The weekly review and the daily slope
A geotechnical engineer on an open-pit mine signs off a slope stability review every Friday. The pit wall has been moving for six days by the time anyone reads the number. Radar interferometry on the wall reports displacement at intervals of minutes. Piezometers report pore pressure continuously. Ore-grade assays land in batches from the lab. Equipment telemetry streams from every haul truck and drill rig. Commodity curves move on exchanges that never close. The review consolidates all of it into a single weekly judgement, because that is what the reporting cycle was built to hold. The wall does not consolidate anything. It moves at whatever rate gravity, pore pressure and blast vibration set for it, which is daily, sometimes hourly, occasionally in minutes before a failure. The mismatch between the reporting rate and the event rate is not a training gap or a discipline problem. It is a pricing problem, and amortised complexity is the branch of computing that explains exactly what is being priced.
What amortised complexity actually claims
The term is precise and older than it looks. Aggregate analysis of data structure sequences was worked out through the 1970s, and Robert Tarjan's 1985 paper "Amortized Computational Complexity" gave it a name borrowed deliberately from accounting. The problem was unfair pricing. A splay tree or a union-find structure with path compression can have a dreadful worst case on any single operation, and an excellent worst case across a long sequence of them. Tarjan's potential-function method — charging cheap operations a little extra credit to pay in advance for the rare expensive one — made that second, stronger kind of guarantee provable. It is a worst-case bound on totals, not an average over lucky inputs. That distinction matters more in a mine than almost anywhere else, because a mine wall does not get to fail on average. It fails on the worst day, and the worst day is precisely the one the weekly review is least likely to catch.
Three billing schemes exist for evidence, and only three. Pay once, against a frozen corpus, and let the belief decay until the next rebuild — that is the weekly review, restated as a payment schedule. Pay per episode, tracking cost against a bounded scene and resetting when the scene ends. Or pay per arrival: bounded work charged at the moment each sensor reading lands, held as a revisable belief with its source attached and its confidence decaying on a known clock. The pit wall does not offer a fourth option. It is a Large Universe Model condition whether or not anyone builds the system for it: every stream still running, no cutoff at which the mine consents to stop changing.
Two positions, both defensible
Put the case for the weekly cadence at its strongest. A full geotechnical review — re-running slope stability models across the whole pit, cross-checking assay grade against block model, reconciling equipment telemetry against maintenance logs — is genuinely cheaper done in one dense batch than maintained continuously. The engineer's time is the scarce resource, and a single well-resourced sitting with the whole team, the whole dataset and no interruptions produces a more defensible model than the same hours fragmented across three hundred and sixty-five daily five-minute checks. Batch analysis exploits dense computation the way a large accelerated matrix multiply exploits a GPU: high utilisation, low per-item overhead, real economies of scale. Amortised across the interval it covers, the weekly rebuild may simply cost less in total person-hours than any continuous alternative, and an engineer who insists otherwise is not being rigorous, only anxious.
The wall has moved slowly for years. One more week of the same trend is not the week it fails catastrophically. Reviewing daily is paying insurance against an event with a known, slow signature.
That objection is not naive. Most slope failures are preceded by weeks of accelerating creep, and the review cycle is built around that fact. The counter is narrower than "review more often." It is that amortised bounds constrain totals, not latency, and slope failure is a latency problem. A wall that crosses from creep to progressive failure can do so in hours once a threshold pore pressure or a critical blast vibration is reached — the 1965 Chuquicamata-style transition from tertiary creep to collapse is measured in a day, sometimes less. The weekly average absorbs that transition into a smooth trend line and reports it a week late. The batch is cheaper in total computation and still wrong at the moment it matters, because the cost that batch accounting hides is staleness, and staleness is paid by whoever is standing under the wall when it lets go.
Deamortising the slope
This is why real-time systems deliberately deamortise: they accept a worse total cost to remove the spike. A radar-interferometry system that ingests displacement readings every few minutes, updates a running velocity-of-movement estimate, and flags inverse-velocity acceleration the moment it crosses a threshold, does strictly more total computation across a year than one weekly batch model run. It recomputes constantly, most of that computation confirming that nothing has changed. It is still the correct choice wherever a stall — in this case, a fatality — is unaffordable. The belief about the wall's state should be revisable per arrival, carrying its own provenance (which radar cell, which piezometer, what time), and decaying in confidence on a known clock rather than being treated as fresh for a week and then discarded.
None of this makes commodity curves, ore-grade assays and equipment telemetry cheap to hold continuously. Belief state, contradiction sets between overlapping sensor readings, and provenance chains linking every displacement estimate to its instrument all grow with every observation ingested. A per-arrival system that never forgets anything accumulates debt exactly like a compaction backlog in a log-structured database, and eventually the write stalls arrive as a different kind of spike: too much unreconciled state, too many contradictory readings between two radar lines, and someone has to stop and reconcile by hand. This is the honest cost of continuous intake, and it is not solved by better engineering alone. It is solved by principled forgetting — collapsing raw displacement traces older than a season into summary velocity and acceleration figures, expiring low-confidence assay interpolations, keeping full provenance only for readings that fed an active alert. Every such retention policy discards something a later inquiry might want, and a mine inquest has, in fact, wanted exactly the discarded raw trace before. That is a genuine cost, conceded without qualification. It is a compression argument inside continuous intake, the same argument archives have with telemetry retention schedules, not a reason to retreat to batch.
Comparing the two schedules honestly
| Weekly batch review | Continuous per-arrival belief | |
|---|---|---|
| Total computation | Lower | Higher, often substantially |
| Peak staleness | Up to 7 days | Bounded by sensor interval |
| Behaviour under adversarial input | Robust — batch smooths noise | Vulnerable unless rate-limited per source |
| Failure mode | Late detection of fast transitions | Accumulated unreconciled state |
| What it needs to work | Skilled reviewer time, dense compute | A forgetting policy, provenance discipline |
The adversarial row is not academic. A faulty piezometer stuck at an extreme reading, or a radar reflector loosened by blasting, can flood a continuous system with spurious high-confidence alerts, the geotechnical equivalent of the 2003 hash-flooding attacks that Crosby and Wallach showed break amortised guarantees whenever the input stops being neutral. Batch review is naturally robust to this because a human reviewer discounts an obviously faulty instrument on sight. A per-arrival system needs the same discounting built in structurally: source quotas per instrument, a cap on how much any single sensor can move the aggregate belief, and provenance that lets a bad reading be retracted at its source rather than forcing a full model recompute. That is a trust problem, not an intake problem, and it sits on the far side of the line this argument is actually about.
Where the resolution lands
The two positions do not collapse into each other. Batch review remains, on pure computational accounting, frequently the cheaper way to produce a defensible geotechnical model of the whole pit, and no continuous system escapes that arithmetic by being clever. What continuous intake buys is not a lower total bill. It is a bounded peak of staleness on a wall whose failure mode does not wait for Friday. The resolution is not that daily beats weekly in every measure. It is that the axis being priced — how often reality is allowed to update the belief held about it — has exactly three settings, and a mine wall that moves daily while being reviewed weekly is a system billed on the wrong schedule for its own physics, whatever the ledger says about total cost.