🔬 applied physics

Relative Focal Mechanism Inversion Using Relative Polarities and S/P Double Ratios

This paper introduces FocMecDR, a relative focal mechanism inversion method that leverages relative P-wave polarities and S/P amplitude double ratios to accurately determine focal mechanisms for small earthquakes, thereby overcoming limitations of traditional methods and revealing uniform stress fields during the nucleation of the 2019 Ridgecrest mainshock.

Original authors: Miao Zhang

Published 2026-03-31

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Miao Zhang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to figure out exactly how a tiny pebble cracked a window. You can't see the pebble, but you can hear the sound it made and see the pattern of the glass shards. In seismology, this "pebble" is a small earthquake, and the "window" is the Earth's crust. Scientists want to know the focal mechanism—essentially, the direction the fault slipped and the angle of the crack—to understand the stress in the ground.

For a long time, figuring this out for tiny earthquakes has been like trying to solve a puzzle in the dark. The tools scientists used (like the HASH method) relied on listening to the very first "pop" of the earthquake wave and measuring how loud the shaking was. But for small quakes, the "pop" is often fuzzy, and the loudness is distorted by the ground under the seismometer (like a microphone sitting on a bumpy table).

Enter FocMecDR: The "Twin Detective" Method

The author, Miao Zhang, introduces a new tool called FocMecDR. Instead of trying to solve the puzzle of a tiny earthquake alone, this method uses a "Twin Detective" strategy. It compares a tiny, messy earthquake (the Target) to a nearby, well-understood earthquake (the Reference).

Here is how it works, using some everyday analogies:

1. The "Echo Chamber" Analogy (Relative Polarities)

Imagine you and your twin are standing in a large, echoey canyon. You both clap your hands.

The Old Way: You try to describe the sound of your clap based on how it sounds to a listener far away. But the wind (noise) and the canyon walls (geology) distort the sound, making it hard to tell if you clapped with your left or right hand.
The FocMecDR Way: You and your twin clap at almost the exact same time. Because you are standing right next to each other, the wind and the canyon walls affect both of you in the exact same way.
The Trick: Instead of listening to the absolute sound, you listen to the difference. If your clap sounds "up" and your twin's sounds "down" to the listener, you know you clapped in opposite directions. If they both sound "up," you clapped the same way.
In Science: The computer uses a technique called "cross-correlation" (like matching two fingerprints) to see if the waves from the two earthquakes are moving in the same direction or opposite directions. This cancels out the "wind" and "canyon walls" (noise and geological distortions).

2. The "Volume Knob" Analogy (S/P Double Ratios)

Earthquakes send out two main types of waves: P-waves (like a quick push) and S-waves (like a side-to-side shake). The ratio of how loud the S-wave is compared to the P-wave tells us the angle of the fault.

The Problem: If you measure the loudness of a tiny earthquake, a small error in the ground model can make the S-wave look twice as loud as it really is.
The FocMecDR Solution: Think of this as a Double Ratio.
- First, you compare the S/P volume of the Target quake.
- Then, you compare the S/P volume of the Reference quake.
- Finally, you compare those two ratios against each other.
The Magic: Because both quakes traveled through the same ground to get to the same microphones, the "static" on the line cancels out perfectly. It's like comparing the volume of two songs played on the same speaker; you don't need to know the exact quality of the speaker to know which song is louder relative to the other.

Why is this a Big Deal?

1. It turns "fuzzy" data into sharp pictures.
Small earthquakes are usually too weak to study accurately. FocMecDR uses the "shadow" of a big, well-studied earthquake to illuminate the tiny ones. It's like using a bright flashlight (the Reference) to see the details of a dark room (the Target).

2. It's a "Truth Detector."
The paper tested this on a pair of earthquakes that were "antisimilar" (they broke in opposite directions). The old method (HASH) got it slightly wrong because it relied on fuzzy first sounds. FocMecDR looked at the relative differences and said, "No, these are definitely opposites," and corrected the solution. It acts as a referee to check if other methods are getting the answer right.

3. The "Stress Uniformity" Discovery.
The author applied this to the foreshocks (the small tremors) right before the massive 2019 Ridgecrest earthquake.

The Finding: All the tiny foreshocks had almost identical focal mechanisms. They were all slipping in the exact same direction.
The Metaphor: Imagine a crowd of people pushing a heavy door. If they are all pushing in random directions, the door won't move. But if they all suddenly start pushing in the exact same direction, the door is about to fly open.
The Implication: This suggests that before a big earthquake, the stress in the ground becomes highly organized and uniform. The fault isn't just "cracking randomly"; it's lining up, ready to rupture. This gives scientists a new way to look for signs that a big earthquake might be imminent.

The Bottom Line

FocMecDR is a clever new way to study small earthquakes by comparing them to their "big brothers." By canceling out the noise and distortions of the Earth, it gives us a clearer picture of how faults are moving. This helps us understand not just where earthquakes happen, but how the Earth is preparing to break, potentially offering a new window into predicting when a fault is about to snap.

1. Problem Statement

Determining focal mechanisms for small earthquakes is essential for characterizing fault geometry and regional stress fields. However, traditional methods face significant challenges:

Waveform Fitting: Requires dense station coverage, high-quality data, and accurate velocity models. It is computationally intensive and often limited to moderate-to-large earthquakes ( $M > 3$ ).
First-Motion/Amplitude-Ratio Methods (e.g., HASH): While computationally efficient, they suffer from:
- Difficulty in picking first-motion polarities for emergent onsets or low Signal-to-Noise Ratio (SNR) events.
- High sensitivity to velocity model inaccuracies and unknown site responses, leading to inconsistent S/P amplitude ratios (often requiring empirical station corrections).
- Ambiguity in distinguishing faulting types (e.g., normal vs. thrust) when azimuthal coverage is limited.

