Broadband Ground Motion Synthesis by Diffusion Model with Minimal Condition
Authors: Jaeheun Jung
Jaehyuk Lee
Chang-Hae Jung
Hanyoung Kim
Bosung Jung
Donghun Lee
Venue: ICML 2025 poster (2025)
Keywords: seismology
earthquake synthesis
diffusion model
Primary area: applications->chemistry_physics_and_earth_sciences

Abstract: Shock waves caused by earthquakes can be devastating. Generating realistic earthquake-caused ground motion waveforms help reducing losses in lives and properties, yet generative models for the task tend to generate subpar waveforms. We present High-fidelity Earthquake Groundmotion Generation System (HEGGS) and demonstrate its superior performance using earthquakes from North American, East Asian, and European regions. HEGGS exploits the intrinsic characteristics of earthquake dataset and learns the waveforms using an end-to-end differentiable generator containing conditional latent diffusion model and hi-fidelity waveform construction model. We show the learning efficiency of HEGGS by training it on a single GPU machine and validate its performance using earthquake databases from North America, East Asia, and Europe, using diverse criteria from waveform generation tasks and seismology. Once trained, HEGGS can generate three dimensional E-N-Z seismic waveforms with accurate P/S phase arrivals, envelope correlation, signal-to-noise ratio, GMPE analysis, frequency content analysis, and section plot analysis.

Broadband Ground Motion Synthesis by Diffusion Model with Minimal Condition
Jacheun Jung
Jachyuk Lee" 1
Changhae Jung'
Hanyoung Kim
Bosung Jung' Donghun Leee
Abstract
grew, various improvements in seismological applications were made by analyzing historically observed seismic wave- forms. For example, the accuracy of earthquake analysis was improved, early warning systems for earthquake-prooe areas were polished, and carthquake-resistee architectural designs became more robust. Recently deep learning found successful applications in seismology (Mousavi & Beroza, 2022), such seismic signal denoising (Zhu al., 2019), fault recognition (An et al., 2021), and earthquake event detection (Mousavi et al., 2020; Saad et al., 2023).
Shock waves caused by earthquakes can be dev- astating. Generating realistic carthquake-caused ground motion waveforms help reducing losses in lives and properties, yet generative models for the task tend to generate subpar waveforms. We present High-fidelity Earthquake Groundmotion Generation System (HEGGS) and demonstrate superior performance using earthquakes from North American, East Asian, and European re- gions. HEGGS exploits the intrinsic character- istics of carthquake dataset and learns the wave- forms using an end-to-end end differentiable genera- tor containing conditional latent diffusion model and hi- hi-fidelity waveform construction model. We show the learning efficiency of HEGGS by train- ing on a single GPU machine and validate its performance using earthquake databases from North America, East Asia, and Europe, using di- verse criteria from waveform generation tasks and seismology. Once trained, HEGGS can generate three dimensional E-N-Z seismic waveforms with accurate P/S phase arrivals, envelope correlation, signal-to-non noiise ratio, GMPE analysis, frequency content analysis, and section plot analysis.
However, the field still faces a shortage of data, particu- larly for large-scale earthquakes as they are rare (Shi et al., 2024; Katsanos al., 2010). Recently, deep-learning based synthesis of seismic waveforms has emerged as potential solution, mostly employing GAN-based generative models conditioned with various geological and seismological infor- mation ( (Wang et al., 2021; Florez et al., 2022; Li et al., 2024; Chen et al., 2024). However, the synthesized waveforms from these models often lack seismological realism, such as phase arrival times and amplitude of ground motions.
We see this problem as a mixed artifact of conditioned gen- eration model architecture and unreliable conditioning in- formation. Hence, we propose adapting diffusion model architecture (Sohl-Dickstein et al., 2015; Ho et al., 2020), which have shown superior realism and stability in image generation, to seismic waveform data in order to present a novel generation system for seismologically realistic ground motions with bare minimum of conditioning informa- tion.
1. Introduction
Broadband ground motion caused by seismic waves is cru- cial in the study of earthquakes and geology since it in- cludes important features related to subsurface structures of the solid Earth. At the same time, is a great challenge from signal processing perspective, as observed ground mo- tion waveform signals cover a wide frequency band and are caused by and unevenly distributed earthquake events.
Our Contribution
We design novel seismic waveform generation model, HEGGS, that requires a bare-minimum set of condi- tional information on the earthquake and the observa- tion point.
As the size of systematically recorded seismic waveforms
We demonstrate constructing datasets for HEGGS us- ing openly available seismic datasets, such that ob- served waveforms are paired with the source earth- quake and time-aligned to the carthquake origin time.
'qqual contribution 'Department of Mathematics, Korea Uni- versity, 145 Anam-ro, Scongbuk- gu, Scoul, Republic of Korea. Correspondence to: Donghun Lee <holy @ kr.>.
Proceedings of the ndnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
We design an end-to-end training method for the model and demonstrate its effectiveness by learning to gen-

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Broadband Ground Motion Synthesis by Diffusion Model with Minimal Condition
erate high-fidelity earthquake ground motions from earthquakes in North America, East Asia, and Europe.
(Han al., 2023) from East Asia, and SCEDC (SCEDC. 2013) from North America. Raw waveforms and corre- sponding metadata including locations, earthquake ID, mag- nitude and earthquake occurrence time are collected. and the processing of each datasets are detailed in Appendix A.
We validate the superior fidelity of generated samples from HEGGS against benchmark models in various per- spectives including seismology-inspired metrics such as GMPE analysis and phase arrival prediction.
3. Method
2. Key Idea
Inspired by conditional music generation method (Ghosal et al., 2023), method first creates spectrograms with diffusion model and then convert spectrograms into wave- forms. Our generative model fully exploits the pair-ability of seismic waveform datasets shown in Section 2 to train both the diffusion process for spectrogram generation and the high-fidelity decoder for waveform generation. We name the method HEGGS, an acronym for High-fidelity Earth- quake Groundmotion Generation System.
Our goal is to robustly synthesize the broadband ground mo- tion data with high level of seismological realism when com- pared actual seismnggaaph-recorde earthquake events. We cast this seismic waveform synthe- sis problem into conditional waveform generation task to learn from seismic waveform databases, with minimal de- pendency on conditional information.
waveforms caused by
We use the following as the minimal conditional info:
3.1. Pair-Exploiting Diffusion Model
1. Stat, Ston latitude and longitude of the station to ob- serve the waveform data.
For each earthquake event, we sample pair of waveforms (W WW*r) from dataset and convert it to spectrograms (X ( and construct conditional vector of target sta- tion Cags by preprocessing.
2. Elat, elon latitude and longitude of the hypocenter.
3. Cdep depth of the hypocenter, in kilometers.
Let q(xi:ri be the forward process of the diffusion model, and consider two trajectories [[x//r[xarr and [[=[")]x0 Recall that X and X shares the prop- erty of earthquake, we may assume that from Xare and Tege we can gather enough features of earthquake to generate Xtst, In this approach, we may consider the transform map na ; Tigs, f) for t > 0 which maps the latent of input Xare to the latent of target Xt as a random variable, with following assumption:
4. My local magnitude of the earthquake.
This set of information is usually considered insufficient for ground motion synthesis. For example, seismological properties of the source earthquake such as focal mechanism or local geological propertics of observation point such as Vs30 are often required in prior works. Instead of addi- tionally demanding the often expensive-to-obtain info, we desire a generative model that learns directly from seismic waveforms with minimal conditional info as metadata.
7( Tege, t) ~ q(x 19g (xto))
(1)
When an earthquake happens, its shock waves propagate and are recorded by nearby seismographs as three-dimensional seismic waveforms named seismograms. Naturally, these seismograms are correlated by the common information about the source earthquake well the information spe- cific to each of the seismographs such geological con- ditions around the observation location. Regional seismic waveform datasets contain multiple waveform observations that can be paired to a source earthquake event, as illustrated in the left panel of Figure 1. The observations paired to the same earthquake would share properties of the same source earthquake, while containing different information pertain- ing their respective observation location. This is how liberate ourselves from asking additional conditional info.
Referring (Salimans Ho, 2022), the loss function LDM of diffusion model in x-space (sample space) is:
CDm =Exxtr Exxxsst :e ( xo(xt Eegt, t)1 (2)
while the SNR weight is simplified.
Considering the Equation (I), we rewrite the loss function
CDM Ex xttx Reye) Xttt mo Eegs, 1)
where
(4)
Hence, we predict my by neural network, which is a com- position of latent transform function and denoising model.
We exploit this intrinsic pair-ability of the seismic waveform datasets, and construct paired leveform-metadant datasets from three earthquake databases from different continents: INSTANCE (Michelini et al., 2021) from Europe, KMA
For the sampling of the reverse process, we exploit the same procedure of the denoising process of diffusion, as
!g! 1 = (e .mo 1ot Eggt, 1)) + o. N(O, (5)
2

