Summary: A level threshold answers how far a structure has moved and says nothing about when it will stop being safe. The methods that forecast the time of failure come from creep theory: Saito’s three stages, Fukuzono’s inverse velocity relation, and the tangential angle criterion that compares a current rate with the steady rate that preceded it. Every one of them is a ratio, and a ratio has no unit, which is why they transfer onto angular rate from a tilt sensor without any conversion to millimetres. The unknown lever arm between angle and displacement scales the vertical axis of an inverse velocity plot and cannot move the point where it crosses the time axis. What decides whether a forecast exists at all is the number of points inside the accelerating window, and what decides whether it can be trusted is the onset of acceleration, the noise floor and the failure mechanism.
1. A Level Threshold Cannot Tell You When
Two slopes can both read two degrees and mean opposite things. One arrived there over five years at a thousandth of a degree per day and has stopped. The other arrived in a week and is still accelerating. The instrument reports the same number in both cases, because the number is a level, and a level carries no time inside it. Everything a forecast needs sits in the shape of the record rather than in its current value.
The published specification for a wireless tiltmeter makes that distinction visible. It lists an alarm angle, three degrees by default, together with an alarm accuracy band and an alarm delay time of 0.3 to 25.5 seconds, defaulting to 2 seconds, described as the time the device must continue to tilt beyond the alarm angle before the alarm fires. That pair is a good detector for a step change, which is what a theft, an impact or a sudden loss of support looks like. It is not a trend detector, and no setting of it will turn it into one.
Threshold design is a subject in its own right, and the alarm threshold guide covers it. The question here is what has to be computed on top of the threshold, and for that the record has to be treated as a curve rather than as a number.
2. The Creep Curve: Three Stages, and Only One of Them Fails
The framework comes from creep theory, which Terzaghi and Haefeli applied to slopes in the middle of the last century and which Saito turned into a prediction method in the 1960s. Working from eighty triaxial compression tests, Saito concluded that displacement was the most useful quantity for predicting the time of failure, and described the process as three stages. Primary creep is decelerating, secondary creep is steady, and tertiary creep is accelerating. The failure happens at the end of the third stage. Saito’s original empirical relation, published with Uezawa in 1961, linked the logarithm of the time to failure to the logarithm of the strain rate, and it is the ancestor of everything that follows.
The consequence that matters operationally is uncomfortable. During secondary creep the record contains no warning at all. A slope that will fail next month and a slope that will stand for another century produce the same straight line. The signal is the departure from that line, and it is the only signal available.
| Creep stage | Angular rate | What the record looks like | What the operator should do |
| Primary | Falling | A curve that flattens out | Log the baseline and do not alarm on it |
| Secondary | Constant | A straight line, indistinguishable from a healthy slope | Fix the steady rate, because every criterion below is referenced to it |
| Tertiary | Rising | A curve that bends upward | The only stage in which a failure time can be estimated. Shorten the reporting interval now |
One complication is worth stating before the methods are introduced. Acceleration is not the same thing as failure. The landslide forecasting literature documents cases in which an initial acceleration is followed by a decrease in velocity and a new equilibrium, without collapse. A rising rate is a demand for attention, not a proof of an outcome.
3. The Tangential Angle: A Threshold That Is Really a Multiple
The first method needs only two numbers: the current rate, and the rate the slope had before. Xu and co-authors published it as the tangential angle criterion, and it was applied in the published analysis of the Baishi landslide in Beichuan County, Sichuan. Call the steady rate of the secondary phase v0. The tangential angle is the arctangent of the current rate divided by v0.
Because both quantities are rates of the same thing, the ratio has no unit. The published reading of the criterion is that once the angle passes 45 degrees the deformation has entered the accelerating phase, that the band from 45 to 80 degrees is the initial acceleration sub-phase, that 80 to 85 degrees is the intermediate sub-phase, that above 85 degrees is imminent sliding, and that the value approaches 89 degrees immediately before failure. Published landslide assessments map those three bands onto three warning levels of increasing severity. The datum makes sense of those numbers: while a slope is creeping steadily, the current rate equals the baseline rate, the ratio is one, and the angle is exactly 45 degrees.
