Optimization and robustness of cost-efficient seismic arrays for Newtonian noise cancellation at the Einstein Telescope

Technical Summary: Optimization and Robustness of Cost-Efficient Seismic Arrays for Newtonian Noise Cancellation at the Einstein Telescope

Problem Statement
The Einstein Telescope (ET), a proposed third-generation underground gravitational wave interferometer, aims to detect gravitational waves down to frequencies of 3 Hz. At these low frequencies, the dominant noise source is expected to be Newtonian noise (NN). NN arises from gravitational forces exerted by density fluctuations in the surrounding rock, primarily caused by seismic body waves (P-waves and S-waves). To mitigate this, an array of auxiliary seismometers is required to measure the seismic wave field, allowing for the prediction and subtraction of NN from the interferometer output. However, drilling boreholes to house these sensors is a significant cost driver. The challenge lies in optimizing the sensor positions to maximize mitigation performance while minimizing costs, specifically by exploring configurations that utilize fewer boreholes with multiple sensors or by utilizing existing infrastructure.

Methodology
The authors employ an optimization framework previously established in Ref. [12] to determine optimal seismometer positions for a full corner of the ET (four correlated mirrors). The optimization minimizes the mean residual of the Newtonian noise signal after subtraction using stationary Wiener filters.

• Optimization Algorithm: The process utilizes Particle Swarm Optimization (PSO) for initialization, followed by the differential optimizer Adam for fine-tuning.
• Performance Metric: The mitigation factor $M(f)$ is defined as the inverse square root of the maximum residual among the four mirrors, serving as a conservative indicator of amplitude reduction.
• Scenarios Tested:
  1. Benchmark: Arrays with one seismometer per borehole, varying the total number of boreholes ($N = 20$ to $70$).
  2. Multi-Sensor Boreholes: Arrays with $N=20$ boreholes containing $n = 1, 2, 3, 5, 10$ seismometers each. Optimization constraints allow $x$ and $y$ positions to vary per borehole, while $z$ positions vary independently within the borehole.
  3. Infrastructure Integration: Addition of 100 fixed-position seismometers (spaced every 100 m) within the interferometer tunnels (arms) up to 5 km from the corner, without further optimization of their specific locations.
  4. Robustness Analysis: The stability of these configurations is tested by introducing random Gaussian displacements ($\sigma = 50$ m) to the optimized positions in all three spatial directions.
• Parameters: Simulations assume a signal-to-noise ratio (SNR) of 15, a P-wave fraction of $p=0.2$, and specific wave velocities ($c_P = 6000$ m/s, $c_S = 4000$ m/s). Arrays are optimized specifically for a frequency of $f_{opt} = 10$ Hz, and their "broadband" performance is evaluated across $1$ to $10$ Hz.

Key Contributions

1. Cost-Effective Multi-Sensor Configuration: The study demonstrates that placing multiple seismometers within a single borehole can achieve mitigation capabilities comparable to arrays with significantly more boreholes (where each borehole contains only one sensor). This addresses the high cost of drilling by maximizing the utility of each borehole.
2. Infrastructure Utilization: The paper quantifies the performance gain of deploying seismometers within the existing ET tunnel infrastructure. This approach requires no additional drilling and significantly enhances broadband performance.
3. Robustness Quantification: The authors provide a detailed analysis of how random deployment errors (up to 50 m) affect mitigation performance, showing that larger arrays and multi-sensor configurations maintain stability better than smaller, single-sensor arrays.

Results

• Broadband Performance: Arrays optimized for 10 Hz generalize well to lower frequencies (1–10 Hz).
  • For $N=20$ boreholes, increasing the number of sensors per borehole ($n$) significantly boosts the mitigation factor. Specifically, $n=5$ (100 total sensors) achieves performance similar to $N=60$ boreholes with $n=1$.
  • Adding 100 tunnel seismometers to the $N=20$ configuration further improves performance. With $n=10$ in boreholes plus tunnel sensors, the mitigation factor exceeds 10 for frequencies $f \gtrsim 3$ Hz, with peak values around 34.
• Robustness:
  • Random displacements reduce performance most significantly at higher frequencies (where the displacement is a larger fraction of the wavelength).
  • Larger arrays exhibit greater stability. For the largest configuration studied ($N=50$ boreholes, $n=10$ per borehole, plus tunnel sensors), the mitigation factor remains $>15$ for all $f \gtrsim 3$ Hz, even with random displacements.
  • For the $N=20, n=5$ configuration with tunnel sensors, the mitigation factor remains $>7$ for $f \gtrsim 3$ Hz under displacement variations.
  • Even with a modest configuration of $N=20$ and $n=3$ (plus tunnel sensors), the array maintains a mitigation factor $>6$ for $f \gtrsim 3$ Hz despite random displacements.

Significance and Claims
The paper claims that optimizing seismic arrays for higher frequencies (e.g., 10 Hz) provides effective broadband mitigation for Newtonian noise down to 3 Hz. The primary significance lies in identifying a cost-effective strategy for the ET: rather than drilling a vast number of single-sensor boreholes, the authors propose using fewer boreholes equipped with multiple seismometers and supplementing them with sensors in the existing tunnel infrastructure.

The authors conclude that such configurations offer "promising broadband performance" and are "particularly stable against variations in the seismometer positions." They suggest that these findings motivate further research into the technical feasibility of multi-sensor boreholes and the integration of sensors into ET infrastructure as a viable strategy for Newtonian noise mitigation. The paper notes that while broadband optimization might yield better trade-offs, the single-frequency optimization strategy tested here is effective. Future work is suggested to include finite-element simulations in realistic geologies and the study of fusion arrays (combining seismometers, strainmeters, and tiltmeters).