Earthquake Location, Depth, and Magnitude

How can station recordings constrain an event in space, time and source size?

Fictional unit investigation. A fictional network surrounds a small event cluster, but two stations have uncertain clocks and one station lies far outside the network. Students compare a best-fit solution with uncertainty and network geometry.

Before beginning, distinguish the scientific question, the measured quantity and the latent parameter that the student hopes to infer. Assign physical units, name a plausible alternate hypothesis, and state what data are fictional. For locating an earthquake, a small residual is not proof of correct origin time or depth. P and S phase association, azimuthal coverage and velocity-model uncertainty can dominate. The experiments are deliberately simplified so their calculations can be independently checked. They are not validated field instruments, real-time forecasts, or site-specific hazard assessments.

Learning goals. Explain the mechanism, translate the problem into measurable quantities, test an alternative interpretation and defend an uncertainty statement.

Physical foundation. Picking P and S onsets converts waveforms to measured arrival times. Association groups related arrivals into candidate events, but low signal-to-noise ratios and overlapping earthquakes can make assignments ambiguous. In the broader unit, Earthquake locations are inverse estimates obtained through physical models, arrival times and network geometry. The best-fitting coordinates may not be unique, while reported magnitude scales address different observables. A complete solution states phase picks, velocity assumptions, error model and station coverage.

Observational method. Label synthetic arrival traces with picks and uncertainty windows; test whether alternate association tables are feasible. Specify the coordinate frame, observation period, relevant units, acquisition and calibration limitations before interpreting the resulting figure.

Common interpretive trap. The first visually prominent wiggle need not be the direct P onset; impulsive noise can bias an event location. A neat numeric solution does not remove this interpretive ambiguity. Ask which further independent measurement could reveal whether the preferred explanation is incorrect.

Applied investigation. Build a pick table including onset quality, phase code and absolute UTC time. Provide a labeled sketch, a short reproducible table and a paragraph separating observed evidence from inferred process.

Interpretation and quantitative example

Worked case 4.1: Direct P–S arrival time separation. Synthetic input: 74.325 km. Apply the explicitly declared relationship x*(1/3.5-1/6) to obtain 8.84821 s. Restrictions: Vp=6 km/s and Vs=3.5 km/s, straight path, same origin time. For this worked exercise, treat the input as a declared hypothetical value rather than a measurement. Copy the formula into a fresh calculation and check the unit conversion independently. Vary the input by ten percent and decide whether the output changes proportionally or nonlinearly. Explain which real-world conditions violate the assumptions listed. If the numeric result seems implausibly precise, round it appropriately and describe a physically meaningful range instead of inventing calibration. The purpose is transparent reasoning, not an official estimate. Select picks before running the optimizer, and make the chosen velocity model explicit. Run a perturbation with one station removed and another with the P-onset shifted. When solutions move, the change reveals sensitivity rather than failure by itself. Distinguish uncertainty caused by poor geometry from uncertainty due to an inaccurate crustal velocity profile.

Evidence design and falsification

A strong evidence chain identifies what the observer physically controls. Specify waveform or map provenance, the acquisition time standard, the instrument or field method, the adopted calibration, and an independent comparison. Is the observed pattern repeatable across genuinely separate records or did it arise after multiple views of the same data? Where an observation is transformed into a calculated quantity, retain its original units and values. Write an audit note on which data would be decisive and which are merely compatible. Here, the principal source of ambiguity is stated explicitly: The first visually prominent wiggle need not be the direct P onset; impulsive noise can bias an event location. Create a four-row evidence matrix: observed quantity, inferred physical quantity, principal confounder, and proposed falsifying observation. Mark an entry as measured only when acquisition metadata is known. Repeat the matrix for a second hypothesis instead of dismissing it without calculation. This discipline prevents a tidy narrative from substituting for testable physical interpretation. End the notebook by explaining how the proposed investigation (Build a pick table including onset quality, phase code and absolute UTC time.) addresses at least one weakness of the primary model.

Knowledge check. (1) Which claim is directly observed and which is inferred? (2) How would an alternate model change the expected observation? (3) Which number must be accompanied by uncertainty? (4) Why is the worked result unsuitable as a real-time safety conclusion?

