Spatial Variation of Sea-Level Sea level reconstruction
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1 Spatial Variation of Sea-Level Sea level reconstruction Biao Chang Multimedia Environmental Simulation Laboratory School of Civil and Environmental Engineering Georgia Institute of Technology Advisor: Dr. Mustafa M. Aral April 8, 213
2 Problem definition Recap of last presentation Fuzzy C-Means Non-spherical shapes in the attribute domain Spatial contiguity in the geographic domain Mean silhoutte value number of clusters Subsequent work Semi-empirical modeling Definition of empirical : adj. based on, concerned with, or verifiable by observation or experience rather than theory or pure logic ( Observation shortage Temporal coverage of spatial sea level data: 19 to 21 Tide gauge stations (observational data):
3 Methodology of reconstruction The basic ideas Filter out noise Capture spatial pattern Fill data gaps Review of previous methods Originated from Singular Value Decomposition (SVD)
4 Methodology of reconstruction The basic ideas Filter out noise Capture spatial pattern Fill data gaps Review of previous methods Originated from Singular Value Decomposition (SVD) t: 1-T (temporal) n: 1-N (spatial) Sea level data matrix H: N T
5 Methodology of reconstruction The basic ideas Filter out noise Capture spatial pattern Fill data gaps Review of previous methods Originated from Singular Value Decomposition (SVD) t: 1-T (temporal) n: 1-N (spatial) (Modified from internet resources)
6 Methodology of reconstruction The basic ideas Filter out noise Capture spatial pattern Fill data gaps Review of previous methods Reduced SVD H U' S V T N T N p p p p T U ' N p H U' Α ht ( ) U' α ( t) N T N p p T N p N 1 p 1 Reconstruction: going beyond T At time t, only R observations, data at N-R points need to be reconstructed ( ') ' α ( ') α ( t ') hr t Ur t R p R 1 p 1 p 1 ( ') U' α ( t ') N p N 1 p 1 Alternative names in climate studies: empirical orthogonal functions (EOFs), reduced space optimal interpolation (Smith et al., 1996; Kaplan et al., 2; Church et al., 24) ht
7 Methodology of reconstruction Our method of data reconstruction Why not reduced SVD (Church et al., 24)? different tasks Data gap vs data famine Construction of spatial pattern Uncertainty issue ( ) ' α ( ) ht U t N p N 1 p 1 Ideas in reduced SVD to serve in the development of new methods Certain spatial relationships do not change over time U Magnitudes of major spatial components can be calibrated during reconstruction The basic ideas Filter out noise Capture spatial pattern Fill data gaps ' N p
8 Methodology of reconstruction Our approach to realize the basic ideas Filter out noise clustering and subsequent spatial averaging within clusters Capture spatial pattern artificial neural network (NN) Fill data gaps utilizing global mean sea level and spatial SST data Neural network architecture Starting from the black box perspective 6 SST 1 Latitude (degree) Indian Pacific Atlantic SST 2 H Longitude (degree) H global H 2 SST 3 H 3 Temporal coverage: Temporal coverage:
9 Methodology of reconstruction Neural network architecture Inside the black box Type of neural network: feedforward Neurons: layer and number Within neuron: weight, bias, transfer function Pre- and post- processing H global H 1 SST 1 H 2 SST 2 H 3 SST 3
10 Methodology of reconstruction Mathematics of NN Weights and biases M 1 H global N W 1 W 1 M1 M2 N1 N1 M 2 2 M 2 1 ( X ) A = TransFcn1 W + B N T M Y 2 T ( W A B ) = TransFcn X M T 1 SST 1 H 1 H 2 M Y 2 T SST 2 H 3 SST Transfer functions 1 2 Log-Sigmoid : y = Tan-Sigmoid : y = 1 x 1 + 2x 1+ e B 1 N B 2 M 1 e Linear transfer : y = x
11 Methodology of reconstruction Training and validating NN NN training is first an optimization problem Gradient descent and related Conjugate gradient and related Levenberg-Marquardt algorithm Other ( ) T 1 T k+ 1 = k Jk J k + I Jk Y f k β β λ β 1 ( ) Y f ( ) T T T βk+ 1 = βk k k λdiag J J + Jk Jk Jk βk Validating NN to improve generalization The best training vs. the best generalization Mean Squared Error (mse) Best Validation Performance is.1461 at epoch Train Validation Best Epochs
