# An Adaptive Piecewise Harmonic Analysis Method for Reconstructing Multi-Year Sea Surface Chlorophyll-A Time Series

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Conventional HANTS Algorithm

_{r}, Y and ε are the reconstructed, original and error series, respectively. T is the characteristic fundamental period (which is typically one year), t

_{i}is the time when Y is observed and k is the harmonic order with a maximum of N. Here, j is the polynomial degree, with a maximum of L, while a

_{k}and b

_{k}are the coefficients of the corresponding cosine and sine components, respectively, and c

_{j}is the coefficient of each polynomial term.

#### 2.2. Algorithm of Adaptive Piecewise HANTS (AP-HA) Method

#### 2.2.1. Step 1: Preprocessing of the Original Data Series

#### 2.2.2. Step 2: Initial Global HANTS Fitting

#### 2.2.3. Step 3: Iterative Piecewise HANTS Fitting

_{r1}and Y

_{r2}are the local functions of subseries from T

_{1}to T

_{3}and T

_{2}to T

_{4}, respectively; α is the weight of each observation.

#### 2.3. Evaluation Strategy

#### 2.4. Sea Surface Chlorophyll-a Dataset

_{rs}) images from the moderate-resolution imaging spectroradiometer (MODIS) on the Aqua satellite with local area coverage (LAC) during the period from 4 July 2002 to 31 December 2018 were downloaded for the geographic area of 117–127°E by 31–41°N from the OceanColor website of the Goddard Space Flight Center (http://oceancolor.gsfc.nasa.gov/). The daily R

_{rs}images were remapped onto a common 1/24° × 1/24° grid based on linear interpolation for convenience and to reduce the data volume for data analysis. A regional statistical GAM (generalized additive model) algorithm was used to calculate the daily sea surface CHL based on R

_{rs}datasets, which have improved accuracy in both magnitude and seasonality when compared with the standard OC3M (ocean CHL three-band algorithm for MODIS) algorithm [33]. The daily CHL images were merged to weekly images using median values, and finally a weekly composite dataset with 240 × 240 spatial pixels and 858 temporal steps (with 52 steps in a complete year) spanning from July 2002 to December 2018 was constructed for further analysis. The weekly composite CHL dataset contained lots of missing observations, with a total percentage of 63.6%. The temporal profiles of sea surface CHL in this area are mainly governed by periodic seasonal patterns and can theoretically be fitted with a series of harmonics [34,35]. Therefore, the harmonic-based algorithms could be used to simulate and reconstruct CHL time series; however, due to their log-normal distribution characteristic, the CHL values should be log-transformed (on the base of 10) before all analyses [36].

## 3. Results

#### 3.1. Illustrating the AP-HA Implementation on a Profile

#### 3.2. Overall Quantitative Evaluation

#### 3.3. Visual Inspection of Typical Data Series

#### 3.4. Results of Reconstructed CHL Images

## 4. Discussion

#### 4.1. Improvements in AP-HA Method

#### 4.2. Limitations and Perspective

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Graphical illustration of the two successive local fittings and their merging. Y

_{r1}and Y

_{r2}are the two local harmonic functions from T1 to T3 and T2 to T4, respectively. Y

_{r}* is their merged function for the overlapped period from T2 to T3.

**Figure 3.**Location of the study area overlaid with a composite image of satellite-derived sea surface chlorophyll-a concentrations during 10–16 September 2013.

**Figure 4.**Example of a CHL time series and the outputs in the different steps of the proposed AP-HA method. Data series result from the three steps: (

**a**) preprocessing of the original data series; (

**b**) initial global HANTS fitting; (

**c**) iterative piecewise HANTS fitting.

**Figure 5.**Example of an original unimodal CHL time series associated with the reconstructed time series, which was processed using different methods: (

**a**) original method; (

**b**) HA31; (

**c**) HA53; (

**d**) HA75; (

**e**) HA97; (

**f**) HA-CV; (

**g**) AP-HA. (

**a**) The points indicate testing data being selected.

**Figure 6.**Example of an original bimodal CHL time series associated with the reconstructed time series, which was processed using different methods: (

**a**) original method; (

**b**) HA31; (

**c**) HA53; (

**d**) HA75; (

**e**) HA97; (

**f**) HA-CV; (

**g**) AP-HA. (

**a**) The points indicate testing data being selected.

Dataset | HA31 | HA53 | HA75 | HA97 | HA-CV | AP-HA |
---|---|---|---|---|---|---|

Training | 0.224 | 0.198 | 0.192 | 0.188 | 0.195 | 0.155 |

Testing | 0.229 | 0.207 | 0.281 | 0.303 | 0.197 | 0.188 |

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**MDPI and ACS Style**

Wang, Y.; Gao, Z.; Ning, J.
An Adaptive Piecewise Harmonic Analysis Method for Reconstructing Multi-Year Sea Surface Chlorophyll-A Time Series. *Remote Sens.* **2021**, *13*, 2727.
https://doi.org/10.3390/rs13142727

**AMA Style**

Wang Y, Gao Z, Ning J.
An Adaptive Piecewise Harmonic Analysis Method for Reconstructing Multi-Year Sea Surface Chlorophyll-A Time Series. *Remote Sensing*. 2021; 13(14):2727.
https://doi.org/10.3390/rs13142727

**Chicago/Turabian Style**

Wang, Yueqi, Zhiqiang Gao, and Jicai Ning.
2021. "An Adaptive Piecewise Harmonic Analysis Method for Reconstructing Multi-Year Sea Surface Chlorophyll-A Time Series" *Remote Sensing* 13, no. 14: 2727.
https://doi.org/10.3390/rs13142727