Learning goals. Explain the mechanism, translate the problem into measurable quantities, test an alternative interpretation and defend an uncertainty statement.

Physical foundation. P–S separation constrains distance to a station in simple velocity models. Multiple ranges or absolute travel times are used for localization; geometry and model error determine ambiguities. In the broader unit, Earthquake locations are inverse estimates obtained through physical models, arrival times and network geometry. The best-fitting coordinates may not be unique, while reported magnitude scales address different observables. A complete solution states phase picks, velocity assumptions, error model and station coverage.

Observational method. Calculate distance from P–S lag and homogeneous Vp, Vs; draw distance circles as geometric intuition only. Specify the coordinate frame, observation period, relevant units, acquisition and calibration limitations before interpreting the resulting figure.

Common interpretive trap. Intersecting circles are not full hypocenter solutions; real travel times vary with depth and crustal structure. A neat numeric solution does not remove this interpretive ambiguity. Ask which further independent measurement could reveal whether the preferred explanation is incorrect.

Applied investigation. Solve a toy three-station localization and repeat after perturbing one pick. Provide a labeled sketch, a short reproducible table and a paragraph separating observed evidence from inferred process.

Interpretation and quantitative example

Worked case 4.2: Moment magnitude from scalar seismic moment. Synthetic input: 108.064 10¹⁸ N m. Apply the explicitly declared relationship (2/3)(Math.log10(x1e18)-9.1) to obtain 7.28912 Mw. Restrictions: Uses SI M0 and standard approximate Mw conversion; not direct Richter intensity. For this worked exercise, treat the input as a declared hypothetical value rather than a measurement. Copy the formula into a fresh calculation and check the unit conversion independently. Vary the input by ten percent and decide whether the output changes proportionally or nonlinearly. Explain which real-world conditions violate the assumptions listed. If the numeric result seems implausibly precise, round it appropriately and describe a physically meaningful range instead of inventing calibration. The purpose is transparent reasoning, not an official estimate. Select picks before running the optimizer, and make the chosen velocity model explicit. Run a perturbation with one station removed and another with the P-onset shifted. When solutions move, the change reveals sensitivity rather than failure by itself. Distinguish uncertainty caused by poor geometry from uncertainty due to an inaccurate crustal velocity profile.

Evidence design and falsification

When comparing scenarios, preserve the same measurement geometry if possible. A change in a source parameter may look like a path effect when stations or filters differ. Design a sensitivity table with columns for adopted value, plausible range, reason for uncertainty and effect on inference. Distinguish an instrumental limitation from a physical ambiguity: improving sampling can solve the former while the latter may need an entirely different experiment. Here, the principal source of ambiguity is stated explicitly: Intersecting circles are not full hypocenter solutions; real travel times vary with depth and crustal structure. Build the result graph only after labelling its axes and quantities. Include a plausible range of inputs rather than a single favorable selection. Annotate any discontinuity, threshold or saturated measurement. Next, state whether the curve demonstrates a general law, a restricted approximation or merely the numerical behavior of an arbitrary toy formula. The caption should not promise resolution that the data cannot support. End the notebook by explaining how the proposed investigation (Solve a toy three-station localization and repeat after perturbing one pick.) addresses at least one weakness of the primary model.

Knowledge check. (1) Which claim is directly observed and which is inferred? (2) How would an alternate model change the expected observation? (3) Which number must be accompanied by uncertainty? (4) Why is the worked result unsuitable as a real-time safety conclusion?

Learning goals. Explain the mechanism, translate the problem into measurable quantities, test an alternative interpretation and defend an uncertainty statement.

Physical foundation. Depth trades off with origin time and velocity structure, particularly when stations are sparse or lie mainly on one side. Formal covariance can understate systematic velocity-model error. In the broader unit, Earthquake locations are inverse estimates obtained through physical models, arrival times and network geometry. The best-fitting coordinates may not be unique, while reported magnitude scales address different observables. A complete solution states phase picks, velocity assumptions, error model and station coverage.

