FRM4SM WORK PACKAGE 5 TECHNICAL NOTE 3
SO-TN-CB-GS-0113 REQ-16: R&D CASE STUDY 3
SAMPLING/REPRESENTATIVENESS/SCALE
ERRORS

François Gibon, Arnaud Mialon, Philippe Richaume, Yann H. Kerr, Ali Mahmoodi, Nemesio Rodríguez-Fernández, Raffaele Crapolicchio, Alexander Boresch

► To cite this version:

François Gibon, Arnaud Mialon, Philippe Richaume, Yann H. Kerr, Ali Mahmoodi, et al. FRM4SM WORK PACKAGE 5 TECHNICAL NOTE 3 SO-TN-CB-GS-0113 REQ-16: R&D CASE STUDY 3 SAMPLING/REPRESENTATIVENESS/SCALE ERRORS. SO-TN-CB-GS-0113, CESBIO. 2024. [hal-04839861](https://hal.science/hal-04839861v1)

HAL Id: hal-04839861

https://hal.science/hal-04839861v1

Submitted on 16 Dec 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Abstract

This technical note summarize investigations relative to the spatial and temporal scale
mismatches between SMOS and the in situ measurement (used as reference for calibration).
First, we present an overview of the different caveats that one should consider when comparing
coarse resolution (>10³ km²) satellite data with in situ measurements, in terms of error budget
and uncertainties. Then, we develop a list of analysis to evaluate the sensitivity of the SMOS
performance to different configuration (spatial and temporal) with the in situ measurement.
Results show that more than just the location of the probe within the footprint, the characteristic
of the surface condition around the probe plays a significant role in the level of agreement
between the SMOS and the probe. In fact, favoring probes with strong similarities between
surface conditions and what the satellite sees on average increases the level of consistency
between datasets. The analysis of SMOS performance as a function of the degree of diversity
of surface states (Shannon, Gini) points in the same direction, i.e. increasing the potential
heterogeneity of the surface condition within the footprint increase the spatial scale mismatch.
Using those outputs added to the representativeness indicator developed by TUWIEN has led to
a subset of SMOS nodes (~ 30) and probes that seems favorable for comparison with satellite.
Other analyses show a sensitivity of the validation scores to the different strategies of colocation,
i.e. spatial (in situ vs. satellite driven) and temporal (autocorrelation sampling window length
and position) but need more investigation to be significant. Finally, a discussion and estimate
of the range of values for the various contributors to the uncertainty budget was made. As not
all the topics could be covered, this document serves as a roadmap for further investigations to
tackle the various issues and better characterize the spatial and temporal scale mismatch.
## Document approval

<table><thead><tr><th></th><th>Role</th><th>Name</th><th>Date</th><th>Signature</th></tr></thead><tbody><tr><td rowspan="5">Written by</td><td rowspan="5">Project engineer</td><td>F. Gibon</td><td>/ /23</td><td rowspan="5"></td></tr><tr><td>A. Mialon</td><td>/ /23</td></tr><tr><td>Y. Kerr</td><td>/ /23</td></tr><tr><td>N. Rodriguez F.</td><td>/ /23</td></tr><tr><td>P. Richaume</td><td>/ /23</td></tr><tr><td>Approved by</td><td>Lead investigator</td><td>A. Mialon</td><td>/ /23</td><td></td></tr><tr><td>Approved by</td><td>Technical officer</td><td>R. Crapolicchio</td><td>/ /23</td><td></td></tr></tbody></table>

## Distribution list

<table><thead><tr><th>Name</th><th>Affil.</th><th>Signature</th></tr></thead><tbody><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr></tbody></table>

<table><thead><tr><th>Name</th><th>Affil.</th><th>Signature</th></tr></thead><tbody><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td></tr></tbody></table>

## Auxiliary documents related to this project

<table><tr><td>ATBD L2</td><td>Kerr, Y.H. and Richaume, P., and Waldteufel, P. and Ferrazzoli, P. and Wigneron, J.P. and Schwank, M. and Rautiainen, K., Algorithm theoretical basis document (ATBD) for the SMOS level 2 soil moisture processor, Technical Report TN-ESL-SM-GS-0001-4b SM-ESL (CBSA), 145p, 2020</td></tr><tr><td>FRM4SM detailed proposal</td><td>FRM4SM - Fiducial Reference Measurements for Soil Moisture Detailed Proposal, Vienna, Austria, AWST TUWIEN CESBIO, January 2021</td></tr><tr><td>MRD</td><td>EEOM-SMOS-MRD, Mission Objectives and Scientific Requirements of the Soil Moisture and Ocean Salinity (SMOS) Mission, Version 5, issue 1.0 040701, https://esamultimedia.esa.int/docs/SMOS_MRD_V5.pdf</td></tr></table>
# Purpose and scope of document

Fiducial Reference Measurement for Soil Moisture (FRM4SM) is an ESA project. The first phase (May 2021-April 2023) was extended to the period June 2023 -December 2024. The main goal is to provide confidence information in satellite-based soil moisture retrievals all along the mission duration towards a maximum Return On Investment (ROI).

## ESA FRM4SM PROJECT CONTRACT N°4000135204/21//I-BG

The present document corresponds to the REQ-16 of the FRM4SM detailed proposal, related to the Case Study 2. It answers, in part, to the different objectives of the Task 5 led by the Working Group 5 which are:

* OBJ-11 – Three SMOS R&D case studies

* OBJ-12 – Sub-footprint variability and vertical stratification influence

* OBJ-13 – Temporal aliasing

* OBJ-14 – In-situ-driven strategy versus satellite-driven strategy

* OBJ-15 – Committed areas

# Document status

<table><thead><tr><td>Issue</td><td>Date</td><td>Details</td><td>Editor</td></tr></thead><tbody><tr><td>0.0</td><td>January 2023</td><td>Draft creation and structure</td><td>François Gibon<br>Arnaud Mialon</td></tr><tr><td>0.1</td><td>May 2023</td><td>First draft version</td><td>François Gibon<br>Arnaud Mialon</td></tr><tr><td>1.0</td><td>February 2024</td><td>Update of the part 3.1.2, 3.1.5,<br>3.2.4, 3.3 and 3.4</td><td>François Gibon<br>Arnaud Mialon</td></tr><tr><td>2.0</td><td>September 2024</td><td>Update of the part 3.1.5, 3.3,<br>3.4, 3.5 and 3.6</td><td>François Gibon<br>Arnaud Mialon</td></tr></tbody></table>

If further corrections are required, please contact François Gibon or Arnaud Mialon (francois.gibon@univ-tlse3.fr, arnaud.mialon@univ-tlse3.fr).
# Contents

## 1 Introduction 6

1.1 Overview of the spatio-temporal scale mismatch between SMOS and the in situ measurements 6

1.2 Which contribution in the validation uncertainty budget ? 10

1.3 Objectives of this technical note 11

## 2 Data & methods 12

2.1 Soil moisture 12

2.2 Auxiliary Data Files 12

2.3 Methods 12

## 3 Results & discussions 12

3.1 Spatial scale mismatch effect on the validation 12

3.1.1 What is the validation score sensitivity with the distance between the SMOS footprint center and the probes' location ? 12

3.1.2 Surface Condition Similarities (SCS) between the SMOS footprint and the probe location 14

3.1.3 Probes representativeness Quality Indicators assessment 18

3.1.4 Spatial colocation strategies investigation: in-situ driven vs. satellite driven ? 19

3.1.5 How to quantify the SMOS sub-footprint level of heterogeneity/uniformity ? 22

3.1.6 In the SMOS L2 retrieval strategy, what is the difference between modeling the average of parameters versus the average of brightness temperatures ? 25