The useful part is not the angle itself but what the angle means once the arctangent is undone. Every boundary is a plain multiple of the rate the slope already had.
| Tangential angle | Rate divided by the steady rate | Published stage | Published warning level |
| 45 deg | 1.00 | Secondary creep ends, acceleration begins | The datum |
| 45 to 80 deg | 1.00 to 5.67 | Initial acceleration sub-phase | First warning level |
| 80 to 85 deg | 5.67 to 11.43 | Intermediate acceleration sub-phase | Second warning level |
| Above 85 deg | Above 11.43 | Imminent sliding | Third warning level |
| Near 89 deg | About 57 | Immediately before failure | Third warning level |
The criterion therefore reduces to one instruction: establish the steady rate, then treat roughly six times that rate and roughly eleven times that rate as the two lines worth acting on. No unit appears anywhere in the calculation, and no conversion to displacement is required.
A second body of work reaches the same order of magnitude from a different direction. An analysis of forty slope failures in coal mines found that the ratio between the acceleration over the last three hours and the acceleration over the last 24 hours was about seven in every case, with the other pairings giving about thirteen and about two. The authors propose issuing an alarm when the three hour figure approaches seven times the 24 hour figure, which indicates failure within about three hours. Note the caveat they attach to it: the forty cases shared broadly similar geological conditions, and calibrating the ratio needs a number of failures in the same area.
4. Inverse Velocity: The Line That Meets the Time Axis
The second method uses the whole record rather than a single ratio. Fukuzono, working from the same creep theory in 1985, showed that the logarithm of the velocity and the logarithm of the acceleration are linearly related during tertiary creep. That is the same statement as a power law between the two, with the acceleration proportional to the velocity raised to an exponent. Integrating that relation gives the inverse of the velocity as a function of the time remaining before failure. The exponent controls the shape. Published experiments put it between 1.5 and 2.2 and in most cases close to 2, and when it is close to 2 the relation becomes linear, which is what makes the method practical: the prediction is then a straight line drawn through the recent data and extended until it crosses the time axis.
Two shapes in that plot are worth recognising before any fitting is attempted. While the slope is at equilibrium the plot is a line parallel to the time axis, because the inverse of a constant velocity is a constant. When acceleration begins the plot descends, and the extrapolation reaches the time axis where the inverse velocity is zero, which is the mathematical statement that the velocity has become infinite. Voight generalised the same relation in 1988 and 1989 and extended it to the terminal stages of failure in other materials, and the method has since been used operationally in open pit mining.
The honest footnote sits on the same page as the formula. The construction assumes that the velocity at failure is infinite. It is not. The velocity at failure varies with rock type, with the volume involved, with the failure mechanism and with the slope angle. One published consequence is that a straight line fit tends to place the failure later than it happens, because the final part of the inverse velocity curve drops steeply and non-linearly, and the straight line cannot follow it without more data than the remaining time allows.
The scale of the problem is best seen in a published case. At the Baishi landslide the steady state phase ran for about three months, the accelerating phase that followed lasted about three and a half months and was itself divided into a 45 day sub-phase and a 58 day sub-phase, and the critical slide phase lasted 12 days. The average movement rate in the main zone was about 78 millimetres per day, against about 20 millimetres per day in the secondary zone. The window in which a forecast has to be produced is measured in days here, which is comfortable, and it is exactly that number which decides whether the method can be run at all.
5. Why an Angle Sensor Can Run Both Methods Without Converting to Millimetres
Both methods are written for displacement, and a tilt sensor reports an angle, so the conversion looks mandatory. It is not, and the reason is a published experimental result rather than an argument. A study using laboratory slope models with pre-designed slip surfaces, followed by a field slope failed by artificial rainfall, established a linear relationship between the tilt of the slope surface and the surface displacement. Its most useful finding is the coefficient: the constant of that linear relation is the real distance between the position of the tilt sensor and the centre of the slip surface. The same work observed an acceleration stage in the surface tilting as the slope approached failure, and concluded that tilting measurements can be used to predict the development or the reactivation of a rotational landslide.