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Broadband Ground Motion Synthesis by Diffusion Model with Minimal Condition
Input waveform
Target meta
1300
-125
-120
nagintude
M,36.4
-118-4488
32,88013
-1780/122
22/8821
14:..00
STFT
Latent Space
30
STC
E.u
Diftusion
M,3.3
U-Net
D
30
Stufon
2
Epicentenoouree
Sourcce 1d
Calculate Loss
3x8000 (S-lyye FPN)
(37600953,
Biatance:
120, d5, 24)
Figi
Egt
22.3)).33
Conditional Vartahles
(1x11)
Matrix Multblliction
Crass Attention
Target waveform
338443183
spasstass
Pixel Space
ACM
CRIIN berashel
Higritaate [c.4,2.4]
Figure 1. Left: Visualization of SCEDC data using paired waveforms. shows that earthquake depending on magnitude. Right: Diagram of the waveform generative model architecture of HEGGS and its training loss.
detected greater distances
where (1t, .o) is mean vector of g( xo), intro- duced in Eq. (7) of (Ho et al., 2020).
loss function shown below: Lours
This is equivalent to conventional denoising process, as
X - (mm(m Fett, 1)) = Bxxor, X (eot
(9)
(a Eigt, 1) nt
where 27 VOLEAB + V1-
Using Cours the loss function, train the waveform generation model end-to-end, covering the encoder, the dif- fusion module, and the decoder with ACM (Amplitude Cor- rection Module), as shown the right panel of Figure 1. For the detailed implementation in the diffusion module, we used U-Net backbone for mg. brought E and D DA More details on the specifications of HEGGS and its training recipe be found in Appendix B.
by assumption and thus
mn Eggt 1) xo(x Teges
Therefore, pair-exploiting training process of HEGGS al- lows the diffusion model to generate t from the Gaus- sian noise #T ~ NO( following conventional reverse fgt
After training diffusion model with HEGGS, we generate waveform with conventional reverse process by setting 2T by Gaussian noise or Z The details with pseudocode of training and generation, can be found in Appendix
process with mg.
.gg
3.2. End-to-end Model Training
From the idea of LDM (Rombach et al., 2022), we consider the autoencoder comprised a downsampling module EAE and an upsampling module DAE. and construct diffusion model on latent space with smaller dimension. If there were suitable pretrained autoencoder, the LDM loss would be
4. Empirical Verification
We showcase the performance of HEGGS in three cases:
generate waveforms of existing earthquakes at existing observation stations Eagt, using an observed carthquake information Were as input waveform.
CLDDM Eizere zt B(zere,zoo, Fe he, gg mo(22Te Legh t)]
(8)
where = ExE(X), zero is latent of diffusion process of zar.
generate at arbitrary locations Elgt= using an observed earthquake information W as input waveform.
There is no suitable encoder-decoder model for seismic waveforms, so we modify Equation (8) into end-to-end
generate waveforms of fictitious earthquake informa- tion Eg (also, without Warc).
3

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Broadband Ground Motion Synthesis by Diffusion Model with Minimal Condition
The first case is designed verify the fidelity of HEGGS, by comparing the generated samples to ground truth wave- forms. We present quantitative results in Section 4.1 and qualitative analyses with visualizations in Sections 4.2.1 to 4.2.3. The results from the second case are presented throughout Sections 4.1 and 4.2, and in Section 4.2.4 we show the results from the third case.
Using the finetuned EQT as the phase picker, we label gen- erated synthetic waveforms earthquake event from the test dataset. The mean absolute error (MAE) metrics of the P wave and the S wave phase arrival times from the ground truth labels are reported in the first two columns of Table 2. The resulting phase arrival times of the P wave and the S wave are considered to be close the ground-truth arrival times on all datasets, as the MAE values measured to be significantly small while other benchmark models failed to generate earthquake event signals on correct arrival time. Notably, generating with imput waveform W gives better results compared to gen- erating without Wore (i.e. with noise), as the observation Were contains earthquake-specific information.
random station for every
4.1. Quantitative Evaluation
To assess and compare the effectiveness of models syn- thesizing seismic waves, we conducted a comprehensive quantitative analysis focusing on key parameters including P-wave and S-wave arrival times, GMPE analysis, and simi- larity measures such envelope correlation, spectrogram image similarity, as well as signal-to-noiss ratio (SNR), and peak signal-to- -oniie ratio (PSNR). Specifically, we compare generated synthetic waveform Worrd from Eigt and com- pare it with corresponding ground truth waveform W7tst to compute each metric.
4.1.2 SIMILARITY MEASURES
We also compare the synthesized waveforms and corre- sponding spectrogram directly to the ground truth wave- forms. We use general purproe similarity measures: the envelope correlation, SNR and PSNR for the waveforms, and MSE for the spectrograms.
For comparison, we also trained the following benchmark models on SCEDC: Seismogen (Wang et al., 2021), Con- seisgen (Li et al., 2024), BBGAN (Florez et al., 2022) and LDM (Rombach et al., 2022). Since the input shape of waveform Whaa or spectrogram X is different from each of the models, we give reasonable modifications to them for the training and evaluation. The detailed changelogs are listed in Appendix E.
Envelope correlation was calculated to measure the simi- larity between the envelopes of synthesized and observed seismic waves, providing insights into the overall waveform fidelity. We applied Savitzky- Filtering (Savitzky & Golay, 1964) technique with polyorder 3 before calculating the envelope correlation. In implementation, we exploit (Beyreuther et al., 2010) implementation of cross correla- tion, which includes the waveform normalization. Further- more, SNR and PSNR metrics were employed to evaluate the quality of the synthesized seismic waves in terms of signal clarity and fidelity to the original data. Addition- ally, we compare the synthesized spectrogram X=en and spectrogram of observed seismic signals Atgf to quantify their similarity using image similarity. We normalized both spectrograms to compare
4.1.1. PHASE ARRIVAL TIMES
The arrival times of P-wave and S-wave are the most basic, but important properties of seismic waveforms, as determin- ing P-wave and S-wave from the seismogram is the first step of earthquake analysis. Since we want to assess the fidelity of the generated waveforms, we fine-tune the SeisBench (Woollam et al., 2022) implementation of EQTranffrrmer (EQT) (Mousavi et al., 2020) on each dataset to use it in labeling phase arrivals times and compare them with the ground truth arrival times. The detailed training recipe we used for EQT is given in Appendix D. We present the perfor- mance measure of the finetuned EQT in the test dataset in Table 1. Note that the waveforms are inherently normalized as preprocessing step of EQT prediction.
The results are summarized in Table 2. The proposed method outperforms the benchmark models on almost all similarity metrics, which may imply that the generated xrred and Wored are more realistic in most cases. These quantitative analyses provide a comprehensive assessment of how similar HEGGS- -eenertte waveforms are to the actual observed seismic ground motion.
Table 1. Performance of EQT picker on each test datasct. Samples with errors less than 0.5s are considered to be positive for Fl score computation.
4.1.3. GMPE ANALYSIS
Ground Motion Prediction Equation (GMPE) is a widely used mathematical modeling in seismology that predicts the ground shaking intensity caused by earthquakes, and it is crucial for seismic hazard assessment and earthquake- resistant structural engineering. The GMPE model re- lates earthquake parameters, like local magnitude M, and hypocentral distance Reypos ground motion characteris-
Dataset
P_MAH(s)
S_MAE(s)
P_F1
S_F1
SCEDC
0.1116
0.2189
0.9728
0.9384
KMA
0.0993
0.1362
0.9635
0.9624
INSTANCE
0.1738
0.3151
0.9797
0.9099
4

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Broadband Ground Motion Synthesis by Diffusion Model with Minimal Condition
Table Results quantitative analysis. Models were evaluated with W when it trained with paired data, otherwise without W
except (+): evaluated without W while the model was trained with paired data.
Waveform
Spectrogram
Dataset
Model
Input
P_MAE (s) S.MAE (s) env.cory
SNR
PSNR
MSE
SeismoGen
w/o Were
1.9558
3.6246
0.4895
-8.6166
23.5431
1.4124
(Wang et al., 2021)
w/ Ware
1.8426
3.3325
0.5454
-8,6282
23.6354
0.8063
ConSeisGen (Li et al., 2024)
w/ Ware
3,9724
6.8992
0.3246
-8.6216
23.6416
0.7461
w/ Ware
3,9102
6,8055
0.2980
-8.5341
23.5329
0.9356
SCEDC
BBGAN (Florez al., 2022)
w/o Were
6,4210
10.416
0.1950
-3,0093
23.7598
1.6150
w/ Ware
diverged
LDM
w/o Ware
1.1142
1.7294
0.6932
-3.0202
24.7573
0.2838
(Rombach et al., 2022)
w/ Were 0.5633
0.7808
0.7726
-3.0015
19.6269
0.2426
HEGGS
(*)w/o Ware 0.5025
0.8003
0.7963
-2.9891
24.6816
0.1531
(ours)
w/ Ware 0.4760
0.5476
0.8187
-2.0051
24.6553
0.1512
LDM
w/o W
1.6233
2.1125
0.7703
-3.0006
25.3883
0.3521
(Rombach al., 2022)
w/ Ware 1,3521
1.6845
0.8076
-2.9989
26.3659
0.3785
KMA
HEGGS
(*)w/o Werc 0.2988
0.5551
0.8769
-3.0034
26.2769
0.1867
(ours)
w/ Were
0.2763
0.4644
0.8785
-2.9768
26.8730
0.1682
LDM INSTANCE (Rombach et al., 2022)
w/o W
0,8417
0.7847
0.7921
-3.0062
22.0767
0.2927
w/ Ware 0.8187
0.7875
0.7898
-2.9904
22.0956
0.2841
HEGGS
(*)w/o Ware 0,5192
0.6804
0.8299
-3.0004
22.3690
0.1376
(ours)
w/ Wore
0.5085
0.6378
0.8301
-2.9976
22.6870
0.1308
ties, such as Peak Ground Acceleration (PGA). Since My is obtained from the peak amplitude of the waveforms, the GMPE analysis shows how the scale of the generated wave- forms from HEGGS matches the real observations.
4.2.1. WAVEFORM ANALYSIS
Synthetic waveforms can be visually inspected alongside real seismic waveforms for similarities in terms of waveform morphology, including amplitude, shape, and duration of seismic signals.
Computing PGA (Emolo et al., 2015; Boore al., 2014; Lanzano et al., 2019a;b) closely related to local magni- tude Mr, which would be calculated by distinet formula (Han et al., 2023; Uhrhammer et al., 2011; Di Bona, 2016) for each region. We follow the conventions in geoscience research to decide PGA computation formula, which de- tailed in Appendix
Figure compares a set of representative synthesized wave- forms and real waveforms from three datasets. Notably, both synthesized waveform and real waveform depict sim- ilar patterns of seismic activity, including distinct seismic phases and their corresponding arrivals. This alignment underscores the effectiveness of our approach in accurately replicating the seismic signal's morphology and temporal evolution. More waveform examples can found Ap- pendix H.
GMPE analysis result, shown in Figure 2, reveals that the synthetic seismic waveforms generated by HEGGS closely resemble how the observed ground truths in distribution. Also, the similarity in PGA values indicates that the mag- nitude of synthesized waveforms is similar to the ground truth.
4.2.2. SPECTROGRAM COMPARISON
We also show the output spectrogram of HEGGS, compared to the spectrogram of the ground truth waveform to examine their fimeffreyeency characteristics. This provides insights into the similarities of temporal distribution energy across di different frequency bands.
4.2. Qualitative Evaluation
We perform qualitative analyses to evaluate the seismologi- cal fidelity and reliability of HEGGS-generaad waveforms.
In Figure 4 show the comparison of spectrograms, where each corresponds to the waveforms in Figure 3. Upon com-
5

