Exceedance Probabilities for Large Earthquakes From DIY Local Earthquake Ensemble Nowcasting and Forecasting

This paper presents a "nowcast transform" method to adjust Gutenberg-Richter statistics within regional earthquake ensembles for improved nowcasting and forecasting of large earthquake exceedance probabilities, demonstrating its consistency and application to the Los Angeles region following the 1994 Northridge earthquake.

The Gist

Imagine you are trying to predict when the next big storm will hit your town. You can't see the storm coming, but you can count the number of tiny raindrops falling every day. This paper is about a new way to use those "tiny raindrops" (small earthquakes) to guess when the "big storm" (a major earthquake) might strike, specifically for the Los Angeles area.

Here is the breakdown of their method using simple analogies:

1. The Core Idea: Counting the Drops

The scientists believe that before a massive earthquake happens, the Earth doesn't just sit still. It gets "tense." As it gets tense, it releases tiny tremors (small earthquakes).

• The Analogy: Think of a rubber band being stretched. Before it snaps (the big earthquake), it makes little creaks and pops (small earthquakes). The more creaks you hear, the closer the rubber band is to snapping.
• The Goal: Instead of guessing based on a calendar (e.g., "It's been 10 years, so it's due"), they count the number of small quakes since the last big one. They call this "Natural Time."

2. The Problem: Not Enough Data

To make a good guess, you need a lot of history. But for Los Angeles, we haven't had a huge earthquake since 1994 (the Northridge quake). That's not enough data to build a reliable model just for that one city.

• The Analogy: Imagine trying to learn how to bake a perfect cake, but you've only baked one cake in your life. You can't be very confident in your recipe.

3. The Solution: The "Ensemble" (The Neighborhood Study)

To fix the lack of data, the scientists looked at Los Angeles and then drew bigger and bigger rectangles around it, like ripples in a pond.

• The Analogy: They looked at the "Los Angeles neighborhood," then the "Greater LA area," then "Southern California," and so on. They assumed that the way earthquakes behave in these bigger areas is statistically similar to how they behave in the small circle around LA.
• The Method: They treated all these different-sized regions as a "training class." They studied how small quakes led to big quakes in all these areas to build a master recipe for predicting the next big one in LA.

4. The "Nowcast" (The Weather Report for Now)

They developed a tool called a "Nowcast." This isn't a long-term forecast (like a 10-year prediction); it's a "current status report."

• The Analogy: Think of a weather app that tells you, "It's currently 40% likely to rain in the next hour."
• How it works: They count the small quakes since 1994. As the count goes up, the "Nowcast" score goes up, telling us the system is getting closer to a "snap."

5. The "Magic Mirror" (The Nowcast Transform)

Here is where it gets clever. The scientists realized that the big regions (the "ripples") might have slightly different rules than the small LA circle. To make sure their training data matched the target, they invented a "Nowcast Transform."

• The Analogy: Imagine you are trying to learn to drive a Ferrari, but you only have a Toyota in your driving school. The Toyota handles differently. The "Transform" is like a magical simulator that adjusts the Toyota's steering and speed so it feels exactly like driving the Ferrari.
• The Result: They adjusted the data from the big regions so it perfectly matched the statistical rules of the LA circle. They found that even with this adjustment, their predictions stayed consistent.

6. The Two Clocks: Calendar vs. Natural Time

The paper compares two ways of measuring time:

1. Calendar Time: How many years have passed since 1994?
2. Natural Time: How many small earthquakes have happened since 1994?

The Surprising Finding:

• Calendar Time: As years pass, the chance of a big quake seems to drop at first, then slowly rise. It's like waiting for a bus that is very late; you start to think it might never come, but eventually, you know it's coming.
• Natural Time: As the count of small quakes goes up, the chance of a big quake rises sharply.
• Why the difference? The Earth gets "stiffer" over time. As the crust heals and hardens after a big quake, it stops making small cracks (small quakes) for a while. So, even if many years pass, if the "count" of small quakes is low, the system isn't actually "charged up" yet. The "Natural Time" clock is a better indicator of the Earth's tension than the wall clock.

Summary of Results for Los Angeles

• Current Status: Since the 1994 Northridge quake, there have been about 448 small earthquakes in the 125km circle around LA.
• The Prediction: Based on this count, there is roughly a 29% to 43% chance (depending on the specific method used) that a magnitude 6.0 or larger earthquake will happen in the next year.
• The Takeaway: The Earth isn't just waiting for a date on the calendar; it's waiting for a specific number of "creaks" (small quakes) to happen before it's ready to "snap."

In a nutshell: This paper gives us a better way to listen to the Earth's "creaks" to understand how close we are to the next big "snap," using data from the whole region to make a smarter guess for the city.

Technical Summary

1. Problem Statement

The paper addresses the challenge of anticipating the local occurrence of future large earthquakes ($M_T \geq 6$) within a specific geographic region (defined as a circle of arbitrary radius surrounding a point of interest, e.g., a city). The core problem is quantifying the probability of such events occurring within a specific natural time (count of small earthquakes) and calendar time (waiting time) following the last major event.

The authors aim to overcome data scarcity in local regions by utilizing an ensemble approach, where statistical data from larger surrounding rectangular regions are used to train and forecast probabilities for the smaller circular region of interest. A critical assumption tested is whether the Gutenberg-Richter (GR) magnitude-frequency statistics of these larger regions are "scaled similar" to those of the target circular region.

2. Methodology

The methodology builds upon a three-part series, integrating Natural Time (event counts), Ensemble Learning, and Receiver Operating Characteristic (ROC) analysis.