Write that distance as r. If tilt and displacement are linearly related with r as the coefficient, then a displacement d produces a tilt of d divided by r, and differentiating gives the angular rate as the velocity divided by r. Invert both expressions and the relationship between them becomes explicit: the inverse angular rate equals r multiplied by the inverse velocity.
That single line settles the question. A plot of inverse angular rate against time is the inverse velocity plot with its vertical axis multiplied by the constant r. Multiplying an axis by a constant stretches or compresses it, and it cannot move the point at which the curve meets the other axis. The predicted time of failure is therefore the same number whether the ordinate is millimetres per day or degrees per day, and the value of r never has to be known. The same argument covers the tangential angle, which is a ratio of two rates and therefore identical whether those rates are angular or linear, and it covers the acceleration ratio method, which is a ratio of two accelerations.
There is one condition, and it carries the whole result. The coefficient r has to stay constant. That holds when the movement is rotation about a fixed centre, which is what the published linearity describes. It stops holding when a new shear surface forms elsewhere, when the moving mass breaks up, or when the mechanism changes from rotation to translation. On a mainly translational slide, where the surface translates rather than rotates, the angular signal is small and the basis for the entire calculation is absent. The mechanism decides, and it has to be established first.
6. The Depth Problem: Sensor Position Sets the Size of the Signal
The same relation, read the other way round, decides where the sensor goes. For a given amount of movement the angle the sensor reports is inversely proportional to its distance from the centre of the slip surface. A sensor far from that centre sees a small angle for a large movement, so the resolution of the instrument buys less and less displacement as the slip surface gets deeper. The published resolution is 0.001 degrees and the small angle constant is 17.46 millimetres per metre per degree, and multiplying the two gives the size of one resolution step at each distance.
| Distance to the slip surface centre | One resolution step at 0.001 deg | Finest velocity over 24 hours | Apparent movement from 0.01 deg of drift |
| 2 m | 0.035 mm | 0.035 mm per day | 0.35 mm |
| 5 m | 0.087 mm | 0.087 mm per day | 0.87 mm |
| 10 m | 0.175 mm | 0.175 mm per day | 1.75 mm |
| 20 m | 0.349 mm | 0.349 mm per day | 3.49 mm |
| 30 m | 0.524 mm | 0.524 mm per day | 5.24 mm |
The last column is the one to read first, because it is set by the published zero point temperature drift of plus or minus 0.001 degrees per degree Celsius, typical. Across a ten degree Celsius diurnal swing that is 0.01 degrees of apparent tilt change. If that change is not cancelled by differencing readings taken at the same time of day, it is worth 5.24 millimetres at a 30 metre radius in a single day, against 10 millimetres for a slide creeping at 10 millimetres a day at that radius. Half the signal, contributed by the instrument, before any other noise is counted. The published worst case figure over the full operating range is plus or minus 0.065 degrees, which at a 30 metre radius is 34 millimetres of apparent movement.
Two conclusions follow. On a shallow slide the angular signal is strong, which is why surface tilt measurement worked in the rainfall triggered tests, and a surface sensor is the right instrument. On a deep seated slide the surface rotates very little for a lot of movement, and a single surface sensor is the wrong instrument. That is the problem an in place chain solves, and it is visible in the published deployment record.
At the TISCO Jianshan iron mine dump in Loufan County, Shanxi, the online monitoring system was upgraded with a deep displacement system. Four key points at the toe of the slope were given boreholes, all 40 metres deep, and eight sensing units were installed in each hole at 5 metre spacing, so 32 units in total and about 35 metres of instrumented column per hole. The site is a large hillside open pit, and the dump is described in the same record as a high potential energy man made debris flow hazard with a dam break risk, which is why the instruments went to the toe.
For that geometry the sensor package matters as much as the principle. The published in place inclinometer family is supplied in standard probe units of 1 metre, 2 metres and 3 metres, with custom lengths available on request, and the probes chain so that one column can cover a deep borehole. A wireless tiltmeter mounted on the surface at the same site would be measuring the rotation of a point 40 metres above the failure surface, which is the least sensitive position available.