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Broadband Ground Motion Synthesis by Diffusion Model with Minimal Condition
M, 5.0
Mi 6.0
PGA (o)
PGA (g)
M. 2.0
3.0
301
Syreneticteeoo input
Syntheticwiio input Syhthelioww inpur
Syntheditwwlo input waveform)
Syrenetictee imput
Synthedctww waveform!
Distance (km)
Distance (km)
Distance (km)
(a) SCEDC
(b) KMA
(c) INSTANCE
Figure 2. Result of GMPE analysis in PGA values with respect to the distance. The points in the figures represent the PGA values calculated from randomly selected waveforms from the test set containing earthquakes filtered by the magnitude range indicated by the black solid lines, and synthetic waveforms using the corresponding metadata. The subfigures correspond to the earthquake source: SCEDC (North America), K.A (East Asia), and INSTANCE (Europe).
SCEDC
2009
1450
RAAA
Tone
(a) SCEDC
NSSAACE
Best
Suncnetic
Figure 3. Comparison of 3-component real and synthetic wave- forms from earthquake datasets SCEDC (North America), KMA (East Asia), and INSTANCE (Europe). For each panel, top shows a real waveform and the bottom shows synthetic waveform gen- crated with the same metadata. The phase arrivals marked as red (P) and blue (S) lines detected by EQT.
Thme
Thmm
(b) KMA
3000
Jom
Time
(c) INSTANCE
paring the synthesized spectrogram with the real spectro- gram, several key observations come to light. Both spec- trograms exhibit remarkable similarities in terms of phase arrival times and frequency band distribution, indicative of the efficacy of our synthesis approach in capturing essential seismic signal characteristics. However, it is discernible that the synthesized spectrogram exhibits a slightly lower resolution compared to the real spectrogram, with some de- tails appearing less defined. This reduction in resolution is particularly evident in the depiction of fine-scale frequency variations and subtle signal features. Despite this differ- ence, the overall agreement between the synthesized and real spectrograms underscores the fidelity of our synthesis method reproducing the fundamental characteristics of seismic signals. More spectrogram examples are shown in Appendix H.
Figure 4. Comparison of real and synthetic spectrograms.
4.2.3. FREQUENCY CONTENT ANALYSIS
We also analyze how the energy released during an earth- quake is retained in different frequencies. This analysis is closely related to the concept of corner frequency (Boore, 1983) and the seismic moment (Mp). The corner frequency is generally associated with the earthquake's magnitude. Specifically, the corner frequency identifies the point at which high-frequency energy begins to decline sharply, in- dicating that larger earthquakes generally have lower corner frequencies. The Mo represents the total energy released by the earthquake, which corresponds to an increase in amplitude on the spectrum as the earthquake's 's magnitude increases. By comparing synthetic and observed seismic
6

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Broadband Ground Motion Synthesis by Diffusion Model with Minimal Condition
101
107
207
10y
2013
10-4
107
10.2
10-3
10.1
10
10-1
10'
30
30
(a) SCEDC
(b) KMA
(c) INSTANCE
Figure 5. (a)-(c).freqeency contents of synthetic waveform compared to the real waveform
Figure 6. The result quite promising: the corner frequency gets smaller and Mo gets larger properly when the magni- tude grows, which is consistent to the theory explained in (Geller, 1976).
18a
Acceleration (mls²)
4.2.5. WAVEFORM ANALYSIS ON SYNTHETIC STATIONS
10-1
By arranging virtual observation stations a linear manner, spatial variations of seismic waves could be observed, fa- cilitating an understanding of seismic event characteristics. The synthesized seismic waves reflected scismic activity at the virtual observation stations, enabling exploration of subsurface structures and seismic wave propagation charac- teristics.
M, 3.0
M, 3.5
M,
M, 4.5
10-1
Real M, = 3.0
Synthetic
The sections in Figure 7 represented the positions of obser- vation stations horizontally and represented the temporal and frequency characteristics of seismic activity vertically. Through such visualization, comparisons between synthe- sized and observed seismic waves could be conducted, as- sessing the fidelity of the synthesized seismic waves in reflecting seismic events. Results from section plots clearly visualized spatial and temporal variations of seismic activ- ity, serving as crucial criteria for evaluating the extent to which HEGGS accurately reproduces actual observed re- sults. More section plot examples of seismic events are provided in Appendix I.
Frequency (Hz)
Figure Magnitude Manipulation
signals, we aim to evaluate the similarity between the two characteristics of corner frequency and My across different magnitudes.
We apply both Fast Fourier Transform (FFT) and Konno- Ohmachi-sman -amachi-smoothing (Konno & Ohmachi, 1998) technique to enhance our comparison. Also, we apply Wood-Anderson simulations (Havskov & Ottemöller, 2010) to compare re- sults from distinct stations. The resulting spectra are shown in Figures 5a to 5c. We observe significant differences in corner frequency and Mo across different earthquake mag- nitudes. We also observe the trend of reduced corner fre- quency reduces and increased Mo as the magnitude grows.
5. Ablation Studies
Compared to the conventional latent diffusion model, we introduced two major components, the efficient learning framework and amplitude correction module, to generate high-quality seismic waveforms. In this section, we present the results of the ablation study to evaluate the role of each component, on SCEDC dataset.
4.2.4. MAGNITUDE MANIPCLATION
Synthesis of waveforms from fictitious earthquake is diffi- cult challenging problem in DL-seismology area, especially with large magnitude, since the seismological features of large earthquake is hard to capture and large magnitude carthquake data is very rare, which requires the extrapo- lation ability of the model. We select Cigt from the test dataset, change Mr, and generate waveform with modified Ceg without Wore and analyze the frequency contents in
To assess the effectiveness of this approach, we conduct training of the LDM (Rombach et al., 2022) with two dis- tinct training schemes: the original and modified one trained by Equation (8) with paired data. Unfortunately, conven- tional LDM training tried to train LDM to generate normalized waveform, which
dataset was diverged. Hence we

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Broadband Ground Motion Synthesis by Diffusion Model with Minimal Condition
feature. The amplitude correction module ACM helps im- prove the quality of seismic waveform synthesis, as shown in the 3rd and 4th rows of Table 3.
0-
Thanks to reviewers, we found that the ACM has ability to allow LDM trainable with unnormalized waveforms, as shown in 5th row of Table 3. The last row of Table 3 shows the remarkable improvement induced by pair-exploiting strategy, but still not better than HEGGS with end-to-end training.
Time
30
30
Input
6. Discussion
Real
Synthetic
60
The HEGGS training method, which takes advantage of the seismic dataset characteristic by training the model with paired data, allows for two modes of generation criteria: with and without W Although HEGGS is trained using Wore, the fidelity of generation without W is promising, and much better than the benchmark models. Generation without Wore allows us to synthesize waveforms of non- existent earthquakes and simulate the ground motion with different magnitude or location, which is big challenge in seismology. As shown in Section 4.2.4, HEGGS shows the theoretically-expeeeed trend of corner frequency and Mo. but may not be perfect since only used minimal condition about location and magnitude. We expect larger success with additional geological features, which we did not included in minimal condition, for this non-existent earthquake synthesis challenge.
0.0
20.0
40.0
60.0
80.0
100.0
120.0
140.0
Offset (kml
Figure 7. Section plots comparing synthetic and real waveforms. The vertical capital letters displayed at the top of the plot are the station ID that observed each real waveform.
Table Results of ablation study. represents the generation of normalized waveform. env.corr refers to the envelope correlation between synthesized waveform and real waveform.
Model
P.MAE (s) S.MAE (s) env.corr
LDM
1.1142
1.7294
0.6932
+paired data*
0.5633
0.7808
0.7726
tend-to-end train
0.8014
1.5367
0.6239
+ACM (HEGGS)
0.4760
0.5476
0.8187
LDM+ACM
1.1131
1.6372
0.6981
The seismic synthesis studies are inevitably build on re- gional dataset, since each observatories operated in- dependently by each government and thus the waveform formats are unaligned. Another challenging problem arises here to build global model by training multiplc models on individual dataset, with consideration of robust consistency, especially on border. We expect HEGGS would be the effective starting point of this research direction.
+paired data
0.7748
0.9402
0.7965
relaxed version of our task.
After that, we tried to generate unnormalized waveforms, by changing the training framework. Preserving the model architecture, we trained the model which has same archi- tecture, but by end-to-end training Equation (9). The only difference between this model and ours is the amplitude correction module ACM.
Compared to other methods, HEGGS shows superior fidelity with minimal conditions, especially for the P/S phase arrival times. We expect HEGGS would applicable to downstream tasks which is sensitive to the phase arrival times,such as early warning systems, earthquake modeling and disaster, hence we are planning to develop algorithms for those down- stream tasks using HEGGS as a near-future research.
The results can be found in Table 3. On first two rows, learning with paired data were very effective to increase the quality of waveform, especially as the phase arrival times were twice as accurate. Comparing 2nd and 3rd rows, the overall scores seem to be worse, but the model 2nd row often generates unrealistic waveforms in qualitative analysis results in Appendix G. Also, note that the result of 3rd row is the result of unnormalized waveform generation while 2nd row generates normalized waveform. Even the difficulty of generation problems were increased, the paired training shows better results, compared to the baseline model LDM. This may indicate the failure of VAE pretraining that pre- trained VAE could not capture the amplitude as important
7. Conclusion
In this paper, we propose HEGGS, an efficient training framework for seismic waveform synthesis utilizing a diffu- sion model and a minimal set of conditions. Our approach generates seismic waveforms using only readily accessi- ble information, such location and magnitude, thereby avoiding the need for extra conditions.
8