3.2 Temporal scale mismatch effect on the validation 26

3.2.1 Influence of the temporal variability of the surface conditions ? 26

3.2.2 Seasonality effect on the validation 27

3.2.3 How to define a common temporal space between SMOS and the probes ? 29

3.2.4 Autocorrelation coherence between SMOS and the probes 30

3.2.5 What is the L-band penetration depth variability due to the soil moisture conditions ? 30

3.2.6 Influence of the different temporal colocation strategies 31

3.3 List of SMOS footprint and/or probe indices 31

3.4 List of preferred SMOS DGG nodes 31

3.5 Uncertainty budget discussion 34

3.6 Step forward 35

## 4 Conclusions & perspectives 36

## References 37
Acronyms

<table><tr><td>- ADF</td><td>Auxiliary Data Files</td></tr><tr><td>- BT</td><td>Brightness Temperature</td></tr><tr><td>- ESA</td><td>European Space Agency</td></tr><tr><td>- FRM</td><td>Fiducial References Measurements</td></tr><tr><td>- FRM4SM</td><td>Fiducial References Measurements for Soil Moisture</td></tr><tr><td>- ISMN</td><td>International Soil Moisture Network</td></tr><tr><td>- MEANWEF</td><td>MEANWEF weighted average function of all the WEFs</td></tr><tr><td>- MRD</td><td>Mission Requirements Document</td></tr><tr><td>- QA4SM</td><td>Quality Assurance for Soil Moisture</td></tr><tr><td>- RFI</td><td>Radio Frequency Interference</td></tr><tr><td>- RTM</td><td>Radiative Transfer Model</td></tr><tr><td>- SM</td><td>Soil Moisture</td></tr><tr><td>- SMOS</td><td>Soil Moisture and Ocean Salinity</td></tr><tr><td>- WA</td><td>Working Area</td></tr><tr><td>- WEF</td><td>WEighted Function</td></tr></table>
# 1 Introduction

## 1.1 Overview of the spatio-temporal scale mismatch between SMOS and the in situ measurements

The evaluation of coarse scale resolution satellite is challenging in many aspects [1]. One of them is the difference of resolution scales in space and time between the satellite and the in situ measurement (reference here): the former being characterized by a footprint of >20 km of radius and a revisit every 2/3 days, the latter characterized by an area of few square meters with a time sampling of minutes.

The order of magnitude of the scale mismatch between SMOS and the probe depends on different features, such as the geostatistic spatial composition of the SMOS footprint or the level of representativeness of the probes. To better understand this issue, here is an overview of the spatial and timescale characteristics for both SMOS and the probes:

* Spatial scale

- SMOS

SMOS measures the thermal emission of the Earth at 1.4 GHz, as brightness temperatures, using aperture synthesis radiometry. Like other passive microwave radiometers, it is characterized by its coarse spatial resolution due to the frequency domain (the L-band, 1.413 GHz). From these observations, SMOS provides soil moisture to data users, such as the level 2 L2SM which are provided on an ISEA grid with a 15-km spatial resolution. This grid paves the Earth spheroid with equal area hexagons, as presented in Figure 1:

<figure><img src="image_2.png" /><figcaption>Figure 1: SMOS Level 2 soil moisture product</figcaption></figure>
Figure 1 presents the retrieved soil moisture along one descending orbit of SMOS. On the zoomed map, one can identify the hexagonal shape of the pixels. The centers (DGG nodes) are ~15 km apart from their nearest neighbors. However, the actual spatial resolution of SMOS is coarser.

<figure><img src="image_3.png" /><figcaption>Figure 2: a) SMOS snapshot from [2], b) SMOS working area, DGG and DFFG grid, mean WEF and L2 pixels c) Brightness Temperature angular profile for a DGG</figcaption></figure>

The Figure 2 a) (from [2]) shows one SMOS Field Of View (FOV), i.e. one snapshot, that is ~1200 km width (and ~800 km for the alias-free area depicted by the black line). For one snapshot, SMOS acquires Brightness Temperature (BT) at different incident angles (displayed as the color scale in the figure a), varying from 0° to higher than 60°. The interception of the antenna gain with the Earth surface is represented by the white ellipses (limit at -3 dB), whose dimensions vary with the incidence angles, from ~35 km at the bore sight to more than 60 km at the border of the FOV. However, if the resolution is well-defined for BTs, it is very difficult to give a value for SM, as the algorithm uses multi-incidence angles.

Figure 2 b) represents one of the ellipses (footprint) at the nadir, which is close to a circle shape centered on one node. These nodes (in red in Figure b), correspond to the Discrete Global Grid (DGG) nodes of the ISEA4H9 grid (~15 km grid) [3], on which the SM retrievals are performed. Due to the grid definition, the pixels have a hexagonal shape, as reported in black in the figure. To proceed to the SM retrieval, another grid system is defined around every DGG nodes. This grid, called the Discrete Flexible Fine Grid (DFFG) [4] [5], is defined as 4km cells. It is used to define the surface conditions with Auxiliary Data Files (ADF), used in the SM retrieval. For the sake of processing, an area of ~123×123 km around each DGG point is defined to cover the different footprint geometries (ellipses' configuration) due to the varying incidence angles. This 123 km box
is referred to as the Working Area (WA) in the level 2 processor. For each DGG node and for each orbit overpass, SMOS acquires BT at different incidence angles, which creates the BT angular profiles, as shown in Figure 2 c). This profile is used to perform the SM/τ retrieval considering the L-MEB (L-band Microwave Emission of the Biosphere) radiative transfer model [6]. This model is initialized with the ADFs which describe as accurately as possible the SMOS footprint content in terms of land cover, soil properties, vegetation properties, and temperatures (ground and vegetation), projected on the DFFG grid.

- **in situ measurement**

The level of representativeness of the probes is a complex topic as many factors are involved, such as the technology, the soil properties, the probes' orientation, the presence of vegetation, the topography, the quality of installation, or the land use [7]. However, the physics under the commonly used probes' technology (TDR, FDR, capacitive), leads to a small volume of measured soil. In [8], the manufacturer provides a theoretical estimation of the volume up to one litre, as represented in Figure 3.

<figure><img src="image_4.png" /><figcaption>Figure 3: a) TEROS-12 probes installed and b) the theoretical measured volume</figcaption></figure>

<figure><img src="image_5.png" /></figure>

**Spatial scale mismatch**

The grid sampling spacing of the SMOS L2SM product is ~15 km, but the SMOS observation footprint is ~43 km of diameter in average (average of the ellipsoids for all the incidence angles). It means that the emission area considered in the BT measurement can vary from ~30 to ~60 km. So, the difference of spatial scale with the in situ measurement representativeness, considered as ~cm², is of a factor of ~10⁵.

- **Temporal scale**

<figure><img src="image_6.png" /><figcaption>Figure 4: Comparison between SMOS L2SM in black (DGG 232283) and an in situ measurement in blue (USCRN-Bronte site).</figcaption></figure>
## SMOS

The temporal resolution of SMOS is defined by its revisit, i.e. the moment of passage over a point. Based on a sun-synchronous orbit, its overpassing time at the equator is 6:00 (solar local time) for ascending overpasses and 18:00 (solar local time) for descending. Its revisit period is variable as a function of the latitude, as shown in Figure 5. In fact, the overpass period decreases with increasing latitude.

As an example, Figure 4 presents two SM time series for a site in the USA, the blue line being the in situ measurements, the black line being the SMOS SM. The two time series depict a different time sampling.

