# What Is the Spatial Resolution of grace Satellite Products for Hydrology?

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## Abstract

**:**

## 1. Introduction

## 2. Spherical Harmonic Coefficients and Their Corresponding Spatial Resolution

#### 2.1. Sampling and Half-Wavelength of the Field

#### 2.2. Ideal Spatial Resolution

#### 2.3. Catchment Averages, Post-Filtering Corrections and Resolution

#### 2.4. Some Exceptions

- 1.
- that have a strong seasonal variation in their water storage [7], or
- 2.
- that are similar in magnitude and temporal phase as their neighboring catchments, i.e., without spatial contrast [23] (it should not be confused with the signal contrast/modulation mentioned earlier), or
- 3.
- that are isolated or dominant in terms of their signal strength, for example huge reservoir volume changes [41]

## 3. Data and Method

Multiplicative: | ${\widehat{f}}_{\mathrm{c}}=$ | $s({\overline{f}}_{\mathrm{c}}-{l}_{\mathrm{c}}^{\mathrm{m}}),$ [19] |

Additive: | ${\widehat{f}}_{\mathrm{c}}=$ | ${\overline{f}}_{\mathrm{c}}-{l}_{\mathrm{c}}^{\mathrm{m}}+{b}_{\mathrm{c}}^{\mathrm{m}},$ [20] |

Scaling: | ${\widehat{f}}_{\mathrm{c}}=$ | $k\phantom{\rule{0.166667em}{0ex}}{\overline{f}}_{\mathrm{c}},$ [24] |

Data-driven: | ${\widehat{f}}_{\mathrm{c}}=$ | ${\overline{f}}_{\mathrm{c}}-{l}_{\mathrm{c}}-{\overline{\delta F}}_{\mathrm{c}},$ [25]. |

- Root Mean Square of Error (RMSE):$$\begin{array}{c}\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{m}\sum _{i=1}^{m}{({f}_{\mathrm{c}}-{\widehat{f}}_{\mathrm{c}})}^{2}}},\hfill \end{array}$$
- Cyclostationary Nash–Sutcliffe Efficiency (${\mathrm{NSE}}_{\mathrm{seas}}$) [45,46]:$$\begin{array}{c}{\mathrm{NSE}}_{\mathrm{seas}}=1-\frac{{\displaystyle \sum _{i=1}^{m}}{({f}_{\mathrm{c}}-{\widehat{f}}_{\mathrm{c}})}^{2}}{{\displaystyle \sum _{i=1}^{m}}{({f}_{\mathrm{c}}-\stackrel{\u02d8}{{f}_{\mathrm{c}}})}^{2}},\hfill \end{array}$$

## 4. Results and Discussion

## 5. Conclusions

- 1.
- The spatial resolution of the band-limited grace spherical harmonics is not the half-wavelength at the equator, but the ideal spatial resolution. The spatial resolution of filtered grace data can also be described by the ideal spatial resolution.
- 2.
- The users have to be wary that the spatial resolution of the corrected dataset is dependent on the adopted method. Furthermore, the spatial resolution is associated with the error tolerance required by the application and it has to be defined by the user.
- 3.
- Given the fact that with enhanced processing techniques grace is able to see some catchments smaller than the spatial resolution and many catchments close to the band-limit resolution, it is worthwhile to provide spherical harmonics up to a maximum degree of 120 or higher.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Appendix A. Filtering on the Sphere

**Figure A1.**The concepts of homogeneity/inhomogeneity and isotropy/anisotropy of the filter weights is illustrated here with the most general form of the filter function (inhomogeneous and ansiotropic) in the top row and the simplest form of the filter function (homogeneous and isotropic) in the bottom row.

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**Figure 1.**Illustration of the ideal spatial resolution computation for filter windows. Two Dirac pulses (gray lines in (

**a**–

**e**)) are set up at points P and Q and the field is filtered (black lines in (

**a**–

**e**)). For a given filter, they are resolved as two different signals at a certain separation (

**c**)—the ideal spatial resolution. However, the signals are more prominent after filtering (contrast), if the distance between the signals is larger (

**d**,

**e**). The contrast is quantified via the modulation transfer function (mtf), which is a function of the modulation transfer (cf. (6)) and signal separation (

**f**).

**Figure 2.**The modulation transfer functions of the Shannon kernels corresponding to the most common bandwidths of the grace spherical harmonic spectra. A modulation transfer of 0 means that the two Dirac pulses separated by those distances will not be resolved, and a modulation transfer of 1 means that the signals are completely resolved and their contrast completely retained. The light-gray lines indicate the ideal resolution of the corresponding kernels.

**Figure 3.**Distribution of 255 catchments used in this study. The catchments are filled in light blue with the boundary in blue. The catchment boundaries were downloaded from Global Runoff Data Centre (GRDC) website.