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Broadband Ground Motion Synthesis by Diffusion Model with Minimal Condition
To empirically validate the proposed method, con- structed seismic dataset from the SCEDC, INSTANCE
on Geoscience and Remote Sensing, 62:1-15, 2024. doi: 10.1
and KMA dataset by collecting simultaneously paired ob- servations aligned with the earthquake's 's origin time. We demonstrate that HEGGS produces more realistic wave- forms than existing benchmark models by applying seismic domain-specific metrics, such as envelope correlation and P/S phase arrival times, for expert-level comparison and applications.
Di Bona, M. A local magnitude scale for crustal earthquakes in italy. Bulletin of the Seismological Society of Amer- ica, 106(1):242-258, 01 2016. ISSN 0037-1106. doi: 10. 10.1785/01201501555 URL https: /ttp::///doi.ooo /rt////00..rr......///////// 1785 0120150155.
Emolo, A., Sharma, N., Festa, G., Zollo, A., Convertito, V., Park, J.-H., Chi, H.-C., and Lim, I.-S. Ground prediction equations for south korea peninsula. Bulletin of the Seismological Society of America, 105(5):2625-2640. 2015.
Impact statement
Our work enables high-fidelity seismic waveform synthesis, enhancing earthquake modeling, early warning systems, and disaster preparedness while promoting Al use in geophysical research.
Esser, P., Rombach, R., and Ommer, B. Taming transformers for high- nigh-resolution image synthesis, 2020.
Florez, M. A., Caporale, M., Buabthong, P., Ross, Z. E., Asimaki, D., and Meier, M. Data-Driven Synthesis of Broadband Earthquake Ground Motions Using Artificial Intelligence. Bulletin of the Seismological Society of America, 112(4):197--1999)) 04 2022. ISSN 0037-1106 doi: 10. 1858010011222 URL https: httpp:://doi.rroo org/ 10.
Acknowledgements
The authors would like to thank Prof. Seongryong Kim and Dr. Jongwon Han of Seismology Lab, Korea University for their professional opinions and feedback on the seismo- logical aspects of this work. The authors are supported in part by the Korea Meteorological Administration Research and Development Program under Grant KM12021-01112 DSS2202--KM211122
Geller, R. J. Scaling relations for earthquake source param- eters and magnitudes. Bulletin of the Seismological Soci- ety of America, 66(5):1511 10 1976. ISSN 0037- 1106. doi: 10. 10.1188/BSSA006605111550 / //doi. /00 10 1 17788///SSS66000000
URL https:
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Broadband Ground Motion Synthesis by Diffusion Model with Minimal Condition
A. Dataset Construction
We used three datasets (SCEDC(SCEDC, 2013), KMA et al., 2023) and INSTANCE(Michelini al., 2021)) from different regions. In this section, we explain how each dataset was constructed. All datasets are collected from corresponding APIs and processed to have 60-seconds duration and applied 1 45Hz bandpass filter.
We split each dataset into training dataset and test dataset, according to the earthquake event, to evaluate the fidelity of generated waveform for the earthquake which is unseen during the training.
dataset
SCEDC
KMA
INSTANCE
Features
Train
Test
Train
Test
Train
Test
#observations
71,488
17,878
237,755
58,925
72,904
19,872
#source event
2,098
525
2,052
514
2,265
593
#station
149
149
134
134
578
534
average #station per events
34.07
34.05
115.87
114.64
24.43
25.29
average magnitude
2.45
2.45
1,45
1,45
3.36
3.36
average epicentral distance
125.25
126.71
235.48
234.22
57.82
57.79
average focus depth
8.51
8.65
11.52
11.73
12.47
11.97
Table 4. Features of each dataset
A.1. SCEDC
We exploit earthquake catalog of SCEDC (SCEDC, 2013) provided by SeisBench (Woollam et al., 2022). We selected waveforms with a sampling rate of 100Hz that included 60 seconds from the earthquake and applied bandpass filter in the 1 ~ 45Hz range to construct our data. Unfortunately, the Seishench-provided dataset had fewer than 13 stations per earthquake events on average, therefore we utilized Obspy APl(Beyreuther et al., 2010) to collect additional observations on more stations in the station list of (Uhrhammer et al., 2011) for each earthquake. Using earthquakes from the catalog during the years 2016 to 2019, we constructed new dataset with approximately 34 stations per source. The Table 4 shows the count of datasets we used.
The V/s0 information was sourced from (McPhillips et al., 2020) and used only during the GMPE analysis, not during the training or model inference processes. The average value was used multiple Vs30 values were present for a single station code. For station codes without Vss00 data, 760m/s was assigned to negate the influence of Vswo during GMPE analysis.
A.2. KMA
KMA data source consist of continuous waveform data were employed, which are operated by KMA (Korea Meteorological Administration) and KIGAM (Korea Institute of Geoscience and Mineral Resources). We exploit the dataset appear in (Han et al., 2023) which constructed from earthquake catalog provided by KMA, spanning from 2016-2020, and used subset consist of observations from broadband sensors. Similarly to SCEDC, the waveforms have a sampling rate of 100hz, duration of 60 seconds, and frequency 45Hz.
A.3. INSTANCE
We used the Seisbench Seishench-provided version INSTANCE dataset and created subset by selecting only the traces satisfying:
1. includes records for 60 seconds from the earthquake occurrence time
2. local magnitude larger than 3.0.
3. P-arrival time is included in the metadata to ensure that the earthquake signal is observed.
For the EQT evaluation in Table 1, we excluded waveforms which include multiple event signals, which are out of our scope.
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Broadband Ground Motion Synthesis by Diffusion Model with Minimal Condition
B. Implementation details
We implement the proposed model with following implementation details.
During the training, W is fixed for specific earthquake source ID, and Wige sampled from earthquakes with the same source ID. Among these, if metadata contained P/S phase labels, samples are randomly selected from those with labels. If P/S phase labels are absent, samples are chosen randomly without considering P/S phase labels. And also we conduct preprocessing of seismic data.
We implement using single NVIDIA-RTX A6000 with 48GB memory. For training, we set the number of epochs to 500 and the training batch size to 4. To enhance training efficiency, we apply an accumulation step 4, resulting in an effective batch size of 16. For the loss, we set the maximum diffusion steps to T 1000 and SNR weight 5. We minimize the loss by AdamW optimizer with learning rate 10-3 5 and pytorch.optim defaults. During the training, we applied learning rate decaying technique with linear scheduler. The total duration of training is approximately 65 hours.
B.1. Neural Network Architecture
We utilize the U-Net backbone with cross-attention architecture similar to (Rombach al., 2022; Ghosal al., 2023), to represent me, with modification in the domain-specific encoder Ta to map Ctgt to hidden feature To(Cige). For the implementation, we construct TT by 5-layer FFN model. The encoded conditional vector Ta (C(ge) will be provided value and key of cross attention module Att K, V) while U-Net feature is provided as query Q.
For AE and DAE, we take same architectures from VAE of (Esser et al., 2020) and give a modification on DAE- With the vanilla module DAE, we find that the proposed model is not effective in accurately predicting the amplitude of the output waveform. Therefore, we propose to attach an additional module ACM after AE e to predict the amplitude correction feature and multiply it to the predicted spectrogram. In detail, we utilize the encoder, TSConformer blocks and Magnitude mask decoder module from MP-SeNet (Lu et al., 2023) and provide output of DAE and auxiliary phase spectrogram induced by GriffinL.im algorithm to correct the amplitude and enhance the quality of generation. Improving the original implementation (Lu et al., 2023) that allows only reducing the output, we add four TSConformer blocks and replace the final sigmoid activation function with Softplus function to provide the ability to increase as well.
C. Pre-processing Recipe
C.1. Conditional Vector Pre-processing
We explain the process of Eggt constuction. Recall the variables that we are used synthesize waveform are:
1. Slat, Ston latitude and longitude of the station to observe the waveform data.
2. Clat, Clon latitude and longitude of epicenter.
3. Cdep depth the hypocenter, unit of kilometers.
4. My magnitude of the earthquake.
We preprocessed those variables to construct an I I-ddimensional condition vector and later provide it to our condition encoder module TØ.
First of all, we encode locational information Sict, Slon; Clat and Cton with the following process:
1. Normalize the values to get s/aar ston état and eton with following:
Stat 4at
Clat - Matt. Slon
Ston Lion and éion 4oo
Clon lion
Sus
élat
(10)
Wiat
liat"
Wat
liat"
uto
Wion lion
where (Letts Utot) and (hin Uto) represent the lower and upper bounds of latitude and longitude, respectively, for the region of interest.
In our datasets, we summarize those bounds in Table 5.
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Broadband Ground Motion Synthesis by Diffusion Model with Minimal Condition
Dataset (region)
hat
Mtat
Iton
SCEDC (Southern California)
32.0
37.9
-121.0
-114.1
KR (South Korea)
33.12
38.60
124.64
131.87
INSTANCE (Italy)
35.00
48.03
5.32
20.01
Table 5. upper and lower bounds of the region interest
2. Motivated from polar coordinate transformation(Mohindee S. Grewal, 2007), which is commonly used in GPS field, we further encode normalized coordinate to following:
(cos (aat)cos((ssn))) siniion)) (os())) sin((soon)
Csta
(11)
Cepi (oossacssssssssolsl sin(e((t)cos((ee sircle(on))
Secondly, we compute the back azimuth angle Asi and encode by
Cazi
(cos(A2)) (iz))
(12)
Lastly, we compute and normalized epicentral distance Repis focus depth d, and magnitude My. Each are normalized by following formula:
SCEDC
KMA
INSTANCE
Reps
(Repi 125.542401)/55.810000
(Repi 219.91) 1119.99
(Repi 077.8158)/31.7445 (d, 12.3680) 13.24466
dg
(dg 44668866 (Mr 2.0)/6.4
(d, (1.59)/5.40
M2
(Mz (.35)/5.22
(M₂ 3..0/6.5
Concatenating the processed features Cata Cepis Cazi, R' pp d' and MG,, we get an 11-dimensional conditional vector Etgt for our problem, the synthesis of seismic ground motion.
C.2. spectrogram construction
The generation target of out model is spectrogram, which is in time-frequency domain. We report the process of spectrogram construction pre- pre-processing. We employed the STFT (Short-Time Fourier Transform) with a hop length 16. Given that the spectrogram's scale is closely related to the earthquake's 'ss amplitude, we used an nf ft and window length of 128 and applied a logarithmic scale transformation for better scale adjustment. Consequently, the original waveform data of size 3 x 6000 was reshaped into 64 x 376.
D. EQT Training Details
We used EQTransformer (Mousavi et al., 2020) provided by SeisBench (Woollam et al., 2022). Starting from pre-trained model provided by SeisBench, we finetune the model with our dataset, with the same training protocol. After standardizing the waveforms, we trained the model using the Adam optimizer, with a batch size of 512 and a learning rate of 10-3, for 100 epochs. Other hyperparameters the optimizer were set to default. For hyperparameter search, the learning rate ranged from 10-2 10-5, and the performance was best when was 10-3
E. Details on Benchmark Models
E.1. SeismoGen (Wang et al., 2021)
SeismoGen is CGAN-based model that generates waveforms conditioned on the presence of seismic events (e.g., P S waves). The Discriminator takes both the waveform and the presence of seismic events as inputs. It then divides the waveform into high and low frequency components, analyzing each to determine waveform is real or synthetic. SeismoGen used data from three stations in Oklahoma: V34A, V35A, and V36A, while we used data from 149 stations from SCEDC. Our synthesis approach used station and earthquake information instead of presence of seismic events. SeismoGen generated