A. Core Assumptions and Definitions

• Gutenberg-Richter (GR) Law: The fundamental assumption is that the GR relation ($N = 10^{a - bM}$) holds for all regions. The $b$-value (slope) is assumed to be similar across the ensemble and the target circle, though it may vary slightly ($\pm 0.05$).
• Natural Time: Defined as the count of small earthquakes ($M_S$) occurring since the last large earthquake ($M_T$).
• Ensemble of Regions: A series of expanding rectangular regions centered on the point of interest (Los Angeles) serves as the training set. These regions are assumed to be statistical "copies" of the target circle when appropriately scaled.

B. The Nowcast Function

The authors utilize a nowcast function $\Phi(n)$ based on Poisson statistics to model the probability of a large earthquake occurring after $n$ small events:
$$\Phi(n) = 1 - e^{-(n/N)}$$
Where $N$ is the expected number of small events between large events, derived from the GR law.

C. The Ensemble Workflow

1. Cycle Construction: The data is segmented into "earthquake cycles," where each cycle begins and ends with a target magnitude event.
2. Scaling: Forecast times ($T_F$) for ensemble members are scaled relative to the target circle based on the ratio of small event rates ($R_i / R_C$).
3. ROC Curve Construction: For each ensemble member, conditional ROC curves are built by sweeping threshold values against the nowcast function.
   • True Positive (TP): Nowcast > threshold AND next large quake occurs within forecast window.
   • False Positive (FP): Nowcast > threshold AND no quake occurs.
   • True/False Negatives: Defined inversely.
4. Ensemble Mean: The mean ROC curve and standard deviation are computed across all ensemble members to determine the Positive Predictive Value (PPV), which represents the forecast probability.

D. The "Nowcast Transform" (Key Innovation)

To rigorously test the assumption that ensemble statistics match the target circle, the authors introduce a Nowcast Transform:

• Purpose: To force the GR statistics (specifically the $b$-value) of the ensemble members to match exactly those of the circular region.
• Mechanism:
  1. Calculate the nowcast value $\Phi(n)$ for an ensemble interval using its local $b$-value.
  2. Invert the function using the target circle's $b$-value ($b_C$) to find a transformed interval $n'$.
  3. Time Adjustment: If $n' \neq n$, the algorithm adjusts the timestamps of small events within the interval:
     • If $n' < n$: Randomly remove events.
     • If $n' > n$: Interpolate new event times between existing neighbors.
  • This preserves the temporal density structure (e.g., aftershock clustering) while enforcing statistical uniformity.

E. Exceedance Probability Calculation

The paper computes conditional exceedance curves (survivor distributions) for:

1. Magnitude: Probability that the next event exceeds a specific magnitude $M$.
2. Natural Time: Waiting time in terms of small event counts.
3. Calendar Time: Waiting time in years, derived by scaling the natural time intervals back to calendar time using the inverse of the rate scaling factor.

3. Key Contributions

1. Local Forecasting via Ensemble: Demonstrated that local earthquake probabilities can be robustly computed using only small earthquake counts from a surrounding ensemble of regions, bypassing the need for long local historical records.
2. Nowcast Transform: Developed a novel algorithm to adjust ensemble statistics to match a target region, validating the "scaled similarity" assumption. The results showed that forecasts using transformed data are generally consistent with non-transformed data, reinforcing the method's reliability.
3. Dual-Time Scale Analysis: Provided a comprehensive framework for calculating exceedance probabilities in both Natural Time (event count) and Calendar Time, highlighting the non-linear relationship between the two.
4. Magnitude and Time Risk Profiles: Generated specific exceedance curves for magnitude (25%, 50%, 75% levels) and waiting times, allowing for dynamic risk assessment as new small earthquakes occur.

4. Results

The methods were applied to a 125 km radius circle around Los Angeles, CA, following the 1994 Northridge ($M6.7$) earthquake.

• Consistency of Methods: The Positive Predictive Value (PPV) curves for the Original, Filtered (outliers removed), and Transformed ensembles were found to be generally consistent.
  • Final PPV values were 29% (Original), 35% (Filtered), and 43% (Transformed).
  • This suggests the ensemble method is robust even if the underlying statistics are not perfectly uniform.
• Magnitude Exceedance: As the number of small earthquakes increases (moving away from the last large event), the probability of a high-magnitude event increases sharply, particularly for the median (50%) and 25% exceedance levels.
• Calendar vs. Natural Time Divergence:
  • Natural Time: As the count of small events increases, the expected remaining natural time to the next large earthquake decreases (intuitively, the system is "closer" to failure).
  • Calendar Time: The expected waiting time in calendar years increases as time passes since the last event.
  • Explanation: The authors attribute this to crustal stiffening. As time passes, small cracks close, increasing crustal rigidity. This shifts the deformation mechanism from unstable stick-slip to stable slip, causing an "anomalous slowing down" in calendar time relative to a simple Poisson process.

5. Significance

• Operational Nowcasting: The paper provides a practical, data-driven tool for "nowcasting" (assessing current state) and short-term forecasting of seismic risk in specific urban centers, even where historical large-event data is sparse.
• Risk Management: By quantifying exceedance probabilities for both magnitude and time, the method offers a nuanced risk profile that can inform emergency preparedness and infrastructure planning.
• Physical Insight: The divergence between natural and calendar time forecasts offers a physical interpretation of fault zone evolution (crustal strengthening) leading up to large earthquakes, bridging statistical seismology with geophysical mechanisms.
• Reproducibility: The authors provide open-source Python code and utilize freely available USGS data, making the methodology accessible for replication and extension to other regions.

In conclusion, this paper validates the "ensemble-of-regions" approach for local earthquake forecasting and introduces a rigorous statistical transform to ensure consistency, offering a robust framework for estimating the probability of future large earthquakes in terms of magnitude, event count, and calendar time.