Existing relative inversion methods often rely on complex ray-theory-based joint inversions or still depend on the HASH framework, inheriting its limitations regarding station corrections and absolute amplitude modeling.

2. Methodology: FocMecDR

The author proposes FocMecDR, a fully automated relative focal-mechanism inversion method. It operates on the principle of double-difference (similar to relative earthquake location), using a well-constrained "reference event" to solve for nearby "target events."

Key Components:

Inputs:
- A reference event with a known, high-quality focal mechanism.
- Waveforms from a target event and the reference event recorded at common stations.
Constraints Used:
1. Relative Polarities: Instead of absolute first-motion picks, the method uses cross-correlation (CC) of P, SH, and SV waveforms between the event pair. If the CC is high (e.g., > 0.7), the relative polarity (consistent or opposite) is determined without needing absolute phase identification. This increases the number of usable constraints and robustness against noise.
2. S/P Amplitude Double Ratios: The method calculates the ratio of S-wave to P-wave amplitudes ($SH/P$ and $SV/P$) for both events. It then computes the double ratio (the ratio of these ratios between the target and reference events).
  - Advantage: This cancels out path effects, velocity heterogeneity, and site responses, eliminating the need for station corrections.
Inversion Process:
- A grid search is performed over strike ( $0-360^\circ$ ), dip ( $0-90^\circ$ ), and rake ($-180$ to $180^\circ$ ).
- An objective function minimizes the misfit between observed and theoretical values for:
  - Relative polarity consistency (based on CC).
  - Logarithmic double-ratio misfits of $SH/P$ and $SV/P$.
- Reference Selection: The method does not fix a single reference event. Instead, it tests multiple nearby eligible reference events for a target and retains the solution with the minimum misfit, optimizing for station azimuthal coverage and data quality.

3. Key Contributions

Novel Constraint Combination: The first method to jointly invert relative polarities (derived from cross-correlation) and S/P double ratios for focal mechanism inversion.
Elimination of Station Corrections: By using double ratios, the method inherently cancels out site effects and velocity model errors, a major source of uncertainty in traditional amplitude-ratio methods.
Automation and Robustness: The workflow is fully automated, requiring no manual polarity picking. It leverages waveform similarity to extract constraints even for small events where first-motion picking is difficult.
Validation Framework: Provides a rigorous synthetic and field-data validation strategy, including the use of "antisimilar" earthquake pairs to test the method's ability to resolve distinct mechanisms.

4. Results

Synthetic Tests:

FocMecDR successfully recovered focal mechanisms for target events with various mechanisms (strike-slip, dip-slip, thrust) using a reference event.
Accuracy: With an accurate reference, 80% of recovered mechanisms had Kagan angles within 2.25°. Even with a perturbed reference (introducing ~10° error), 80% of target solutions were within 6.36°, indicating the method does not fully inherit reference uncertainties.
Robustness: The method performed well even with significant station azimuthal gaps (up to 210°) and with single-component (vertical) data.

Field Verification (2019 Ridgecrest Sequence):

Antisimilar Pair: Applied to a known pair of antisimilar earthquakes ( $M$ 2.01 and $M$ 1.65). FocMecDR recovered the target mechanism with a Kagan angle of 31° relative to the catalog solution but demonstrated a significantly better fit to the observed relative polarities and double ratios than the catalog (HASH) solution, which showed polarity mismatches in SH and SV phases.
Aftershock Cluster: Applied to 85 aftershocks in the northwestern Ridgecrest zone. FocMecDR resolved 47 events. 80% of these solutions had Kagan angles < 29° compared to refined HASH solutions. Detailed analysis of a mismatched pair revealed that the catalog HASH solution contained errors in S-wave polarities that FocMecDR correctly identified and corrected.

Application to Foreshocks:

Applied to 39 foreshocks preceding the $M_w$ 6.4 Ridgecrest mainshock.
Successfully resolved 15 events (plus the $M$ 4.0 reference).
Finding: All 16 events exhibited highly uniform strike-slip mechanisms (mean strike $\approx 140^\circ$ , dip $\approx 87^\circ$ , rake $\approx 170^\circ$ ).
Implication: This suggests a highly uniform local stress field during the nucleation phase, contrasting with the more variable mechanisms often seen in post-mainshock sequences.

5. Significance

Stress Field Characterization: FocMecDR enables high-resolution tracking of temporal stress variations in the local stress field, particularly for small earthquakes where traditional methods fail.
Nucleation Insights: The application to Ridgecrest foreshocks suggests that a transition to a uniform stress state and consistent focal mechanisms may be a precursor to large ruptures, offering a potential framework for evaluating fault-zone rupture readiness.
Quality Control: The method serves as an independent tool to verify and improve existing focal mechanism catalogs by identifying inconsistencies in S-wave data that absolute polarity methods might miss.
Future Potential: The approach lays the groundwork for cluster-based inversions (without a single reference) and could be integrated with machine learning for real-time stress monitoring.

In summary, FocMecDR represents a significant advancement in seismic source analysis by leveraging relative measurements to bypass the limitations of absolute amplitude modeling, providing a robust tool for studying small earthquakes and earthquake nucleation processes.