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Broadband Ground Motion Synthesis by Diffusion Model with Minimal Condition
waveforms as 40 seconds at 40Hz, but we aimed for 60 seconds 100Hz. We used an input noise length of 1 1500 and added upsampling at the end of the first convolution layer. The basic training used noise input, and for comparison with HEGGS, we also trained using waveform. When using waveforms, we modified each pipeline to utilize one ENZ channel. The hyper-parameters we used included the Generator learning rate and Discriminator learning rate are set to 10 and 10-6, using the RMSprop optimizer over 3000 epochs. The À is set to 10 when using noise and 15 when using the input waveform. We saved the best model based on envelope correlation. We experimented with learning rates ranging from 10 to 10-7, using both Adam and RMSprop optimizers. The value of À was tested at 5, 10, and 15. The best-performing combination of these parameters was selected for the final model. Additionally, the results reported in Table 2 reflect the best performance achieved across 30 iterations. Addressing the instability of the original method, we added the L.1 loss Equation (13) from pixopix(Isola et al., 2017) an additional loss term to improve training stability.
E.2. ConSeisGen (Li et al., 2024)
ConSeisGen is ACGAN-based model that generates waveforms conditioned on the epicentral distance. The Discriminator consists of two components: Dp, which learn determining whether the waveform is real synthetic, and Do. which learn regression estimating the distance between the epicenter and the station. While ConSeisGen generated waveforms with 3 channels and length of 4096, we aimed to generate waveforms with 3 channels and a length of 6000. We modified the first linear layer and removed upsampling in the final layer. ConSeisGen used KiK-net data, which began recording shortly before the arrival of the P-wave. However, the SCEDC data utilized in this model was recorded from the onset of the earthquake for a duration of 60 seconds. ConSeisGen generates waveforms based on the epicentral distance. However, waveforms can vary even at the same distance due to factors like magnitude and geological conditions. To generate waveforms for specific locations, utilized minimal additional condition such as station data and source data along with the epicentral distance. The hyper-parameters we used included the Generator learning rate and Discriminator learning rate are set to 2 x 10-1 and 10 10-5, using the Adam optimizer over 5000 epochs. Referring eq.4 of (Li et al., 2024), the loss function consists of Adversarial Loss, Regression Loss(Lreg), and Diversity Improvement Las)( The Lreg computes the loss between Do's 's output and the condition vector, with the Areg set to 1. The Las aims to prevent mode collapse by maximizing the distance between feature maps, with Xas set to 10 when using noise and 5 when using waveforms. We experimented with learning rates ranging from 10-4 to 10-6, using both Adam and RMSprop optimizers. The value of Aa; was tested at 5, 10, and 15, while Areg was fixed at 1. The best best-performing combination of these parameters was selected for the final model. Additionally, the results reported in Table 2 reflect the best performance achieved across 30 iterations. Addressing the instability of the original method, we added the L1 loss Equation (13) from pix2pix(Isola et al., 2017) as an additional loss term to improve training stability.
(G) -rrgg
[lKeae ((
(13)
E.3. BBGAN (Florez et al., 2022)
BBGAN is a conditional generative model within the Wasserstein GAN framework. The original conditions of BBGAN are Vss0. earthquake magnitude, and epicentral distance. We modified conditional vector to ours, add conditional vector encoder Te to both generator and discriminator, modified the last upsample layer of generator to have scale factor 3 (original: 2), and lastly increased the number of hidden features of last convolution block of discriminator, corresponding to our waveform shape (3, 6000). Those changes allows the model to generate (3, 6000) shape waveform from the provided conditional vector. To further improve the performance, we replaced all relu activations of generator and leaky relu activations of discriminator to gelu activation. Additionally, while the original BBGAN paper utilized data from Japanese networks K-NET and KiK-net with earthquake magnitudes larger than 4.5, our approach employed data from the SCEDC (SCEDC, 2013) with earthquake magnitude larger than 2.0 for training. In the training process, we set 500 training epoch and batch size 32, and Adam optimizer with learning rate 5 X 10-7 and B = (0.9, 0.999). Also the final loss function is composed of adversarial loss, LI reconstruction loss, and a KL divergence term. The L1 regularization term was set to 25, and the KL regularization term was set to 0.01. For evaluation during the validation loop, envelope correlation was used as the performance metric. During the training, the linear learning rate decay technique was applied.
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Broadband Ground Motion Synthesis by Diffusion Model with Minimal Condition
E.4. LDM (Rombach et al., 2022)
E.4.1. VAE (ESSER ET AL., 2020) PRETRAINING
Due to lack of pretrained weights of VAE trained on seismic spectrogram, we first need train VAE encode X and X to latent vector Z and Z
Employing equation (25) of (Rombach al., 2022), we set the loss function for VAE training is:
Lady Litotal Pore (xE(),))))))() (La((s)))))Lataaaaaaaaaaaa (DAnE((A((E))) (o0
logDx(x) + AMKL]
(14)
where dgz is low weighted Kullhack-L.blle regularization term by factor 10-6.
Unfortunately, the VAE training on our spectrogram diverged, due difficulty on magnitude processing. Therefore, we apply standardization on spectrogram to relax the problem. And the latent space size is 64 16 94.
We report reconstruction performance of the Auto-encoder model using the proposed metrics. The reconstruction performance results as follow in Table 6.
Table 6. Reconstruction result
waveform
spectrogram
Model
P_MAE (s)
S_MAE (8)
envelope corr
SNR
PSNR
MSE
VAE
0.5155
0.7066
0.7567
-2.9984
25.1800
0.2459
E.4.2. LDM (ROMBACH ET AL., 2022)
We train LDM using the pretrained VAE Appendix E.4.1 and DDPM(Ho et al., 2020) scheduler. Additionally, the overall model architecture is adapted and modified base on the TANGO (Ghosal et al., 2023) model and code. But, while TANGO models incorporate text-encoded conditions through Large Language Model, the seismic data does not exist text conditions. Therefore, we employ our preprocessed conditions and apply our conditional vector encoder T9 for training. During model training, the learning target set the samples from the DDPM scheduler. Training conduct using two methods and training losses.
Equation (15): not utilizing the characteristic of paired data
Equation (16): utilizing the characteristic of paired data
We set the hyperparameters for the AdamW optimizer as follows: an initial learning rate 10-5 and 3 = (0.9, 0.999), and a weight decay of 10 2 and adam epsilon 10-8, 8 Also, we apply the learning rate decaying technique with the linear scheduler. The training batch size is set with an accumulation step 4, resulting in total effective batch size of 16. The model trained for 500 epochs. The results indicate that training with paired data outperforms training without paired data.
LLDM = e, ttst xx xx(2 Cegt, t)ll
(15)
LLDM
Eizore 21xt de mg( Eegt f)ll
(16)
F. GMPE formula
In this section, we express the PGA formula for GMPE analysis. Given waveform W, we obtain local magnitude M2 first, and compute the PGA value later.
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Broadband Ground Motion Synthesis by Diffusion Model with Minimal Condition
E.I. SCEDCISCEDC, 2013)
Given the waveform W, the local magnitude Mr of SCEDC can be computed by using the following formula (equations I to 6 of (Uhrhammer et al., 2011))
Mr = logA Aog (Rhypo) where
6 Σ TP(n) x T(n,2) 2).
(17)
Ao (Rhypo) = 1. 1.11log RAypo + 0.00189 x Riyyo + 0.591 +
n=l
where A is amplitude of W and (r()) is attenuation function of southern california region. The station adjustment term was not applied due to a lack of values for recently installed stations.
The TP(1) coefficients are
TP(1) = +0.056,
TP(2) = -0.031,
TP(3) 0.053,
TP(4) +0.080,
(18)
TP(5) -00228,
(P(6) +0.015,
When z is
<(r) 1.11366 log(r) 2.00574,
(19)
< r S 500 to - -1 7,, =) is the Chebyshev polynomial
T(n, 2) cosin arccos(2)].
(20)
After determining the local magnitude Mr. we obtain the PGA value by equation I of (Boore et al., 2014), with pynga (Wang, 2012) implementation. Since HEGGS doesn't exploit the focal mechanism information, we set mech and rake to be which represents unspecified.
F.2. KMA(Emolo et al., 2015)
For the KMA dataset, the local magnitude M,, can be computed by the following equation (equations and of (Sheen al., 2018)):
Mt Iog - logAA S
(21)
Ao 0.5869log( Repi 100) + 0.001680(Repi 100) +
where A is the peak amplitude of the Wood-Anderson simulated waveform and S is station-wise correction and Repi epicentral distance in kilometers.
After determining local magnitude My we obtain PGA value Y by (Emolo et al., 2015) with following formula for South Korea peninsula:
logY = 3.16 + 0.75ML
[VRa R2 + 3.72
(22)
0.72log
-0.0034
F.3. INSTANCE(Lanzano et al., 2019a;b)
the local magnitude My on the INSTANCE dataset can be computed by the following equation (equation I to 14 of (Di Bona, 2016)):
Mr log log Ao
(23)
log.A + 1.749log( RAypo/ B0y) 100) 0.0016(Rnype) 100) 2.9445 C
where A is the peak amplitude of the Wood-Anderson simulated waveform and the station-wif correction term.
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Broadband Ground Motion Synthesis by Diffusion Model with Minimal Condition
After determining the local magnitude ML, we obtain the PGA value by equations to 5 of (Lanzano et al., 2019b) and 8 of (Lanzano al., 2019a). In these equations, Moment Magnitude(Mwy) 2016) proposed formula that satisfies Mr .Mw average. Therefore, this study used Mr instead of Mw. Also, HEGGS doesn't I exploit the focal mechanism information, we set the style of faulting SOF to O, representing the normal fault type.
used compute PGA. However, (Di Bona,
G. Qualitative Analysis on Ablation Models
This section is dedicated to the qualitative analysis of the SCEDC of the models mentioned in Table 3. The figure compares the Real observation (Real), HEGGS(w/ ACM), end-to-end train (w/o ACM), and LDM + paired data. The human- labeled P/S arrival times of the earthquake are indicated by orange and black lines, while the P/S arrival times detected by EQT for each waveform are shown in red and blue.
G.1. Positive samples
We first list the positive samples, which are the results that all models generated realistic waveforms with accurate phase arrivals.
Real
N
Z
33
50
10
33
40
50
50
30
27
43
50
Ours
w/o ACM
40
LDM paired data
ST.P
Figure Positive synthesis results of our model and ablation models, compared to real observation.
G.2. Negative Samples
also include the results of the synthesis that at least one of the models failed to generate realistic and accurate waveforms.
Real
Z
Time (s)
Ours
33
50
33
40
50
w/o ACM
LDM paired data
ST.P
GT.5
P.nkk
5.olck
Figure 9, Negative synthesis results of our model and ablation models, compared to the real observation.
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Broadband Ground Motion Synthesis by Diffusion Model with Minimal Condition
H. Additional Figures: Waveform and Spectrogram
This section presents the waveforms and spectrograms shown Figure and Figure 4. The seismic data we used consist of 3-components, ENZ. Each pair displays the same waveform and spectrogram, with the top representing the real observation and the bottom representing the synthetic generated HEGGS. The red and blue lines on the waveforms indicate the P/S arrival times.
H.I. SCEDC
SCEDC
Synthetic
waveform
Real
2506
2 550
5500
Synthetic
2 cove
(b) spectrogram
Figure 10. Synthesis results of our model compared to the real observation.
19