**Figure 4.**RMSE and corresponding fit with respect to the catchment area for different correction methods. Subplots (

**a**) to (

**e**) show RMSE and corresponding fit for one method in colour and others in gray. Subplot (

**f**) compares the fit for different methods.

**Figure 5.**${\mathrm{NSE}}_{\mathrm{seas}}$ and corresponding fit with respect to the catchment area for different correction methods. Subplots (

**a**) to (

**e**) show ${\mathrm{NSE}}_{\mathrm{seas}}$ and corresponding fit for one method in colour and others in gray. Subplot (

**f**) compares the fit for different methods.

**Table 1.**Comparison of the half-wavelength and ideal spatial resolution values for typical grace bandwidths. Also shown are the typical area associated with them. The area of an equi-angular grid cell centered on the equator (${A}_{\mathrm{g}}$) is associated with the half-wavelength, and the spherical area (${A}_{\mathrm{s}}$) is associated with the ideal spatial resolution. N/A stands for not applicable.

L | Half-Wavelength | Ideal Spatial Resolution | |||||
---|---|---|---|---|---|---|---|

${\mathbf{\psi}}_{\frac{\mathbf{1}}{\mathbf{2}}}$ | ${\mathit{A}}_{\mathrm{g}}$ | $\sqrt{{\mathit{A}}_{\mathrm{g}}}$ | ${\mathbf{\psi}}_{\mathbf{0}}$ | ${\mathit{A}}_{\mathrm{s}}$ | $\sqrt{{\mathit{A}}_{\mathrm{s}}}$ | ||

[1000 km${}^{\mathbf{2}}$] | [km] | [1000 km${}^{\mathbf{2}}$] | [km] | ||||

60 | 3 | ${0}^{\circ}$ | 111.5 | 334 | $4.{33}^{\circ}$ | 182.4 | 427 |

90 | 2 | ${0}^{\circ}$ | 49.6 | 223 | $2.{90}^{\circ}$ | 81.9 | 286 |

96 | 1 | ${875}^{\circ}$ | 43.6 | 209 | $2.{72}^{\circ}$ | 72.0 | 268 |

120 | 1 | ${5}^{\circ}$ | 27.9 | 167 | $2.{18}^{\circ}$ | 46.2 | 215 |

180 | 1 | ${0}^{\circ}$ | 12.4 | 111 | $1.{46}^{\circ}$ | 20.7 | 144 |

Gauss 400 km | n/a | n/a | n/a | $6.{11}^{\circ}$ | 363.2 | 603 |

**Table 2.**The approximate resolvable catchment size (in 1000 km${}^{2}$) for a repair method performing better than a given RMSE. The values given here are obtained from the fit, and one should be more careful while carrying out studies for catchments close to the limit as the scatter is wide. Our suggestion is to avoid any interpretation after the fit tends to become noisy. When the fit for a method becomes noisy, we represent it by writing N/A.

RMSE ($\mathbf{cm}$) | Filtered | Multiplicative | Additive | Scaling | Data-Driven |
---|---|---|---|---|---|

1 | 750 | 1563 | 1103 | 563 | 250 |

1.5 | 337 | 1323 | 360 | 260 | 152 |

2 | 152 | 1103 | 152 | 90 | 63 |

2.5 | 90 | 951 | 1223 | 76 | N/A |

3 | N/A | 810 | N/A | N/A | N/A |

**Table 3.**The approximate resolvable catchment size (in 1000 km${}^{2}$) for a repair method performing better than a given ${\mathrm{NSE}}_{\mathrm{seas}}$. The values given here are obtained from the fit, and one should be more careful while carrying out studies for catchments close to the limit as the scatter is wide. Our suggestion is to avoid any interpretation after the fit tends to become noisy. When the fit for a method becomes noisy, we represent it by writing N/A.

${\mathbf{NSE}}_{\mathbf{seas}}$ | Filtered | Multiplicative | Additive | Scaling | Data-Driven |
---|---|---|---|---|---|

0.9 | 2000 | – | – | 750 | 450 |

0.8 | 1100 | 1750 | 1100 | 600 | 200 |

0.7 | 380 | 1200 | 450 | 500 | 150 |

0.6 | 280 | 900 | 250 | 250 | 90 |

0.5 | 200 | 480 | 180 | 180 | N/A |

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**MDPI and ACS Style**

Vishwakarma, B.D.; Devaraju, B.; Sneeuw, N.
What Is the Spatial Resolution of grace Satellite Products for Hydrology? *Remote Sens.* **2018**, *10*, 852.
https://doi.org/10.3390/rs10060852

**AMA Style**

Vishwakarma BD, Devaraju B, Sneeuw N.
What Is the Spatial Resolution of grace Satellite Products for Hydrology? *Remote Sensing*. 2018; 10(6):852.
https://doi.org/10.3390/rs10060852

**Chicago/Turabian Style**

Vishwakarma, Bramha Dutt, Balaji Devaraju, and Nico Sneeuw.
2018. "What Is the Spatial Resolution of grace Satellite Products for Hydrology?" *Remote Sensing* 10, no. 6: 852.
https://doi.org/10.3390/rs10060852