### SMOS revisit frequency versus latitude

<figure><img src="image_7.png" /><figcaption>Figure 5: Comparison of SMOS revisit period for two points of different latitudes.</figcaption></figure>

## in situ measurement

The sampling period of in situ measurements is related to the data logger characteristic. For example, the ZL6 data logger of MeterGroup records the measurements every minute, and then average them as a function of the interval of sampling parametrized. The resulting time series, in blue Figure 4, is able to track the high frequency variation of soil moisture, as the daily cycle.

<figure><img src="image_8.png" /></figure>

### Temporal scale mismatch

Even if the SMOS revisiting time is relatively low (2/3 days) compared to the in situ measurements, the low frequency soil moisture fluctuation are well tracked, as shown in Figure 4. However, the high frequency soil moisture fluctuation, as the daily cycles or the spike (as the 30/06/18), are missed.
## 1.2 Which contribution in the validation uncertainty budget ?

In our case, “validation” means characterizing the uncertainty of a dataset, by comparing it to a reference. For SMOS soil moisture, the comparison with ground measurement (used as a reference) bears several sources of uncertainties:

* $u_{SMOS}$: the one we want to quantify, that depends on many factors, such as the surface conditions nature and **heterogeneity**, the RFI contamination or the ADF uncertainties

* $u_{insitu}$: depends on the sensor technology representativeness, the quality of installation, the calibration...

* $u_{scale}$: the spatio-temporal scale mismatch between the two datasets, highly dependent on the land surface configuration, especially the level of **heterogeneity** and uniformity of the SMOS footprint

An example of the variability of the SMOS performance is highlighted in Figure 6. In this case, 21 probes were compared to the same SMOS DGG. The result of this comparison leads to a strong variability in the scores displayed in the Taylor diagram, in Figure 6. This example shows us that for one DGG node, depending on the selected in situ site, the SMOS performances vary significantly: $\Delta R \approx 0.3$, $\Delta Bias \approx 0.14 \text{ m}^3 \cdot \text{m}^{-3}$ and $\Delta STDD \approx 0.04 \text{ m}^3 \cdot \text{m}^{-3}$. So, in regards with the SMOS accuracy target of $0.04 \text{ m}^3 \cdot \text{m}^{-3}$ [9] there is already a variability of scores of the same order only with the probe's selection within the footprint. So, the conclusion of whether SMOS reaches the specification should be considered with care. The aim of the future studies is to better estimate these uncertainties due to the location of the probes to extract the absolute SMOS performance.

<figure><img src="image_9.png" /><figcaption>Figure 6: Taylor Diagram of one SOS node (DGG 170178) compared to several probes that are within the SMOS footprint.</figcaption></figure>

The community use different strategies to reduce the scale mismatch between the two datasets. Strategies are used to upscale the reference, such as the soil texture-weighted method, the arithmetic average, Thiessen Polygon, area averaging using Voronoi diagram, soil texture-weighted on Voronoi diagram etc... [10]. Others focus on reducing the SMOS spatial resolution, such as the disaggregation
[11]. As for the temporal scale, methods such as time average, wavelet analysis or autocorrelation analysis are undertaken to deal with this scale mismatch.

## 1.3 Objectives of this technical note

The performance of passive microwave soil moisture retrieval is defined by the level of agreement with the in situ measurements. However, due to the high difference in spatial resolution (two different mesurand), the uncertainty component on the scale mismatch is non-negligible.

The objective of this technical note is to document the scale mismatch order of magnitude to better understand the *u*<sub>scale</sub> component of the uncertainty budget in the SMOS validation context. The objectives of this study are to i) define a subset of probes minimizing the *u*<sub>scale</sub> and ii) derive recommendations to minimize the *u*<sub>scale</sub>.

To do so, we propose to investigate this issue through different point of view:

*   **Spatial scale mismatch**

- Analysis of the SMOS performances sensitivity to the distance between the SMOS DGG center and the probe location

- Analysis of the differences/similarities between the probes surface conditions and the SMOS footprint in relation to SMOS performance

- Assess the probes Quality Indicator (QI) "representativeness": TU WIEN developed an index of representativeness based on triple colocation that we want to assess in this analysis

- Quantify the different spatial colocation strategies: here, we want to investigate the differences of SMOS performances regarding different spatial colocation methods (in situ driven vs. satellite driven)

- Analysis the level of heterogeneity of the SMOS footprint: by this study, we want to i) quantify the spatial distribution of the SMOS footprint and ii) analyze the SMOS performance sensitivity

- Quantify the error due the SMOS retrieval procedure: the idea is to model BTs using 2 different strategies of aggregation within the SMOS footprint and analyse the impact on the SMOS performance

*   **Temporal scale mismatch:**

- Map the surface condition variability amplitude: the SMOS performances depend on the surface conditions [12]; in this part, we focus on mapping the areas where the surface conditions have a high temporal dynamic (due to vegetation seasonal cycle, or snow...)

- Quantify the seasonal effects on the validation: to quantify the temporal sensitivity of the SMOS performances, we propose to analyse the score using different time windows

- Investigate the common time-space between SMOS and the probes: as in [13], spectral methods can be used to better understand the common scale between the two data sets

- Autocorrelation: can it be used to analyse the temporal dynamics captured by the in situ and SMOS ?

- Analyze the L-band penetration depth dependence with the seasons: the penetration depth is sensitive to the soil moisture condition; we plan here to analyse this sensitivity

- Quantify the different temporal colocation strategies: due to the difference of sampling period between SMOS and the probes, the different temporal colocation have to be analysed
# 2 Data & methods

## 2.1 Soil moisture

* SMOS L2 Soil Moisture v700

The analysis is based on the SMOS SM v700, which is detailed in the ATBD [14] and in the technical note DT5-1 [12].

* ISMN as reference

The analysis is based on the ISMN network, which is considered as the reference for our analyses and is detailed in the technical note DT5-1 [12].

## 2.2 Auxiliary Data Files

The ADFs used in the present analyses are detailed in the technical note DT5-1 [12].

## 2.3 Methods

The validation procedure used here, is described in details in the technical note DT5-1 [12]. The SMOS filtering criteria (i.e. SMOS obs. kept) used here are: $CHI2P \ge 0.05$, $RFI$ and $RFIP \le 0.1$, $FLTOPO$ set to 0 and SM range [0:1] $m^3 m^{-3}$. For the ISMN probes, the filtering criteria (in situ obs. kept) are i) a depth $\le 10$ cm and ii) GLFAG set to 1. For the more specific analyses, the methods are developed directly in the following section.

# 3 Results & discussions

## 3.1 Spatial scale mismatch effect on the validation

### 3.1.1 What is the validation score sensitivity with the distance between the SMOS footprint center and the probes' location ?

The influence of the distance between the probes and the SMOS footprint center on the validation scores is investigated. Two cases are presented in Figure 7, to illustrate that this distance can vary.

<figure><img src="image_10.png" /><figcaption>Figure 7: Example of two SMOS footprints (nodes 2007112 and 153325) and the closest in-situ sites: on the left panel, the probe is closer to the center of the SMOS footprint than in the right panel.</figcaption></figure>
The validation scores are presented in Figure 8 as a polar projection, and in Figure 9 as a function of the distance between the SMOS DGG centers and the probe locations. One can notice that the x-axis shows a range of distances from 0.15 to 8.60 km. No clear trends are suggested from these representations, as no specific pattern appears.

<figure><img src="image_11.png" /><figcaption>Figure 8: a) normalized position of the 1991 probes used in this validation experiment within the SMOS mean WEF; polar representation of the scores for each probes' localization with R in panel b), ubRMSD in panel c), bias in panel d), RMSD in panel e), absolute bias in panel f).</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>Figure 9: Scores as a function of the distance between the probes and the SMOS DGG centers</figcaption></figure>

<figure><img src="image_13.png" /></figure>

Effect of the distance SMOS DGG ↔ in situ probes

This analysis shows that there is no relationship between the score and the distance between the probe and the center of the SMOS antenna gain.