Observational method. Compare residual curves for fixed trial depths and plot how network azimuth affects depth resolution. Specify the coordinate frame, observation period, relevant units, acquisition and calibration limitations before interpreting the resulting figure.

Common interpretive trap. An event depth given to a decimal kilometre may still be poorly constrained despite a precise numerical optimizer. A neat numeric solution does not remove this interpretive ambiguity. Ask which further independent measurement could reveal whether the preferred explanation is incorrect.

Applied investigation. Report a depth range justified by perturbations rather than unexplained decimals. Provide a labeled sketch, a short reproducible table and a paragraph separating observed evidence from inferred process.

Interpretation and quantitative example

Worked case 4.3: Direct P–S arrival time separation. Synthetic input: 148.075 km. Apply the explicitly declared relationship x*(1/3.5-1/6) to obtain 17.628 s. Restrictions: Vp=6 km/s and Vs=3.5 km/s, straight path, same origin time. For this worked exercise, treat the input as a declared hypothetical value rather than a measurement. Copy the formula into a fresh calculation and check the unit conversion independently. Vary the input by ten percent and decide whether the output changes proportionally or nonlinearly. Explain which real-world conditions violate the assumptions listed. If the numeric result seems implausibly precise, round it appropriately and describe a physically meaningful range instead of inventing calibration. The purpose is transparent reasoning, not an official estimate. Select picks before running the optimizer, and make the chosen velocity model explicit. Run a perturbation with one station removed and another with the P-onset shifted. When solutions move, the change reveals sensitivity rather than failure by itself. Distinguish uncertainty caused by poor geometry from uncertainty due to an inaccurate crustal velocity profile.

Evidence design and falsification

Choose one prediction that would differ under a competing mechanism. A useful prediction has a measurable sign, timing, spectral or spatial signature rather than an undefined expectation that a map should look different. Explain how the predicted quantity would be calculated before looking at the fictional test record. If both explanations remain consistent with the evidence, report the unresolved alternatives and the next practical observation needed. Here, the principal source of ambiguity is stated explicitly: An event depth given to a decimal kilometre may still be poorly constrained despite a precise numerical optimizer. Keep a processing log with coordinate reference, UTC times, units, processing order and exclusion rules. Compare pre- and post-processing records and explain discrepancies. In the conclusion, identify which claim survives the most consequential alternative parameterization. Reproducibility means another researcher could regenerate the result and find the same limitations, not simply the same decimal answer. End the notebook by explaining how the proposed investigation (Report a depth range justified by perturbations rather than unexplained decimals.) addresses at least one weakness of the primary model.

Knowledge check. (1) Which claim is directly observed and which is inferred? (2) How would an alternate model change the expected observation? (3) Which number must be accompanied by uncertainty? (4) Why is the worked result unsuitable as a real-time safety conclusion?

Learning goals. Explain the mechanism, translate the problem into measurable quantities, test an alternative interpretation and defend an uncertainty statement.

Physical foundation. Local magnitude ML, body-wave mb, surface-wave Ms and moment magnitude Mw use different measurement windows and can saturate or be inappropriate across ranges. Mw connects logarithmically to scalar seismic moment. In the broader unit, Earthquake locations are inverse estimates obtained through physical models, arrival times and network geometry. The best-fitting coordinates may not be unique, while reported magnitude scales address different observables. A complete solution states phase picks, velocity assumptions, error model and station coverage.

Observational method. Convert synthetic M0 to Mw using a declared SI relation; compare it with a conceptual local-magnitude estimate. Specify the coordinate frame, observation period, relevant units, acquisition and calibration limitations before interpreting the resulting figure.

Common interpretive trap. A magnitude change of one unit is not a change of one joule, and intensity differs across sites for one earthquake. A neat numeric solution does not remove this interpretive ambiguity. Ask which further independent measurement could reveal whether the preferred explanation is incorrect.

Applied investigation. Translate two moment magnitudes into a seismic moment ratio while keeping magnitude and intensity separate. Provide a labeled sketch, a short reproducible table and a paragraph separating observed evidence from inferred process.