---

Broadband Ground Motion Synthesis by Diffusion Model with Minimal Condition
H.2. KMA
KMA
Z
Real
Synthett
aa) waveform
Roat
134
340
Synthbttcc
T5mal 1a]
Time Oe)
Time (a)
(b) spectrogram
Figure 11. Synthesis results of our model compared to the real observation.
H.3. INSTANCE
INSTANCE
Synthetid
a waveform
Real
7000
2 550g
5500
Synthetic
2 core
100
Time
Timrs
(b) spectrogram
Figure 12, Synthesis results of our model compared to the real observation.
20

---

Broadband Ground Motion Synthesis by Diffusion Model with Minimal Condition
I. Additional figures: Section plot
The section plot is constructed by following process. Initially, a specific earthquake event chosen, and input data randomly selected (indicated by the blue line). We virtual stations established equidistant intervals from the epicenter, generate waveforms, and plot together with real observations. The red lines represent ground truth observations and black lines the synthesized waveforms. Note that the azimuth angle of observations varies, while the synthetic stations set to have same values. This potentially affect the P/S wave arrivals and lead to mismatch in visualization, but the effect is not considered to be significantly large.
I.1. SCEDC
Ract
10.0
-0.5
08.5
1203
1930
6.30
8.50
10..0
1230
1800
45.0
60.00
40.0
38520
(a) Section plot axis
(b) Section plot N axis
(c) Section plot Z axis
Figure 13. Section plot on synthetic stations.
23
PG.
Syrthat
Symthert
Defacdaro
0rfe1
(a) Section plot on axis
(b) Section plot on N axis
(c) Section plot on Z. axis
Figure Section plot on synthetic stations.
1.2. KMA
20:222
45 1.222
30,
20
Offeet 250
Offer
(a) Section plot on axis
(b) Section plot on N axis
(c) Section plot on Z axis
Figure Section plot on synthetic stations.
21

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Broadband Ground Motion Synthesis by Diffusion Model with Minimal Condition
091t0 (60)
12304 prnn
(a) Section plot on axis
(b) Section plot on axis
(c) Section plot on Z axis
Figure 16. Section plot on synthetic stations.
L33. INSTANCE
30
21
43.0
00.
188.
40.0
08.3
100.3
1200
3400
20.0
62.5
105.3
1800
(a) Section plot on axis
(b) Section plot on axis
(c) Section plot on Z axis
Figure 17. Section plot on synthetic stations.
23
53
120.
40.2
125.0
13.3
100.0
12800
(a) Section plot E axis
(b) Section plot on N axis
(c) Section plot on Z axis
Figure 18. Section plot on synthetic stations.
22

---

Broadband Ground Motion Synthesis by Diffusion Model with Minimal Condition
J. On pair-Exploiting Diffusion Model
In this appendix, we provide more detailed explanations about the training and inference HEGGS for the clarification.
J.1. Remark:Prediction targets of diffusion model
Referring equation (2).(4)(6) and (7) of (Ho et al., 2020), the forward process g(11::3 X) would be given by
3, -N N(Vosso, Tz)
(24)
and thus we have
It Vārao + vI- Tre where (O,
(25)
The backward process q(xx-1/0t, 2o) would be
(01-11 x2) ((( B.I)
(26)
where
xo), 1 20 +
var(1 a(-1) 1 Cte
24 and B:
1 04-- 8r.
(27)
In implementation. it is required to find (=,, term in Equation (27). There are several methods for the prediction, with replacing zo by estimate ny (2t, t). The HEGGS directly predicts æ in sample space, as it would be more natural since we want to learn the morphology between paired data, compared to the alternatives which predicts the noise c (Ho et al., 2020) or v-predictionnSallmann & Ho, 2022).
Algorithm 2 Generation
Input: Diffusion steps T.condition vector Tgtt source waveform W (optional) if Ware is given then
Algorithm 1 HEGGS training
Input: Seismic dataset D, diffusion steps T repeat
convert Were to spectrogram Xarc
= EA (x)(X
(Ware, Weat Cagx) IB
else
convert (Warre Wratt) to (tor)
sample 2T ~ N(0,1) end if
t Uni "nnffrrm(1...... .... T) c~ (0.1)
for T T..... do sample N'00 1)
Take gradient descent step on
V x DA ( (ma (ztTC Tegt, t))l
compute mø(2 &gt:t) t)
where 24 A (X 0r)e
+VI Tiec
compute 2--1 = T4e (2t, 2) + v Bez (Eq. 27)
until converged
end for Xtet DAB(20)
Convert X to waveform Wigt
Return: W
J.2. Training with pairs
As described in Section 3. we consider the paired data (x Xtst) with corresponding condition vector Eyre and Etgt- Note that Eeve is not in use.
Since X and gtt are the observations of same earthquake, we make assumption that there exist morphology using Cegt, random variable. We formulate this assumption with
which maps the latent æere of X at time i, to x lgt
Equation (1), as follows:
na are Tege, 1) ~ q(> (
(1)
23