## 3.1.2 Surface Condition Similarities (SCS) between the SMOS footprint and the probe location

We propose to develop an index which quantifies the differences/similarities of surface conditions between the probes' location and the global SMOS footprint. For the sake of illustrating the concept, Figure 10 presents two cases where the surface condition similarity between the probe and the SMOS footprint can vary. In case a), the probe is located in a more forested area than the global SMOS footprint while in case b), the probes are in the same conditions as the global SMOS footprint.
The main ideas here are to:

1. Define the Surface Condition Similarities (SCS) index, which quantifies the differences/similarities of land surface condition between SMOS and the in situ probes

2. Analyze the sensitivity of the SMOS uncertainty to the SCS
<figure><img src="image_14.png" /><figcaption>Figure 10: Examples of a SMOS nodes with their footprint (background from the Matlab basemap tool hosted by ESRI) and the in situ probes: a) the probe is in a more forested area than the global SMOS footprint, b) the probe is in the same condition as the global SMOS footprint</figcaption></figure>

# 1. Surface Condition Similarities (SCS) definition

We propose here to use the SMOS ADFs on the DFFG grid to quantify the similarities/differences of surface condition between the SMOS footprint and the probe location. We used 9 land covers based on the IGBP and SoilGrids databases: forest, low vegetation, sand, clay, bulk density, SOC, high topography, low topography and open water bodies. For each of the 1991 in situ probes compared with the nearest SMOS DGGs, two vectors of 9 values are defined, respectively $\mu$ and $\tau$, as presented Figure 11:

* $\mu$: value corresponding to the landcovers WA (35x35 4km pixels) fractions x meanWEF

* $\tau$: values at the 4x4 km pixel containing the probe

<figure><img src="image_15.png" /><figcaption>Figure 11: On the left side, the concept, on the right side an example for the forest condition where the SMOS footprint is at $\mu$=20%, and the probe location at $\tau$=90%, which makes an absolute difference of 70%.</figcaption></figure>

To derive an index which represent the level of difference of geophysical condition between the probe and the satellite footprint, we chose to normalise the difference between $\mu$ and $\tau$ by their maximum regarding the 1991 in situ probes:

$$SCS = \frac{\tau - \mu}{\max_{allprobes}(|\tau - \mu|)} \quad (1)$$
The normalized differences are represented in Figure 12 for all surface conditions (i.e. each lines). The blue/white/red colors represent the situation where the probe is in the higher/similar/lower value of the respective surface characteristic than the SMOS footprint.

<figure><img src="image_16.png" /><figcaption>Figure 12: For each comparison (x-axis), blue/red color shows that the probe is in higher/lower value of the respective surface characteristic (y-axis) than the SMOS footprint.</figcaption></figure>

Figure 13 details the SCS distribution for each surface condition:

<figure><img src="image_17.png" /><figcaption>Figure 13: a) Histograms of the normalized difference and b) summary of the differences values.</figcaption></figure>

* Forest/Nominal: In two-thirds of the cases, the SMOS footprint shows a higher/lower value of forest/nominal than the one at the probe location, suggesting that the probes are primarily situated in a less densely forested region than the footprint content of SMOS.

* Sand/clay/bulk density/SOC: the values are always higher at the SMOS footprint scale than at the probe one.

* Topography: probes are primarily situated in the flatter region regarding the footprint content of SMOS.
* Open water: probes are not often situated close to open water when there is a presence in the footprint content of SMOS.

## 2. Analyze the sensitivity of the SMOS uncertainty to the SCS

In this second step, the idea is to investigate the potential of using the SCS to classify the probes regarding their level of SCS. The assumption is that selecting probes that are in the same surface condition as the SMOS footprint can reduce the uncertainty of the spatial scale mismatch.

To do so, different ways to use the SCS can be considered, such as the sum, the average, ... One proposition is to sum all the SCSs in their absolute form, for the 9 surface descriptors (i.e. row-wise in Figure 12). Per consequence, the values of this sum represent the level of difference between the probe and the SMOS footprint. Then, thresholds have to be defined. For example, the Figure 14 represents the results using the following thresholds:

* High level of similarity: $SCS \le 0.3$

* Middle level of similarity: $0.3 < SCS \le 0.7$

* Low level of similarity: $0.7 < SCS$

The histograms (Figure 14, on the left) represent the proportion of probes classified with a similarity of surface condition High, Middle or Low. The correlation R and ubRMSD scores are improving when considering higher SCS conditions. It is not clear though for the bias and RMSD.

<figure><img src="image_18.png" /><figcaption>Figure 14: Test of probes classification based on their level of SCS.</figcaption></figure>

<figure><img src="image_19.png" /></figure>

### SMOS vs. in situ probes surface condition mismatch

Classifying the probes using the index developed here to represent the similarity of surface condition with the SMOS footprint shows a clear effect on the validation scores (R and ubRMSD). Further investigations are needed to refine the definition of the SCS by adjusting the thresholds.
### 3.1.3 Probes representativeness Quality Indicators assessment

The main objective here is to investigate the use of the Quality Indicators of representativeness developed by TU Wien. The triple colocation method, used to derive this indicator, is described in detail in [15] and class the probe as a function of their level of representativeness. One should notice that some validation points (n=469) have not been flagged, i.e. 'not found' (Further exchanges are needed with the FRM4SM team to fix this issue). The results, presented on the boxplot Figure 15 show a clear improvement of all the scores when the probes flagged 'representative' are considered and even better with the 'very representative' ones.

<figure><img src="image_20.png" /><figcaption>Figure 15: Boxplots representing the different SMOS validation scores by QIs of representativeness.</figcaption></figure>

<figure><img src="image_21.png" /></figure>

**Representativeness Quality Indicators classification ?**

The first results show that the scores improve when using the 'representative'/'very representative' probes. Further investigations are needed to include the 'not found' one.
### 3.1.4 Spatial colocation strategies investigation: in-situ driven vs. satellite driven ?

The main objective of this part is to determine the effect, on the SMOS evaluation, of the different strategies of spatial collocation with the in situ probes. Different examples of strategies are illustrated in Figure 16:

**A - in situ driven:** the commonly used "one probe - one node", also called "point-scale", when only one probe is considered.

**B - Dense network:** when there are several probes within the footprint, with different merging strategies: average, Voronoi triangle,...

**C - Multi-DGGs:** using several SMOS nodes (average) to be compared to one in-situ site.

<figure><img src="image_22.png" /><figcaption>Figure 16: Three different spatial colocation strategies</figcaption></figure>

**A - in situ driven strategy**

Figure 17 corresponds to the case "in situ driven". An evaluation of the SMOS product is processed for each of the 27 probes. The corresponding Taylor diagram (right Figure 17) shows a high variability in the score. The scores vary significantly from one site to the others, with a difference between the best and the worst cases of $\Delta R \approx 0.6$, $\Delta Bias \approx 0.24 \ \text{m}^3 \cdot \text{m}^{-3}$ and $\Delta \text{STDD} \approx 0.09 \ \text{m}^3 \cdot \text{m}^{-3}$.

<figure><img src="image_23.png" /><figcaption>Figure 17: Validation case in a dense network (Hobe network)</figcaption></figure>

This example illustrates how diverse the scores can be from one site to the others, even though there are colocated to the same node.
## B - "Dense network" strategy

The map in the Figure 18 shows the different SMOS DGG node where we can find more than five ISMN in situ probes within a footprint. Those sites are candidates to apply the method used in [16] to estimate the *u*<sub>scale</sub> component.