Interpretation and quantitative example

Worked case 4.4: Moment magnitude from scalar seismic moment. Synthetic input: 183.039 10¹⁸ N m. Apply the explicitly declared relationship (2/3)(Math.log10(x1e18)-9.1) to obtain 7.4417 Mw. Restrictions: Uses SI M0 and standard approximate Mw conversion; not direct Richter intensity. For this worked exercise, treat the input as a declared hypothetical value rather than a measurement. Copy the formula into a fresh calculation and check the unit conversion independently. Vary the input by ten percent and decide whether the output changes proportionally or nonlinearly. Explain which real-world conditions violate the assumptions listed. If the numeric result seems implausibly precise, round it appropriately and describe a physically meaningful range instead of inventing calibration. The purpose is transparent reasoning, not an official estimate. Select picks before running the optimizer, and make the chosen velocity model explicit. Run a perturbation with one station removed and another with the P-onset shifted. When solutions move, the change reveals sensitivity rather than failure by itself. Distinguish uncertainty caused by poor geometry from uncertainty due to an inaccurate crustal velocity profile.

Evidence design and falsification

Reconstruct the inference backwards. Start with the headline result and locate every conversion, classification and assumption required to obtain it. An estimate based on a chosen velocity, phase pick or site parameter should be labelled conditional on those inputs. Review the same steps after making one intentional mistake in the input metadata; document which output would become misleading and why. Here, the principal source of ambiguity is stated explicitly: A magnitude change of one unit is not a change of one joule, and intensity differs across sites for one earthquake. Method A: compute a baseline result using the stated relation, preserving all units. Method B: perturb one relevant input and record the change. Method C: replace the simplest assumption with an alternative that would be plausible at the study scale; if you cannot recalculate it with the available information, state what is missing. Method D: compare with an independent kind of observation, if available. This order avoids hiding assumptions under additional significant figures. End the notebook by explaining how the proposed investigation (Translate two moment magnitudes into a seismic moment ratio while keeping magnitude and intensity separate.) addresses at least one weakness of the primary model.

Knowledge check. (1) Which claim is directly observed and which is inferred? (2) How would an alternate model change the expected observation? (3) Which number must be accompanied by uncertainty? (4) Why is the worked result unsuitable as a real-time safety conclusion?

Learning goals. Explain the mechanism, translate the problem into measurable quantities, test an alternative interpretation and defend an uncertainty statement.

Physical foundation. First-motion polarity and full-waveform methods constrain source geometry, often yielding a pair of nodal planes. Finite rupture directivity can change shaking patterns even for similar source magnitude. In the broader unit, Earthquake locations are inverse estimates obtained through physical models, arrival times and network geometry. The best-fitting coordinates may not be unique, while reported magnitude scales address different observables. A complete solution states phase picks, velocity assumptions, error model and station coverage.

Observational method. Interpret a fictional polarity distribution, and compare expected forward and backward directivity with a two-sided rupture assumption. Specify the coordinate frame, observation period, relevant units, acquisition and calibration limitations before interpreting the resulting figure.

Common interpretive trap. A beach-ball plot does not reveal the actual fault plane without geological or aftershock constraints. A neat numeric solution does not remove this interpretive ambiguity. Ask which further independent measurement could reveal whether the preferred explanation is incorrect.

Applied investigation. Prepare a mechanism interpretation with two candidate fault planes and the evidence needed to choose. Provide a labeled sketch, a short reproducible table and a paragraph separating observed evidence from inferred process.

Interpretation and quantitative example

Worked case 4.5: Direct P–S arrival time separation. Synthetic input: 221.825 km. Apply the explicitly declared relationship x*(1/3.5-1/6) to obtain 26.4077 s. Restrictions: Vp=6 km/s and Vs=3.5 km/s, straight path, same origin time. For this worked exercise, treat the input as a declared hypothetical value rather than a measurement. Copy the formula into a fresh calculation and check the unit conversion independently. Vary the input by ten percent and decide whether the output changes proportionally or nonlinearly. Explain which real-world conditions violate the assumptions listed. If the numeric result seems implausibly precise, round it appropriately and describe a physically meaningful range instead of inventing calibration. The purpose is transparent reasoning, not an official estimate. Select picks before running the optimizer, and make the chosen velocity model explicit. Run a perturbation with one station removed and another with the P-onset shifted. When solutions move, the change reveals sensitivity rather than failure by itself. Distinguish uncertainty caused by poor geometry from uncertainty due to an inaccurate crustal velocity profile.