---

Broadband Ground Motion Synthesis by Diffusion Model with Minimal Condition
This assumption includes the intuition that the broadband waveform signal is combination of earthquake information, which is considered to be included in X and local geological features near observatory, encoded by positional information from Crgt.
For training, we aim to train the neural network mg which is composition of and denoising model Xg. Precisely, mg would be written by
ma (2, 2, (,),(2,,,1),,,,)
(28)
Since n (Cgg,, xt we have ma Cegt: = (x(x (a Ergt, 1) for the paired latents (2f at 2 the loss function of diffusion model Equation (2) would be equivalent Equation (3):
DM
E (x Xtot dege), el ma(xt"e ,eggs t)l
(29)
After that, we consider same procedure in latent space (the =t for the clarification) with autoencoder consist of the encoder EaR and decoder ID DAE, we obtain the loss function Equation (9), with end-to-end training.
Lours Exx Xts or DA (mo (Dam(mo(> Catt, t)
(9)
In Algorithm 1, present training algorithm for the HEGGS training with Cours. The paired waveforms and corresponding condition vector of target waveform would be sampled from the dataset, and the gradient descent would update all modules ma, EAB and DAE together.
Remark J.1. For the training process of diffusion model with Equation (9), several details below are considered for the loss and model design.
1. During the training. the noise is designed to be added the z instead of Z This would provide robustness against site-specific noise which already included in observation W and its latent vector Zerc
2. When t is small, 2/-h would be almost same Zere (this is also because X itself is already noisy) and thus the model would learn the transformation ry with more attention.
3. Regarding the intuition that 20 and 2, will be identified (in distribution) when is sufficiently large, the training loss Equation (9) would be equivalent to the conventional training loss for Xy training when we disregard the end-to-end training. Hence, the model learns to generate from the noise w/o Warre too, during the training.
4. Since 1) and my does not take Egre as input. Therefore the model learns to extract common information from z:re through multiple pairs of observations of same earthquake during training, regardless the local information (encoded by location) of observatory. This makes the model can handle z( as input too, since it shares the information of earthquake.
J.3. Inference w/o W
Although the diffusion model is trained with paired data and takes Waree as an input, our model is capable to synthesize seismic waveform without the observation Wasre
Since is defined to map the source latent zero to target latent 2,tt, it also maps the target latent itself, in distribution. Precisely, we can write
lgt Tege, 1) zt
lge
(30)
and thus the output of neural network would be
m= 1gt -Tege t) = (0(n tor Teges t), Tege, = xo(2 Teges t)
(31)
Therefore, we can use conventional reverse process
2 tgt
(1 tgt mo
1ot Eege, t)) + O N
(32)
even if Agt is the gaussian noise sampled from (0,
In Algorithm 2, we summarize the generation process of our model. Note that the diffusion steps are equivalent to L.DMRombach et al., 2022) when Ware is not given.
24