<figure><img src="image_24.png" /></figure>

Figure 18: Map of SMOS node where there is more than 5 probes (from the ISMN network), within the footprint.

Figure 19 compares SMOS node 221032 with the 21 probes present within the footprint. The Taylor diagram shows a large variability of scores when looking at the different probes individually. However, comparing the SMOS estimation (blue line on the time series) to the average of the in situ probes (red line on the time series),

Dense network case study: SOILSCAPE - SMOS DGG ID 221032 - 21 probes

<figure><img src="image_25.png" /></figure>

Figure 19: Example of dense network case in the DGG 221032
Dense network case study: SOILSCAPE/AMERIFLUX - SMOS DGG ID 170178 - 20 probes

<figure><img src="image_26.png" /><figcaption>Figure 20: Example of dense network case in the DGG 170178</figcaption></figure>

## C - "Multi-DGGs" strategy

In Figure 21, the Taylor diagram (right panel) presents the scores of all the SMOS DGG nodes that are located around the Hobe site (Denmark), and the particular probe Hobe 2.08. The result of this comparison, shows a large variability between all the SMOS nodes with a range of $\Delta R \approx 0.07$, $\Delta Bias \approx 0.03 \ \text{m}^3 \cdot \text{m}^{-3}$ and $\Delta STDD \approx 0.03 \ \text{m}^3 \cdot \text{m}^{-3}$.

<figure><img src="image_27.png" /><figcaption>Figure 21: Validation case with a multi-DGG approach for the particular site of Hobe in Denmark</figcaption></figure>

<figure><img src="image_28.png" /></figure>

### Effect of the spatial colocation strategy

The first results show a strong variability of scores using the in-situ driven strategy. Further analyses are needed to extract the potential of using the dense network approach in order to approximate the *u*<sub>scale</sub> component.
### 3.1.5 How to quantify the SMOS sub-footprint level of heterogeneity/uniformity ?

In [17], we saw that SMOS footprint content can influence its performance. In this part, we focus on how the spatial distribution of the surface conditions within SMOS footprint can also influence the SMOS performance. Here are some possible issues/causes:

* **Increasing of the auxiliary data files (ADFs) uncertainties**

ADFs are used in the SMOS L2 processor to parametrize the radiative transfer model L-MEB. Based on different database. These ADFs define the surface conditions at the DFFG grid resolution with a spatial resolution of 4x4 km. Original datasets are usually available at higher resolution, but are aggregated in the context of SMOS retrieval. This adds an uncertainty to their native uncertainties, as represented in Figure 22. One can notice the loss of spatial definition between the original optical image (left) and the derived SMOS ADF landcover classification (right).

* **Increasing of the scale mismatch with the probes**

In a situation where the surface condition diversity is high, the risk of having a probe installed in a "non representative" area compared to the SMOS footprint context is also high.

* **Increasing of the error in the retrieval modelling**

In the SMOS version 700, the soil moisture retrieval is processed based on modeled BTs, that are obtained by averaging surface conditions (i.e. ADF) within SMOS footprint. However, another strategy would be to model BTs for every surface conditions and then average the modelled BTs. The impact of this strategy has not been quantified yet (see Section 3.1.6).

<figure><img src="image_29.png" /><figcaption>Figure 22: Native optical view (left) compared to the corresponding SMOS ADF landcover (right), with soil moisture probes marked with red point/crosses.</figcaption></figure>
We investigate the effect of the landcover sub-footprint spatial distribution on the performance with two steps:

## 1. Characterization and quantification of the level of heterogeneity

We try here to define and quantify the heterogeneity of the land cover classes within the SMOS footprint. The heterogeneity of the footprint can be defined as its level of diversity of classes but also the spatial distribution of classes: for example, in Figure 23, the case of 2 classes equally represented, but with two different spatial distributions: a) the mix is homogeneous and in b) the mix is heterogeneous.

<figure><img src="image_30.png" /><figcaption>Figure 23: Two different spatial distributions of two classes. a) homogeneous distribution, b) heterogeneous distribution.</figcaption></figure>

To characterize the surface heterogeneity, four indicators are tested and evaluated:

* **Number of classes:** quantify the number of different classes within the footprint

* **Standard deviation of classes:** quantify the standard deviation of the pixels, i.e. if all the classes had the same number of pixels within the footprint, the std will be equal to 0.

* **Shannon's diversity index:** quantify the entropy within the footprint [18]. The higher this indicator, the greater the diversity of surface conditions in the footprint.

$$H = - \sum_{i=1}^{n \text{class}} p_i \ln(p_i) \qquad (2)$$

with $p_i$ the proportion of pixels $n_i$ within the total N corresponding to the class i:

$$p_i = \frac{n_i}{N} \qquad (3)$$

* **Gini-Simpson index** The Gini indicator [19],

$$H = 1 - \sum_{i=1}^{n \text{class}} p_i^2 \qquad (4)$$

Based on a landcover map<fnref n="1" /> (the top left image in Figure 24), we generalize the computed statistics above for each DGG node.

<footnote n="1">¹GLC-SHARE, Global Land Cover SHARE. Database Beta-Release Version 1.0 - 2014, by John Latham, Renato Cumani, Ilaria Rosati and Mario Bloise. FAO document.</footnote>
<figure><img src="image_31.png" /><figcaption>Figure 24: Global workflow</figcaption></figure>

Figure 24 presents the approach for the landcover classification, for one node (top row) and the four indicators for France. On those maps, we can notice that the maximums of diversity are located at the boundaries between different environments: earth/sea, land/mountains, crops/city, land/forest.

## 2. Analyse of the SMOS performance sensitivity to the heterogeneity

The relationship between the SMOS performances vs. ISMN and the different footprint statistics are presented in Figure 25 and Figure 26. SMOS performance decreases slightly (i.e. increase of ubRMSD and decrease of the R correlation) when the entropy increases.

<figure><img src="image_32.png" /><figcaption>Figure 25: Summary of the ubRMSD SMOS performance sensitivity to different heterogeneity index.</figcaption></figure>
<figure><img src="image_33.png" /></figure>

Figure 26: Summary of the correlation SMOS performance sensitivity to different heterogeneity index.

NB: The results with or without the WEF application are very similar.

<figure><img src="image_34.png" /></figure>

### SMOS footprint characterization

First analysis shows that an increase in the level of diversity of landsurface conditions within the footprint, lightly decreases the performance metrics.

### 3.1.6 In the SMOS L2 retrieval strategy, what is the difference between modeling the average of parameters versus the average of brightness temperatures ?

Two strategies of retrieval presented in Figure 27. In the Case 1, the BT are first modeled for each high resolution cells (DFFG), and then averaged to obtain the SMOS-like BT. In the Case 2, which corresponds to the SMOS operational retrieval, the BT is modeled using the averaged of the different surface parameters. Not in the framework of FRM4SM project, this analysis is dedicated to SMOS ESL tasks.
Case 1: BTs modeling for each pixels

<figure><img src="image_35.png" /></figure>

Case 2: BT modeling for an average pixel

<figure><img src="image_36.png" /></figure>

Figure 27: Two different strategies to obtain a modeled BT: i) on the left, by modeling BT considering all the surface conditions, and then average all the BT; ii) on the right, by averaging the surface conditions, and then modeling one BT.

<figure><img src="image_37.png" /></figure>

**Average the BTs effect ?**

This issue is mentioned as it contributes to the uncertainty budget of SMOS. It can not be completed within the framework of FRM4SM project, but has to be reported along with all the potential sources of uncertainties.