Evidence design and falsification

A strong evidence chain identifies what the observer physically controls. Specify waveform or map provenance, the acquisition time standard, the instrument or field method, the adopted calibration, and an independent comparison. Is the observed pattern repeatable across genuinely separate records or did it arise after multiple views of the same data? Where an observation is transformed into a calculated quantity, retain its original units and values. Write an audit note on which data would be decisive and which are merely compatible. Here, the principal source of ambiguity is stated explicitly: A beach-ball plot does not reveal the actual fault plane without geological or aftershock constraints. Create a four-row evidence matrix: observed quantity, inferred physical quantity, principal confounder, and proposed falsifying observation. Mark an entry as measured only when acquisition metadata is known. Repeat the matrix for a second hypothesis instead of dismissing it without calculation. This discipline prevents a tidy narrative from substituting for testable physical interpretation. End the notebook by explaining how the proposed investigation (Prepare a mechanism interpretation with two candidate fault planes and the evidence needed to choose.) addresses at least one weakness of the primary model.

Knowledge check. (1) Which claim is directly observed and which is inferred? (2) How would an alternate model change the expected observation? (3) Which number must be accompanied by uncertainty? (4) Why is the worked result unsuitable as a real-time safety conclusion?

Research scenario: A fictional network surrounds a small event cluster, but two stations have uncertain clocks and one station lies far outside the network. Students compare a best-fit solution with uncertainty and network geometry. The goal is to produce a reasoned comparison of at least two defensible models, not to reverse-engineer a predetermined answer. Start by recording each input variable with units, a confidence grade and a plausible range. Name the forward calculation associated with each inference and outline how the inputs enter the formula. Preserve the fabricated dataset in its supplied CSV form and cite its generation method; these records do not correspond to real earthquakes or real sensor networks.

Procedure. First, draw a map or schematic that makes geometric assumptions visible. Second, check station or sample identifiers and the relationship of the observation period to the question. Third, compute baseline statistics and at least one worked relation by hand before using the browser model. Fourth, investigate a competing hypothesis using an independently chosen measurement or a perturbation of a consequential parameter. Fifth, repeat after withholding one doubtful observation. Compare whether the scientific conclusion survives that choice. Select picks before running the optimizer, and make the chosen velocity model explicit. Run a perturbation with one station removed and another with the P-onset shifted. When solutions move, the change reveals sensitivity rather than failure by itself. Distinguish uncertainty caused by poor geometry from uncertainty due to an inaccurate crustal velocity profile.

Required outputs. Submit a two-page technical brief with a hypothesis table, raw and corrected values, an equation inventory, an uncertainty budget, a labelled graphic and a short limitations section. Include all model parameters. Do not omit unsuccessful tests and do not represent computer-generated data as measured field observations. Students should record their own explanations rather than copying one another's interpretation. Discuss how different station distributions or sampling choices could cause an apparently significant spatial pattern without a corresponding physical change.

Extensions. Earthquake locations are inverse estimates obtained through physical models, arrival times and network geometry. The best-fitting coordinates may not be unique, while reported magnitude scales address different observables. A complete solution states phase picks, velocity assumptions, error model and station coverage. Write an explicit counterfactual: what observation would undermine the main hypothesis? Consider what a larger dataset could resolve and what it would leave uncertain. Finally, design an accessible presentation of the result with a descriptive figure caption, a plain-language explanation of each symbol, and a table of numerical values for readers who cannot interpret a colored graph. Re-evaluate the result against professional source and public-safety boundaries before suggesting a consequential application.

Peer-review rubric. Assign 0–4 points each for correctness of units, explicit assumptions, data provenance, comparison with alternatives, uncertainty treatment, computational reproducibility and ethical communication. A polished figure without units, or a convincing narrative without a falsifiable test, should not receive full credit. Explain in prose why the least certain input affects confidence, and identify at least one result that could change with additional observations.