12 Results of reconstruction Training and validation: example SLR-199 (cm) SLR-199 (cm) SLR-199 (cm) - Observation: validation Reconstruction Observation: training Instances Output ~=.97*Target Error Histogram with 2 Bins Data Fit Y = T Errors = Targets - Outputs Training: R=.9872 Training Zero Error Target
13 Results of reconstruction Issue 1: local minimum + initial weights/biases zero initialization Error Histogram with 2 Bins random [-1, 1] initialization Error Histogram with 2 Bins Training Zero Error Training Zero Error 4 4 Instances 3 2 Instances Errors = Targets - Outputs Errors = Targets - Outputs Solution: Multiple trainings with random initial weights/biases (1 reps)
14 Results of reconstruction Issue 2: generalization Training without validation check 1 Best Training Performance is.2627 at epoch 2 Train Best Goal Training with validation check 1 Best Validation Performance is.149 at epoch 11 Mean Squared Error (mse) Mean Squared Error (mse) Train Validation Best Goal Epochs Epochs
15 Results of reconstruction Issue 2: generalization Training without validation check Training with validation check SLR-199 (cm) - SLR-199 (cm) SLR-199 (cm) -1 SLR-199 (cm) SLR-199 (cm) -1 SLR-199 (cm)
16 Results of reconstruction Issue 2: generalization Training without validation check : reconstructed regional means vs global observation Training with validation check : reconstructed regional means vs global observation 1 Sea level relative to 199 (cm) -1-2 Reconstructed mean of Cluster 1-3 Reconstructed mean of Cluster 2 Reconstructed mean of Cluster 3 Global mean of observations Year Sea level relative to 199 (cm) - -1 Reconstructed mean of Cluster 1-1 Reconstructed mean of Cluster 2 Reconstructed mean of Cluster 3 Global mean of observations Year
17 Results of reconstruction Issue 2: generalization Validation dataset used as training data : reconstructed regional means vs global observation Training with validation check : reconstructed regional means vs global observation Sea level relative to 199 (cm) - -1 Reconstructed mean of Cluster 1-1 Reconstructed mean of Cluster 2 Reconstructed mean of Cluster 3 Global mean of observations Year Solution: Training with validation check (1%) Sea level relative to 199 (cm) - -1 Reconstructed mean of Cluster 1-1 Reconstructed mean of Cluster 2 Reconstructed mean of Cluster 3 Global mean of observations Year
18 Results of reconstruction Impact of region division Division based on ocean basins Division based on clustering SLR-199 (cm) - SLR-199 (cm) SLR-199 (cm) - SLR-199 (cm) SLR-199 (cm) - SLR-199 (cm)
19 Results of reconstruction Impact of SST as input: clustering H global and 3 SST s as inputs H global as the only input SLR-199 (cm) - Reconstruction Observation: training Observation: validation SLR-199 (cm) SLR-199 (cm) - SLR-199 (cm) SLR-199 (cm) - SLR-199 (cm)
20 Results of reconstruction Impact of SST as input: ocean basin H global and 3 SST s as inputs H global as the only input SLR-199 (cm) - SLR-199 (cm) SLR-199 (cm) - SLR-199 (cm) SLR-199 (cm) - SLR-199 (cm)
21 Results of reconstruction Final results: clustering : reconstructed regional means vs global observation SLR-199 (cm) SLR-199 (cm) SLR-199 (cm) - Reconstruction Observation: training Observation: validation Sea level relative to 199 (cm) Sea level relative to 199 (cm) Reconstructed mean of Cluster 1 Reconstructed mean of Cluster 2 Reconstructed mean of Cluster Year : global mean - observation vs reconstruction Global mean of observations Area-weighted mean of 3 reconstructed regions Year
22 Results of reconstruction Final results: ocean basin : reconstructed regional means vs global observation SLR-199 (cm) SLR-199 (cm) - Reconstruction Observation: training Observation: validation Sea level relative to 199 (cm) Year 4 2 Reconstructed mean of Cluster 1 Reconstructed mean of Cluster 2 Reconstructed mean of Cluster : global mean - observation vs reconstruction Global mean of observations Area-weighted mean of 3 reconstructed regions SLR-199 (cm) Sea level relative to 199 (cm) Year
23 The End Thanks!
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