## 3.2 Temporal scale mismatch effect on the validation

### 3.2.1 Influence of the temporal variability of the surface conditions ?

SMOS performances are sensitive to the surface conditions as reported in the document [12]. However, those surface conditions are dynamic in space and time, involving a time dependency of the SMOS performances. Some surface condition changes are presumed to be important, such as the snow cover, the land surface temperature seasonality, the anthropic land cover change, or the vegetation seasonality. Figure 28 represents the vegetation dynamic at the global scale. The colors correspond to the respective period of maximum of greenness. Areas with the highest color intensity are characterized by a high seasonality of activities, meaning a potential high variation of the SMOS performances regarding the chosen time window.
<figure><img src="image_38.png" /><figcaption>Figure 28: Global map of land-cover dynamics, data from the Copernicus Sentinel-2, processed by GeoVille.</figcaption></figure>

### 3.2.2 Seasonality effect on the validation

In order to determine the scores' sensitivity to the seasonality, an analysis has been done by computing the scores for different time window configuration, as presented in Figure 29. The scores are performed for varying time window, defined by a starting date ("Window start" from 1st January to 31st December) and an ending date ("Window stop" from "Window start" to 31st December).

<figure><img src="image_39.png" /><figcaption>Figure 29: Method used to process the validation scores for a variable time window. Scores are processed and plotted on the 2D figure for every beginning (y-axis right figure)/end (x-axis right figure) of the time window.</figcaption></figure>

Two sites are analyzed having very different surface and climate conditions. The first one is located in a semi-arid area with a strong seasonality (Niger [13.6°N;2.6°W], Figure 30), whereas the second one (Figure 31) is located in a more temperate area (US [32°N;-100.2°W]). The minimum time step defined here is the month.

In Figure 30, depending on the time window configuration, the correlation R varies from -0.4 to 0.8 (ΔR=1.2), the RMSD from 0.06 to 0.01 m³.m⁻³ (ΔRMSD=0.05 m³.m⁻³), the bias
from -0.003 to 0.03 m³.m⁻³ (ΔBias=0.033 m³.m⁻³) and the ubRMSD from 0.005 to 0.05 m³.m⁻³ (ΔubRMSD=0.045 m³.m⁻³).

In Figure 31, depending on the time window configuration, the correlation R varies from 0.67 to 0.87 (ΔR=0.2), the RMSD from 0.078 to 0.05 m³.m⁻³ (ΔRMSD=0.028 m³.m⁻³), the bias from 0.015 to 0.053 m³.m⁻³ (ΔBias=0.038 m³.m⁻³) and the ubRMSD from 0.05 to 0.068 m³.m⁻³ (ΔubRMSD=0.018 m³.m⁻³). Note that the ΔubRMSD=0.018 m³.m⁻³ is lower in this case than in the previous one (Figure 30).

<figure><img src="image_40.png" /><figcaption>Figure 30: Validation time window analysis for the Banizoumbou AMMA-CATCH site</figcaption></figure>
<figure><img src="image_41.png" /><figcaption>Figure 31: Validation time window analysis for the Bronte USCRN site</figcaption></figure>

<figure><img src="image_42.png" /></figure>

### Seasonality effect on the validation

There is a high variability in the validation scores in the context of climate marked with strong seasonality. We need to extend our analysis to all the in-situ probes to be able to generalize and quantify the effect of seasonality on the validation scores.

### 3.2.3 How to define a common temporal space between SMOS and the probes ?

This task is about the investigation of spectral analysis to identify common time/space between the SMOS observations and the in-situ measurement ones.

<figure><img src="image_43.png" /></figure>

#### Spectral analysis

At the time of writing, this analysis has not been processed yet.
### 3.2.4 Autocorrelation coherence between SMOS and the probes

The temporal coherence between the SMOS observation revisiting period, and the in situ measures can be quantified by comparing the autocorrelation scheme of the two datasets. This autocorrelation could improve our understanding on the temporal scale mismatch error between the two information. In Figure 32 two "extrem cases" are represented. The correlogram are in good agreement in the right graph when it is not as obvious on the left one. To go further, one should find a method to quantify this agreement and generalize the approach to all the probes.

<figure><img src="image_44.png" /></figure>

Figure 32: Two autocorrelation case study, in Argentina the two correlogram are not very consistent, by opposite to the Benin one where the two correlogram are well in-line.

<figure><img src="image_45.png" /></figure>

**Autocorrelation analysis**

At the time of writing, the correlogram can be computed, however, more investigation is needed to quantify their agreements.

### 3.2.5 What is the L-band penetration depth variability due to the soil moisture conditions ?

The L-band penetration depth is highly dependent on the soil moisture conditions. Analyzing the validation scores for different probes depth was initiated in the TN1 of the project. The results show a worsening of the scores with increasing depth (from surface to 1m). To go further, this analyses have to be extended to different soil moisture conditions (from dry to wet) to help to quantify this sensitivity.

<figure><img src="image_46.png" /></figure>

**L-band penetration depth analysis**

At the time of writing, this analysis has not been processed yet.
### 3.2.6 Influence of the different temporal colocation strategies

The aim of this task is to determine the different temporal colocation strategies. Indeed, the most common approach is to compare satellite observations with the closest in time in situ measurements. However, considering the spatial mismatch, one can expect the common information to be included in temporal synthesis. For example, SMOS Level 3 products are provided with several temporal aggregation strategies: 3 days, 10 days (min/median/max) and monthly averages. Methods to compute in situ measurements (mean, media, quantiles) over corresponding periods are to be analyzed.

<figure><img src="image_47.png" /></figure>

**Temporal colocation strategies analysis**

<figure><img src="image_48.png" /></figure>

At the time of writing, this analysis was not processed yet.

## 3.3 List of SMOS footprint and/or probe indices

Table 2 reports the indices that are used to analysis the spatial and temporal mismatches between satellite observations and in situ measurements.

<table><caption>Table 2: List of indices used to evaluate the temporal and spatial mismatch between satellite SM and in situ measurements</caption><thead><tr><td>Indice</td><td>Name</td><td>Description</td><td>Range</td><td>Maturity</td></tr></thead><tbody><tr><td>A</td><td>Distance</td><td>Distance from the probe location to the DGG node</td><td>[0:8] km</td><td>High</td></tr><tr><td>B</td><td>SCS</td><td>Level of surface condition similarities between the probes location and the SMOS footprint</td><td>not similar<br>similar<br>very similar</td><td>Moderate</td></tr><tr><td>C</td><td>Shannon</td><td>Level of classes diversities (entropy) (8 classes for the moment)</td><td>[0:2]</td><td>High</td></tr><tr><td>D</td><td>Gini</td><td>Level of inequality of classes</td><td>[0:1]</td><td>High</td></tr><tr><td>E</td><td>Autocorr.</td><td>Level of coherence between the SMOS and probe autocorrelation correlogram</td><td>?</td><td>Low</td></tr><tr><td>F</td><td>Seasonality</td><td>Risk of high score variability due to seasonality</td><td>?</td><td>Low</td></tr><tr><td>G</td><td>TUWIENQI</td><td>Probes representativeness based on SNR analysis</td><td>not representative<br>representative<br>very representative</td><td>High</td></tr></tbody></table>

## 3.4 List of preferred SMOS DGG nodes

We propose to classify the probes from ISMN based on the different indices. To do so, we attribute a value to each probe based on the sum of the different level of similarity/representativeness of the SCS/TUWIENQI (Table 3). For example, the probes considered as the best to evaluate satellite SM, are classified "0", as they are very similar (SCS=0) and very representative (TUWENQI=0), see Table 3.

