per m | Percent of the one in 500 criterion |
| Resolution | 0.001 | 0.017 | 0.9 |
| Accuracy | 0.005 | 0.087 | 4.4 |
| Zero deviation | 0.05 | 0.873 | 43.6 |
| Zero drift, typical | 0.08 | 1.396 | 69.8 |
| Zero drift, maximum | 0.13 | 2.269 | 113.4 |
Against the design criterion the ranking is comfortable until the last row and then it is not. Accuracy consumes 4.4 percent of the allowance and the zero deviation 43.6, both of which a careful installation can live with. The maximum zero drift consumes 113.4 percent on its own, and the typical figure 69.8, which leaves nothing for the movement being measured. Against the cracking threshold of one in 300 the same four terms consume 2.6, 26.2, 41.9 and 68.1 percent, and every one of them fits with a third of the budget to spare.
That is the sixth position this series has recorded on this axis and the first where the verdict flips inside one family of published limits. Elsewhere the answer was outside, or well inside, or in the middle, or all inside. Here the same hardware is non-compliant against the design criterion and compliant against the cracking threshold, so a specification that says the instrument is suitable for settlement monitoring without naming the rung has said nothing at all. The rung has to be in the sentence, and so does the base, because the rung is a ratio and the datasheet is an angle.
What the channel can honestly be told to do here is detect a change. The absolute judgement belongs to the survey, and the reason is not caution but arithmetic: 113 percent of a budget cannot certify compliance with it, while 113 percent of a budget is a perfectly good offset against a signal that grows to several times the drift. A sag of 20 mm in 3 m of cover is 0.695 degrees, five times the maximum drift, and 50 mm in the same cover is 1.74 degrees, thirteen times it. The late stage is well inside the channel. The early stage is not.
The instrument class comparison makes the boundary concrete. The review that tabulates tiltmeters used on sinkhole sites quotes a detection threshold of one arc second, which is 0.00028 degrees, eighteen times finer than the accuracy figure here and a different class of device with a different installation. The Bayou Corne precursor, 260 mm over 500 m, is 0.03 degrees, which is a quarter of this channel’s maximum zero drift and less than half its typical one. The class of instrument that saw that precursor is not this class, and claiming otherwise would be the one thing this article must not do.
One further number belongs in the specification and is usually left out. A 27 month precursor sampled hourly is about 20,000 readings per axis, and no threshold applied to that series raw will be stable, because the diurnal and seasonal components are larger than the trend being sought. The alarm has to act on a filtered rate, which makes the filter part of the instrument specification rather than a detail of the platform, and a channel that cannot export raw samples for a later re-filter cannot be retuned after the event. The same applies to the resolution figure: 0.001 degrees is meaningless if the hourly record is a sawtooth of 0.05 degrees driven by sun on the enclosure.
No karst deployment is described here. The nearest published geometry the company holds is a sleeper array in a railway tunnel and a tower base watch over a mined void, both single-structure geometry and neither of them areal, so the placement rules above are derived from the published mechanics and not demonstrated on a company site. That distinction belongs in the proposal, not in a footnote.
9. What to Specify
Write the alarm as an angle over a stated base. A settlement trigger of 20 mm should be written as 20 mm over a stated distance, or converted to millimetres per metre and to degrees. If the client insists on millimetres, the base goes in the same sentence, because a millimetre figure without one is an assumption and the assumption will not be the site’s.
Derive the grid from the cover, not from the fence line. The half width is roughly a third to a half of the soil thickness over the rockhead, the radius a unit can see is about twice that, and the spacing should not exceed it. On a site with 5 m of cover that is a spacing of 3 to 5 m, which is a different order of cost from the 20 m grid usually proposed, and saying so early is the honest thing to do.
Put the units on a ring at the expected inflection radius rather than at the centre of the depression, and use at least four. Three on a line can detect the reversal of slope that distinguishes a bowl from a slope; a fourth off the line locates the centre in plan; a pair alone is a comparator that cannot tell the two states apart.
Tie the array to a survey baseline and repeat it. The profile is the length that turns millimetres into an angle, and it is also the only way to recover the total settlement, which the array cannot see because a uniformly settling site settles flat.
Watch the water on a separate channel and decide from the rock which one is primary. In limestone the water is primary and the tilt array confirms; in gypsum and salt the rock is primary and the tilt array is a rate instrument on a process that may complete within the design life. A plan that lists only one of the two channels has decided the geology without saying so.
Alarm on the change of rate as well as on the level. The one long-lead case in the published record fired on an acceleration from 0.2 to 0.4 mm per day, which is a doubling rather than a threshold, and a level alarm set to catch that would also have caught every seasonal cycle on the site.
Log the interventions. Grouting, void filling, re-levelling and drainage works all reset the geometry, and an unsegmented trend through such a record reads the repair as recovery and the recovery as stability, which is the same failure mode recorded on re-levelled machine rails.
State the rock and therefore the clock in the monitoring plan, in the first paragraph. Limestone, gypsum and salt do not share a rate, do not share a precursor time and do not share a monitoring object, and a plan that says karst without naming the rock has left the most consequential variable unset.
Keep the boundary in writing. The array dates and locates a change around a known feature. It does not detect an unknown void, does not measure a cavity, does not size a remediation and does not replace radar interferometry, ground penetrating radar, drilling or a geotechnical assessment of the site.
Frequently Asked Questions
- Can a tilt sensor detect a sinkhole before it opens? It depends which stage and which clock. The published precursors on the Dead Sea shore ran 32 to 92 days and the one at Road 90 ran 27 months, but the movement at Bayou Corne worked out at about 0.03 degrees once it was spread over the area that deformed, which is below the zero drift of a millidegree-class instrument. What such an instrument sees well is the sag stage, where a 20 mm drop in 3 m of cover is 0.695 degrees. So the honest answer is that a tilt array dates and locates the late stage reliably and cannot be sold as an early warning system for cover collapse on its own.
- What spacing should a sinkhole tilt array use? The spacing is set by the cover, not by the site area. The half width of the depression is roughly a third to a half of the soil thickness over the rockhead, a unit sees slope out to about twice the half width, and the grid spacing should not exceed that radius. For 5 m of cover that means 3 to 5 m, and for 30 m of cover it means 20 to 30 m. Because that cost is unaffordable across a whole site, the practical arrangement is a dense ring around a known feature or a mapped cavity, and areal methods for everything else.
- How does a settlement limit convert into a tilt limit? Divide the settlement by the distance over which it occurs. The break-even for the common 20 mm figure is a half width of 10 m, where it lands exactly on the design criterion of one in 500, which is 2 millimetres per metre or 0.115 degrees. Narrower depressions make the same 20 mm permissive by a factor of up to six, wider ones make it conservative. This is why a settlement trigger and an angular trigger should be written together, and why the base belongs in the specification.
- Does the rock type change the monitoring plan? It changes the object being monitored. Limestone dissolves at 0.01 to 0.03 mm per year, so the void is inherited and the hazard is a trigger, which makes groundwater the primary channel. Gypsum dissolves about a hundred times faster, 100 mm of opening in 43 years at a measured rate and far faster in flowing water, so generation and triggering become one question and the rock is primary. Salt is another order again, with cavities that develop in a few hundred years. Same word, three plans.
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