Reviews:
[{"review_id": "ukl2begp4k", "reviewer": "Reviewer_agtf", "summary": "They tackle the task of generating earthquake-caused ground motion waveform (something I know nothing about). The propose a end-to-end generation method", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely appreciate the reviewer’s time and effort in evaluating our work. We acknowledge that the reviewer may not have direct expertise in seismology, and we appreciate their openness about this. Nevertheless, we are grateful that they found our use of diffusion models in this context interesting and that they recognized the validity of our experimental results.\n\nSeismic waveform synthesis plays a critical role in earthquake engineering, hazard assessment, and seismological research. Accurate synthetic waveforms enable better modeling of earthquake effects in regions with limited observational data, improving preparedness and mitigation strategies. Traditional approaches rely on complex physical simulations or empirical models that require extensive geological and seismological information.\n\nOur work introduces HEGGS, a data-driven generative model that synthesizes high-fidelity seismic waveforms with minimal conditioning information, making it more practical and scalable than existing methods. As our experimental results demonstrate, HEGGS achieves state-of-the-art fidelity both in ML-metrics and key seismological metrics, including phase arrival accuracy and GMPE consistency, further validating its potential impact.\n\nWe collaborate with seismologists to design, model, and validate our research with in-field professional opinions. We hope all readers of this work gain knowledge and insight on high-fidelity time series generation as well as earthquake research revolutionized by machine learning.\n\nThank you again for your review and consideration.", "claims_and_evidence": "- novel method: HEGGS (I don't have the expertise to say)\n- HEGGS demonstrate its superior performance using earthquakes from North American, East Asian, and Europe (accurate based on the results)\n- HEGGS superior against benchmark models in seismology-inspired metrics such as GMPE analysis and phase arrival prediction (accurate based on the results)", "methods_and_evaluation_criteria": "I know nothing about seismology, but the diffusion process in spectrogram space seems reasonable. Diffusion models are quite good at generating high quality and diverse data.", "theoretical_claims": "All theory relates to seismology which I know nothing about.", "experimental_designs_or_analyses": "They seems good and diverse, but again I cannot assess the quality of the chosen metrics and dataset given my lack of expertise.", "relation_to_broader_scientific_literature": "I don't know the literature on seismology.", "essential_references_not_discussed": "I don't know the literature on seismology.", "other_strengths_and_weaknesses": "Interesting usecase.", "other_comments_or_suggestions": "My recommendation doesn't mean much, I'll defer it to other experts reviewers, I'll change it later. Meta-reviewer, please take my review with a grain of salt because I don't know anything about the subject.", "questions_for_authors": ".", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "7GjBNJkrJT", "reviewer": "Reviewer_BpqJ", "summary": "This paper presents a diffusion model for synthesizing earthquake ground motion data at a specific location with minimal conditioning information. Unlike previous approaches that rely on extensive geological and seismological data, the proposed method leverages the inherent information embedded in ground motion signals observed at a given location. The authors recognize that these signals inherently encode details about both the target location and the earthquake source. Based on this insight, they formulate a conditional diffusion model that generates ground motion signals for the target location using meta-information about the target location and the observed signal at the source. Experimental results demonstrate that the proposed model achieves state-of-the-art fidelity, both quantitatively and qualitatively, across various evaluation criteria.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely appreciate the reviewer’s thoughtful feedback. Below, we address the main concerns raised.\n\n> the scale of the metrics can vary significantly depending on the magnitude\n\nIn implementation, each metrics includes the waveform normalization process in default. Here are the details:\n\n- P/S phase arrival: We use the EQTransformer (EQT) model (Mousavi et al., 2020), which inherently includes normalization in its preprocessing steps. \n- Envelope correlation: We utilize the ObsPy implementation (https://docs.obspy.org/packages/autogen/obspy.signal.cross_correlation.correlate.html), which applies normalization by default.\n- SNR and PSNR: Both metrics are inherently scale-invariant.\n- Spectrogram mse: Our comparisons are performed at the log-spectrogram level, where normalization is applied prior to computing differences. \n\nGiven these normalization strategies, our reported improvements remain meaningful and are not artificially amplified by variations in signal magnitude.\n\n> potentially making the improvement marginal in practice\n\nBeyond metric scaling, we emphasize that our improvements are practically significant in seismological applications. In earthquake signal analysis, two of the most critical factors are arrival time accuracy and amplitude distribution:\n\n - P/S arrival time accuracy: In literature of P/S phase picking problems, even human expert-labeled data exhibit unavoidable errors, with deviations under 0.5 seconds typically considered acceptable. HEGGS is the first deep learning synthesis approach to achieve this level of accuracy, making it the most precise generative model for earthquake waveform synthesis to date.\n\n- Amplitude Distribution: Due to unknown geological features and interference effects, real-world amplitude variations can differ by a factor of up to 10, even for well-calibrated models. As demonstrated in our GMPE analysis, the synthesized waveforms from HEGGS align closely with real observations, further validating the fidelity of our approach.\n\nThese aspects reinforce that our results are not only statistically significant but also practically valuable for seismic applications.\n\n> Could the authors provide additional qualitative comparison results, including more examples from the baseline methods?\n\nWe acknowledge the reviewer’s request for more qualitative comparisons. Due to manuscript file size limitations (including the appendix), we had to limit the number of visual examples included. \nHowever, we are more than happy to provide additional qualitative comparisons. To comply with the double-blind policy, we release an [anonymous GitHub repository](https://anonymous.4open.science/r/Seismic-waveforms-4630) containing supplementary qualitative results, allowing for more in-depth comparisons against baseline methods. \n\nPlease be patient: loading may take time, as the figures are fairly large in size due to its resolution. We recommend to download whole repository and seek.", "claims_and_evidence": "This reviewer is not deeply familiar with the literature on ground motion synthesis. However, the formulation of the problem using minimal conditioning information (e.g., meta-information about the target location and the observed signal from the source) appears reasonable and well-motivated. No apparent issues were identified with the claims presented in the paper.", "methods_and_evaluation_criteria": "I may not be the best candidate to assess this aspect in depth. However, after reviewing several recent works cited in the paper, I found the evaluation criteria to be consistent with existing approaches and saw no obvious issues.", "theoretical_claims": "Please refer to the “Claims And Evidence” section.", "experimental_designs_or_analyses": "The experimental designs appear valid overall. However, one minor concern is that while most of the comparison results show a notable gap between the proposed method and the baselines, the scale of the metrics can vary significantly depending on the magnitude of the ground truth, probably except for PSNR. This suggests that, in certain scenarios, the observed gap may be less meaningful than it appears, potentially making the improvement marginal in practice. This suggests that, in certain scenarios, the observed gap may be less meaningful than it appears, potentially making the improvement marginal in practice. In this regard, incorporating more qualitative comparisons could provide stronger support for the validity of the proposed method.", "relation_to_broader_scientific_literature": "Diffusion models have become a widely adopted framework for generative modeling across various domains. This work extends their application to the domain of ground motion synthesis, exploring their potential in this context. By leveraging the strengths of diffusion models, the paper contributes to the growing body of research that applies generative techniques to scientific and engineering problems, particularly in seismology.", "essential_references_not_discussed": "I’m not aware of such references.", "other_strengths_and_weaknesses": "Seismological data analysis is both an interesting and crucial task. I agree that generative modeling of such data has the potential to contribute to disaster preparedness and mitigation, ultimately helping to reduce loss of life.", "other_comments_or_suggestions": "Could the authors provide additional qualitative comparison results, including more examples from the baseline methods?", "questions_for_authors": "I believe the previous discussions have already sufficiently addressed my concern.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "8vca1EQg00", "reviewer": "Reviewer_UGgN", "summary": "This paper proposes a High-fidelity Earthquake Groundmotion Generation System (HEGGS), which is a new solution for generating realistic earthquake-induced ground motion waveforms. The major contributions contain: 1) a new benchmark using openly available seismic datasets, with paired observed waveforms of the source earthquake and time-aligned to the earthquake origin time; 2) a new seismic waveform generation model based on LDM that requires a bare-minimum set of conditional; 3) end-to-end training architecture with supervision on spectrograms and an ACM module to improve the performance; 4) sufficient experiments for earthquake waveform generation and ablation study.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the detailed and constructive feedback. Below, we address the key concerns raised.\n\n> Motivation for using a diffusion model\n\nIn seismic synthesis, high fidelity is most important and desirable. However, waveform in high frequency (>1Hz) is very hard to generate especially without local geological features. Prior works, mostly GAN-based, failed to achieve high fidelity  and was very hard to train on private data due to extremely unstable training.\nConversely, diffusion models have demonstrated superior stability and high-fidelity synthesis in vision-based conditional generation, attributes that we leverage for seismic applications.\nUnfortunately, vanilla Latent Diffusion Models (LDMs) alone were insufficient for our task; our methodological innovations were crucial in making them effective for seismic waveform generation.\n\n\n> Clarification of diffusion process in HEGGS\n\nFor better readability, we provide a rendered PDF containing algorithm blocks in [Anonymous Github](https://anonymous.4open.science/r/Seismic-waveforms-4630). The two-page PDF, showcasing what we will add to the appendix to address your feedback, also contains some motivations and considerations summarized as a remark. We provide a highlight of the PDF to address concerns you raised as follows:\n\n- HEGGS training adds noise to $Z^{src}$ rather than $Z^{tgt}$, since the real observation $W^{src}$ itself is very noisy and we wanted to provide robustness against this noise. Note that this noise is clearly differs from the gaussian noise since it often caused by human activity.\n- Also, we emphasize that our neural network $m_\\theta$ is a composition of $\\eta$ and $x_\\theta$. Since we defined $\\eta$ as a transformation which maps latent $z_t^{src}$, not $Z^{src}$, to target latent $z_t^{tgt}$, adding noise to $Z^{src}$ is required for the theoretical consistency.\n- For the generation scenario without $W^{src}$, we set $z_T^{tgt}$ as a Gaussian noise as conventional diffusion models do. Due to property of $\\eta$ in Eq. (6), $m_\\theta$ becomes equal to $x_\\theta$ and hence the trained model would work equally to the conventional diffusion model's reverse process. We also provide pseudocode of generation in Latex form:\n\n```latex\n\\usepackage{algorithm,algorithmic}\n\\begin{algorithm}[H]\n   \\caption{Generation}\n   \\label{alg:inference}\n\\begin{algorithmic}\n   \\STATE {\\bfseries Input:} Diffusion steps $T$,condition vector $\\vec{c}_{tgt}$, source waveform $W^{src}$ (optional)\n   \\IF{$W^{src}$ is given} \n   \\STATE convert $W^{src}$ to spectrogram $X^{src}$\n   \\STATE $z_T=\\mathcal{E}_{AE}(X^{src})$\n   \\ELSE\n     \\STATE sample $z_T\\sim\\mathcal{N}(0,1)$\n    \\ENDIF\n   \\FOR{$t=T,\\cdots,1$}\n   \\STATE sample $\\mathbf{z}\\sim\\mathcal{N}(0,1)$\n   \\STATE compute $\\tilde{z} = \\mathbf{m}_\\theta(z_t,\\vec{c}_{tgt},t)$\n   \\STATE compute $z_{t-1}=\\tilde{\\mu}_t(z_{t},\\tilde{z})+\\sqrt{\\tilde{\\beta}_t}\\mathbf{z}$ (Eq. \\ref{eqn:diffusion_backward})\n   \\ENDFOR\n   \\STATE $X^{tgt} = \\mathcal{D}_{AE}(z_0)$\n   \\STATE Convert $X^{tgt}$ to waveform $W^{tgt}$\n   \\STATE {\\bfseries Return:}  $W^{tgt}$\n\\end{algorithmic}\n\\end{algorithm}\n```\n\n> Generated images are blurred\n\nThe observed relative blurriness in Figure 4 arises due to both issue on real observations and diffusion model.\nThe nature of real seismic observations often contain local noise, which are hard to synthesize without specialized method obtained from the long-term noise observations.\n\nSeismologically, the most fundamental fidelity of earthquake \"event\" signal regards P/S phase arrival times and magnitude decay pattern over wave propagation distance, which are the most important aspects influencing earthquake location and intensity identification.\n\n> Integrating ACM into VAE pretraining\n\n\n| Model | P_MAE (s) | S_MAE (s)| _env.corr_|\n|--|--|--|--|\n|LDM*| 1.1142 | 1.7294 | 0.6932 |\n|LDM + paired data* | 0.5633 | 0.7808 | 0.7726|\n|LDM + paired data + end-to-end-train| 0.8014| 1.5367| 0.6239|\n|LDM + paired data + end-to-end-train + ACM (HEGGS)| 0.4760| 0.5476| 0.8187|\n|**LDM + ACM**| 1.1131 | 1.6372 | 0.6981|\n|**LDM + paired data+ ACM**| 0.7748 | 0.9402 | 0.7965 |\n\n(*): Results with normalized waveform. \n\nWe conduct additional experiment with differenct ablation recipe, **LDM+ACM** and **LDM + paired data+ ACM** to address your concern.  The results are listed on last two rows of table above, which is extended from the Table 3 of manuscript.\n\nThe advantage of ACM enables the LDM trainable with unnormalized waveforms, and showing much better fidelity compared to prior GAN-based methods, but there is a significant performance gap compared to HEGGS.\n\nThe pair-exploitting training shows significant improvement, but still not better than HEGGS. Possibly, the VAE pretraining fails to capture important features for the correct synthesis.", "claims_and_evidence": "Yes, I think most of the claims in this submission are supported by the experiments.", "methods_and_evaluation_criteria": "I think the proposed methods and evaluation criteria make sense for the earthquake waveform generation problem.", "theoretical_claims": "No proofs for theoretical claims need to be checked as it is an application-driven submission.", "experimental_designs_or_analyses": "This submission provides experiments on the phase arrival times, similarity measures, GMPE analysis, and Qualitative Evaluation to prove the effectiveness of the proposed methods. There are also sufficient ablation studies to prove the performance of each key module.", "relation_to_broader_scientific_literature": "Not related to the broader scientific literature.", "essential_references_not_discussed": "From my point of view, this submission has cited most related works that are essential to understanding the key contributions.", "other_strengths_and_weaknesses": "Some weaknesses: \n1) The motivation for using a diffusion model is not clearly explained. The paper mentions that previous GAN-based models could not generate sufficiently realistic seismic waveforms with minimal conditional information (as they require seismological properties of the source earthquake, such as the focal mechanism or local geological properties of the observation point), but it does not provide a thorough analysis of why the diffusion model is able to address this limitation.\n\n2) The diffusion process presented in this paper differs from traditional diffusion processes and requires further clarification. Based on Figure 1, it appears that the paper adds noise to $z^{src} $, then uses a U-Net to simultaneously perform the latent transform function and the denoising model to predict $z^{tgt}$. This differs from traditional diffusion models, which typically involve a single trajectory. In conventional diffusion models, noise is added to $z^{tgt}$ and then denoised, with $z^{src} $ serving as a condition to guide the denoising process of $z^{tgt}$. I believe the authors need to provide a more detailed explanation of why they add noise to $z^{src}$ and the potential advantages of this approach.\n\n3) There is some ambiguity regarding the inference process in the scenario without $W^{src} $. From the Methods section, it is unclear how $x_T^{\\text{tgt}} $ should be initialized in the absence of $W^{src} $. I would recommend that the authors provide a comprehensive description of the overall training and inference algorithms to clarify this point.", "other_comments_or_suggestions": "1. From Figure 4, it seems that the generated images are blurred compared to the ground truth. Could you please clarify why this happen?\n2. As the ACM module helps to improve the quantitative results a lot, I wonder whether it is possible to add the ACM module to the VAE\npertaining to the LDM so that you may obtain a good pre-trained encoder and a good performance of the classical LDM?", "questions_for_authors": "See above.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This submission addresses the task of generating ground motion waveforms caused by earthquakes using a latent diffusion model. All three reviewers praised the importance and relevance of the task, the effectiveness of the proposed method, and its potential for notable societal impact. The rebuttal successfully addressed the raised questions, and no major concerns were found.

It is worth noting that while all reviewers, including the AC, are familiar with diffusion models, they have limited expertise in the specific domain of seismic data synthesis.

Given the overall positive consensus, the AC recommends accepting this submission.
Decision: Accept (poster)