<table><caption>Table 3: Values attributed to each classes of the similarity index (SCS) and representativeness (TUWENQI)</caption><thead><tr><td>SCS</td><td>TUWIENQI</td></tr></thead><tbody><tr><td>0: very similar</td><td>0: very representative</td></tr><tr><td>1: similar</td><td>1: representative</td></tr><tr><td>2: not similar</td><td>2: not representative</td></tr><tr><td></td><td>3: unknown</td></tr></tbody></table>
The result of this classification can be found in the boxplot Figure 33. It shows that the classification gives valuable information on the expected SMOS performances. For example, there are 56 probes classified as "0", for which the scores (R, bias, RMSD and ubRMSD) are significantly better than for the other classes. These probes are represented by red points on the map Figure 33. It corresponds to the 30 SMOS DGG nodes described in Table 4.

<figure><img src="image_49.png" /></figure>

Figure 33: Top: boxplot of the scores as a function of classes of probes (from 0= representative, to 4 being the worst probes for evaluating satellite SM, 5=unknown) the classified probes. Bottom right: number of probes per class. Bottom left: map of the 56 probes (30 SMOS DGG nodes) respecting the highest criteria.
<table><caption>Table 4: List of the SMOS DGGs containing in situ measurements which are either i) in the same condition as the SMOS footprint and ii) with a good SNR. **One has to be careful that this list is not corresponding to the location where SMOS is supposed to perform the best, there are only location of DGG/probes where the comparison seems to be the fairest.**</caption><thead><tr><td>DGGID</td><td>Lat.</td><td>Lon.</td><td>DGGID</td><td>Lat.</td><td>Lon.</td></tr></thead><tbody><tr><td>144598</td><td>46.970</td><td>-118.552</td><td>217966</td><td>37.155</td><td>-97.251</td></tr><tr><td>144604</td><td>47.465</td><td>-117.567</td><td>220014</td><td>36.603</td><td>-97.488</td></tr><tr><td>150781</td><td>48.361</td><td>-111.958</td><td>222059</td><td>35.882</td><td>-98.227</td></tr><tr><td>154829</td><td>43.460</td><td>-119.783</td><td>222063</td><td>36.104</td><td>-97.567</td></tr><tr><td>158986</td><td>47.070</td><td>-109.899</td><td>222068</td><td>36.374</td><td>-96.739</td></tr><tr><td>160974</td><td>41.890</td><td>-119.516</td><td>223110</td><td>37.021</td><td>-93.860</td></tr><tr><td>162101</td><td>49.028</td><td>-102.208</td><td>243645</td><td>34.130</td><td>-87.431</td></tr><tr><td>165158</td><td>47.277</td><td>-105.205</td><td>244161</td><td>34.154</td><td>-86.828</td></tr><tr><td>165595</td><td>41.705</td><td>-117.529</td><td>248242</td><td>32.332</td><td>-89.043</td></tr><tr><td>172276</td><td>41.826</td><td>-113.764</td><td>248793</td><td>33.859</td><td>-83.449</td></tr><tr><td>174900</td><td>45.512</td><td>-103.311</td><td>249306</td><td>33.765</td><td>-83.361</td></tr><tr><td>179533</td><td>45.719</td><td>-99.181</td><td>2069365</td><td>13.644</td><td>2.596</td></tr><tr><td>184523</td><td>35.140</td><td>-119.595</td><td>2069877</td><td>13.510</td><td>2.660</td></tr><tr><td>191814</td><td>41.995</td><td>-101.428</td><td>2072925</td><td>9.743</td><td>1.565</td></tr><tr><td>204156</td><td>41.562</td><td>-93.305</td><td>2073438</td><td>9.739</td><td>1.692</td></tr></tbody></table>
## 3.5 Uncertainty budget discussion

Based on the GUM (Guide to the Expression of Uncertainty of Measurement) [20] nomenclature we can develop the result of the SMOS ubRMSD, using ISMN as a reference [21], as:

<figure><img src="image_50.png" /></figure>

### Ranges of values of each uncertainty budget components:

<table><caption>Figure 34: Global overview of the uncertainty budget with ranges of the different components.</caption><thead><tr><th></th><th>Low<br>m<sup>3</sup>m<sup>-3</sup></th><th>High<br>m<sup>3</sup>m<sup>-3</sup></th><th>Description/reference</th></tr></thead><tbody><tr><td>U<sub>total</sub></td><td>0.06<sup>a</sup><br>0.04<sup>b</sup></td><td>0.09<sup>a</sup><br>0.06<sup>b</sup></td><td><sup>a</sup> SMOS vs. ISMN ubRMSD (Gibon et al. 2024)<br><sup>b</sup> with "bests" sites only (TN3)</td></tr><tr><td>U<sub>SMOS</sub></td><td>0.02<sup>c</sup></td><td>0.056<sup>d</sup></td><td><sup>c</sup> Lower bound based on average DQX (TN3)<br><sup>d</sup> estimated from the max values of U<sub>total</sub>, U<sub>insitu</sub> and U<sub>scale</sub></td></tr><tr><td>U<sub>insitu</sub></td><td>0.01</td><td>0.05</td><td>in situ probes accuracy review: depend on the technology and calibration quality (Susha Lekshmi et al. 2014)</td></tr><tr><td>U<sub>scale</sub></td><td>0.03<sup>e</sup><br>X<br>0.056<sup>g</sup></td><td>0.05<sup>e</sup><br>0.051<sup>f</sup><br>X</td><td><sup>e</sup> estimation from TC (Dorigo et al. 2021)<br><sup>f</sup> estimation with dense network (Benninga et al. 2020)<br><sup>g</sup> computed from the min values of U<sub>SMOS</sub>, U<sub>insitu</sub> and U<sub>total</sub></td></tr><tr><td>Cov.</td><td>X</td><td>X</td><td>Heterogeneity within the SMOS footprint influence:<br>- U<sub>SMOS</sub>: by adding more complexity of modelling<br>- U<sub>scale</sub>: by increasing the mesurand difference</td></tr></tbody></table>
In Figure 34, we propose a representation of the different components of the uncertainty budget with:

* $u_{total}$ is the total uncertainty resulting from the comparison of SMOS with ISMN. The study [17] shows a median of 0.0699 m$^3$.m$^{-3}$ (ubRMSD) with an IQR (Inter Quartile Range) of [0.058, 0.086] m$^3$.m$^{-3}$. When restricting to the best sites (Table 4), the ubRMSD improves with a median of 0.0539 m$^3$.m$^{-3}$ and an IQR of [0.0413, 0.0647] m$^3$.m$^{-3}$.

* $u_{SMOS}$ corresponds to the SMOS uncertainty, the one we want to determine. It is influenced by many factors, such as surface conditions within the footprint [17], the modelling intrinsic error, ADF uncertainties, the radiometric noise or the RFI contamination. In a perfect world, we would deal only with the radiometric noise, which can be estimated with the SMOS DQX distribution over the ISMN sites (see histogram $\leftarrow$). The average of the histogram show an average of estimation of 0.02 m$^3$.m$^{-3}$. This value can be interpreted as lower bound of $u_{SMOS}$. The upper bound can be estimated by isolating the $u_{SMOS}$ component from the global equation in Figure 34 and use the maximal value of $u_{insitu}$, $u_{scale}$ and $u_{total}$, respectively 0.05, 0.051 and 0.09 m$^3$.m$^{-3}$. This result in an upper bound of $u_{SMOS}$ equal to 0.056 m$^3$.m$^{-3}$.

<figure><img src="image_51.png" /></figure>

* $u_{insitu}$ is the uncertainty of the in situ measurements, used as reference. Due to many factors, such as the quality of calibration or installation, it can vary significantly from 0.01 to 0.05 m$^3$.m$^{-3}$ according to the review done in [22].