7. How Many Points the Forecast Needs, and What Shortening the Window Costs
A straight line needs points, and inside the accelerating window the number of them is set entirely by the reporting interval. On a wireless tiltmeter that interval is a configuration value rather than a property of the hardware. The published range for the heartbeat interval is 60 seconds to 131,071 seconds, which is about 36 hours, and the default is 86,400 seconds, which is one day. That range is wide enough to change the method completely. The published landslide case above gives a critical phase of 12 days and two acceleration sub-phases of 45 and 58 days, and the interval decides how many measurements fall inside them.
| Heartbeat interval | Points in a 12 day window | Battery life from the published model | Points in a 3 day window |
| 24 h (default) | 12 | about 6 years | 3 |
| 6 h | 48 | about 6 years | 12 |
| 1 h | 288 | about 3.7 years | 72 |
| 15 min | 1,152 | about 1.6 years | 288 |
| 1 min (floor) | 17,280 | about 48 days | 4,320 |
The battery column is calculated from the manufacturer’s own published model, which states its assumptions: a capacity of 19,000 mAh, a loss factor of 0.7, an average current of about 31 milliamps during a transmission lasting about 22 seconds, and a standby current of about 0.22 milliamps while the alarm channel is enabled. What the column shows is that moving from one day to six hours costs almost nothing, that the cost only becomes material below about an hour, and that the floor is a state to enter rather than a setting to install. The point for this guide is narrower still, and it is that the reporting interval is the resolution of the forecast.
Read against the case, the default interval works. Twelve points in the critical window is a thin basis for a regression run under time pressure, but it is a basis, and 45 to 58 points in each sub-phase are comfortable. But that is a slow landslide. Where tertiary creep is short, the same default delivers one or two points and there is nothing to fit. Because the duration of the accelerating phase is not known in advance, the interval has to be chosen from the mechanism rather than accepted as a default, and it has to be shortenable remotely once the first warning level is crossed.
The second half of the decision is the part that catches people out. A shorter averaging window is not a free increase in information, because terms that cancel over a long window stop cancelling over a short one, and the largest of those terms is the diurnal thermal cycle. Over a 24 hour interval the two readings being differenced sit at the same point in the cycle and the term largely cancels. Over a one hour interval it does not, and the sensor’s own zero point temperature drift stops being an offset and becomes a rate. The published drift of plus or minus 0.001 degrees per degree Celsius, applied to the warming half of a morning, produces an apparent angular velocity that competes with the real one.
Two cheap measures address it. Report at the same time of day, which the displacement monitoring literature recommends for exactly this reason and which is a configuration setting rather than a hardware requirement. And carry a temperature channel, so that the day’s drift can be removed rather than argued about. There is also a published cost to the short window that has nothing to do with temperature: rates computed over short time periods have been shown to produce a wider spread in the predicted failure time than rates computed over longer periods. Running at the one minute floor without filtering therefore buys more points and a worse forecast, which is the opposite of the intuition that faster sampling is always better.
8. What Failure Prediction Cannot Do
The historical record is the best discipline available to anyone who has just learned a prediction method. In 1932 a rock slope above the village of Linthal in the Swiss Alps was monitored, and two separate forecasts of catastrophic failure were issued. Neither of them happened. The explanation the engineer gave afterwards was lack of experience. The reviewed limitations of the method itself are more specific, and worth listing without softening.
It applies to one mechanism. The method is valid for failures that occur by accelerating creep. It does not apply to brittle or structural failure, where the mass gives little precursor deformation before it moves. One published strain criterion set the threshold for planar instability at 0.03 to 0.06 percent strain and for wedge instability at 0.2 percent, which is a very different regime from one whose only input is a trend.
It needs an identified onset. Fitting must begin after the onset of acceleration, and the later literature adds a trend update point for a significant change in the acceleration. Both have to be identified, and identifying them early enough to be useful is not automatic.
Its central assumption is false. The velocity at failure is assumed to be infinite. The guidance states plainly that this is not verified, and that the velocity at failure varies with rock type, volume, mechanism and slope angle.
It assumes the load does not change. Rainfall, groundwater, snowmelt and human activity all change the rate, and none of them is in the fit. The literature classifies them together as natural noise.
It has two noise classes and only one of them is yours. Instrumental noise is the accuracy of the instrument. Natural noise is everything else, and only the first of the two is under the control of the person specifying the system.
Two points on one slope can disagree by an order of magnitude. This is a published observation about forecasts from different instruments, or from different locations, on the same unstable slope. It is not a worst case.
Filtering helps and is not neutral. The coefficient of determination of the fit rises as an acceleration phase develops, which is a useful signal in itself, but the filter that raises it also moves where the line crosses the axis.
The operational conclusion from the same literature is that long term forecasts are never entirely definite and that monitoring has to continue up to the point of failure. A forecast is not a substitute for a watch, and the same literature that supplies the number is the literature that reports its uncertainty.
9. From a Forecast to a Trigger Action Response Plan
The reason prediction matters in procurement is that a safety management system asks for it by name. The Global Industry Standard on Tailings Management, published in 2020 by the International Council on Mining and Metals together with the United Nations Environment Programme and the Principles for Responsible Investment, sets 77 auditable requirements across six topic areas and requires instrument monitoring across the whole lifecycle of a facility. Its operating mechanism is a pair of definitions that translate directly into what a monitoring system has to produce.
The first is the critical operating parameter, defined as a parameter whose exceedance carries a real risk of a dam safety incident. The second is the trigger action response plan, one for each critical parameter, which is the pre-agreed action taken when the parameter leaves its range. The requirement is that any condition outside that range is addressed promptly through the plan, or through a critical control.
That structure answers the question of what a forecast is for. A warning level with no action attached is not a monitoring system, it is a subscription. The published warning levels map onto the plan without modification, because there are three of them and they already mark three different commitments: the initial acceleration band is the level that raises attention and shortens the reporting interval, the intermediate band is the level that changes how the site is worked, and the imminent sliding band is the level that stops work and moves people.
Two closing constraints come from the same source. Instrument monitoring is an addition to inspection and not a replacement for it, and the guidance states that visual inspection should never be fully replaced by instrumentation on an active facility. And the trigger has to be a number somebody can act on inside the time the method gives them, which for a fast slope can be hours rather than days. A plan whose first step is a meeting is not yet a plan.
Frequently Asked Questions
- Can a tilt sensor replace displacement monitoring for failure prediction? For movement that is rotation about a fixed centre, yes, and the prediction is mathematically identical, because the lever arm between angle and displacement scales the vertical axis of the plot and cannot move the point where it crosses the time axis. For movement that is mainly translation there is little rotation to measure, so the answer is no. Establish the mechanism first, because it decides whether a signal exists at all.
- What heartbeat interval should I set on a wireless tiltmeter for this? The published default of 24 hours is a power setting, not a monitoring decision. Set the interval so that a usable number of points falls inside the accelerating phase, report at the same time of day so that the diurnal thermal term cancels, and shorten it remotely once the first warning level is crossed. The published floor is 60 seconds, and running there continuously costs most of the battery, so it is a state to enter and leave rather than a setting to install.
- My inverse velocity prediction keeps moving later. Why? There are two published reasons. The method assumes the velocity at failure is infinite, and the final part of the curve drops more steeply than a straight line can follow, which pushes the extrapolated crossing to the right. And the fit is usually run over data that still contains part of the steady phase, which flattens the line. Refit only on data after the onset of acceleration, and refit again when the trend changes.
- How accurate does the sensor need to be? Resolution is rarely the binding constraint and the drift term usually is. At the published resolution of 0.001 degrees, one step is 0.035 millimetres at a two metre radius and 0.524 millimetres at a thirty metre radius, so the resolution floor sits well below the rates that matter. The published zero point temperature drift of plus or minus 0.001 degrees per degree Celsius, across a ten degree Celsius day, is worth 0.35 millimetres at two metres and 5.24 millimetres at thirty metres, which is comparable with a slow rate and is the term to design around.
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