* $u_{scale}$ is the uncertainty due to the spatio-temporal scale mismatch between the SMOS data and the in situ measurement one. Highly complex to estimate, some assumption can be done based on different method. From [15], authors use the triple colocation method to approximate the $u_{scale}$, with a range of 0.03-0.05 m$^3$.m$^{-3}$. In [16], authors use the dense networks to evaluate the variability of the individual probes regarding the average of the other, resulting on an estimation of 0.051 m$^3$.m$^{-3}$. We can also provide a lower estimation using the minimal value of the other, the result gives 0.056 m$^3$.m$^{-3}$.

* Cov is the contribution of all the factors which can influence the others, dashed purple arrow in Figure 34. One of them is related to the “heterogeneity” of the footprint. It can impact strongly the complexity of modelling, the uncertainty on the ADF, and increase $u_{SMOS}$. In the same manner, it can also enhance the disparity between the SMOS footprint and the in situ measurement, ultimately leading to an increase in $u_{scale}$. However, it is not straightforward to estimate, hence the X's in the table in Figure 34.

## 3.6 Step forward

* $u_{scale}$: The methodology described in the supplementary material of [16] to estimate $u_{scale}$ can be applied to the dense networks of the ISMN data. To do so, one could compute the variability of the probe regarding the average of the closest probes within the footprint.

* One can also classify the probes which are representative of different classes. For example, by combining the value of the fraction of the probe and optimizing the SCS, more representative per land cover classes (i.e. forest, crop), by optimizing the SCS indices, to finally get an estimation of $u_{SMOS}$ per classes.
* The uncertainty global mapping in Figure 35 from [17], can be refined. To do so, one can use the best identified sites (red triangle in the map) to perform the regression. However, there is a risk of too low number of points to derive significant relationships.

<figure><img src="image_52.png" /><figcaption>Figure 35: Map of predicted SMOS uncertainty, from [17], and red triangle the "bests" sites identified in this analysis.</figcaption></figure>

# 4 Conclusions & perspectives

In this document, the spatio-temporal scale mismatches between SMOS and the probes are investigated. To do so, the performance of SMOS, using the in situ measurements from the ISMN as a reference, are analyzed through different point of view, such as the distance between the SMOS node and the probe or the level of diversity of land surface condition within the footprint. The main outputs of the analysis are:

* Distance between probes and SMOS footprint center does not have an impact on the SMOS performance

* Similarity of surface conditions between the probes near environment and the average SMOS footprint improves the scores of SMOS evaluation

* Classification of the probes using the SNR optimization method from TUWIEN shows a clear improvement of SMOS performances

* Spatial colocation methods "in situ driven" vs. "satellite driven" were initiated but more investigations are needed to get significant results, and to better use dense networks to estimate $u_{scale}$

* Levels of diversity of classes within the footprint (Shannon and Gini index) show a very slight worsening of the score with increasing diversity
* Temporal aspect (seasonality, autocorrelation,...) was initiated but needs more investigations to get clear results

* A list of 30 DGGs points was identified as being in the best conditions to evaluate SMOS. Results showed an ubRMSD IQR from [0.058, 0.086] m³.m⁻³ using the ISMN to [0.04, 0.06] m³.m⁻³ using these 30 DGGs. This improvement can be interpreted as a reduction of the *u*<sub>scale</sub> component and a reduction of the heterogeneity

* An attempt to dissect the uncertainty budget $u_{total}^2 = u_{SMOS}^2 + u_{scale}^2 + u_{insitu}^2 + cov$ have been done to estimate the upper and lower bound of the members (see the table in Figure 34)

The previous analysis leads us to consider the future activities:

* To investigate methods to better define future mission requirements thresholds for surface and root zone soil moisture.

* To better characterize soil moisture validation results between in situ FRM and satellite products (active and passive, coarse and high resolution) based on selected heterogeneity indicators (e.g. Shannon, Gini, topology models). Different spatial scales shall be considered in the definition of heterogeneity, as well as the impact of the geophysical auxiliary files used to characterize the heterogeneity of the satellite footprint.

* To propose and evaluate indices, and how they can be used together to classify the in-situ measurement sites as FRM and included in the QA4SM service.

* To revisit the approach to characterize the surface condition nearby the probes developed during the phase 1 of the project for the high resolution soil moisture products and to derive uncertainty sensitivity to the spatial resolution by cross comparing soil moisture product at different spatial resolution (e.g. Sentinel-1, SMOS L4).

* To better understand what the common temporal scale between satellite products is and FRM by analysing the autocorrelation between FRM time series and soil moisture products level 2 and composite Level 3 (e.g. CATDS 3/10 days, monthly, etc.)

* Improve the knowledge of the satellite soil moisture uncertainty budget considering the 3 main components: *u*<sub>ref</sub> for in-situ dataset, *u*<sub>scale</sub> for spatial representatives error, and *u*<sub>sat</sub> for sensor and retrieval errors. Improve *u*<sub>scale</sub> characterization using dense network (e.g. Hobe in Denmark, Remedhus in Spain, SMAPVEX in USA, etc.).

* Use the case of SMOS level 2 operational processor to draft a method to derive uncertainty traceability diagram associated to the level 2 retrieval algorithm.

## References

[1] Land Product Validation Subgroup (Working Group On Calibration And Validation Committee On Earth Observation Satellites). Soil moisture product validation good practices protocol, 2020.

[2] N. Reul, S.A. Grodsky, M. Arias, J. Boutin, R. Catany, B. Chapron, F. D'Amico, E. Dinnat, C. Donlon, A. Fore, S. Fournier, S. Guimbard, A. Hasson, N. Kolodziejczyk, G. Lagerloef, T. Lee, D.M. Le Vine, E. Lindstrom, C. Maes, S. Mecklenburg, T. Meissner, E. Olmedo, R. Sabia, J. Tenerelli, C. Thouvenin-Masson, A. Turiel, J.L. Vergely, N. Vinogradova, F. Wentz, and S. Yueh. Sea surface salinity estimates from spaceborne l-band radiometers: An overview of the first decade of observation (2010–2019). *Remote Sensing of Environment*, 242:111769, June 2020.
[3] Kevin Sahr, Denis White, and A Jon Kimerling. Geodesic discrete global grid systems. *Cartography and Geographic Information Science*, 30(2):121–134, 2003.

[4] Yann H Kerr, Philippe Waldteufel, Philippe Richaume, Jean Pierre Wigneron, Paolo Ferrazzoli, Ali Mahmoodi, Ahmad Al Bitar, François Cabot, Claire Gruhier, Silvia Enache Juglea, et al. The smos soil moisture retrieval algorithm. *IEEE transactions on geoscience and remote sensing*, 50(5):1384–1403, 2012.

[5] Y.H. Kerr, P. Richaume, P. Waldteufel, P. Ferrazzoli, J.P. Wigneron, M. Schwank, and K. Rautiainen. Algorithm theoretical basis document (ATBD) for the smos level 2 soil moisture processor. Technical Report TN-ESL-SM-GS-0001-4b SM-ESL (CBSA), page 145p, 2020.

[6] J-P Wigneron, Y H Kerr, P Waldteufel, K Saleh, M-J Escorihuela, P Richaume, P Ferrazzoli, P de Rosnay, R Gurney, J-C Calvet, J P Grant, M Guglielmetti, B Hornbuckle, C Mätzler, T Pellarin, and M Schwank. L-band Microwave Emission of the Biosphere (L-MEB) Model : Description and calibration against experimental data sets over crop fields. *Rem. Sens. Environ.*, 107:639–655, 2007.

[7] Muhammad Waseem Rasheed, Jialiang Tang, Abid Sarwar, Suraj Shah, Naeem Saddique, Muhammad Usman Khan, Muhammad Imran Khan, Shah Nawaz, Redmond R Shamshiri, Marjan Aziz, et al. Soil moisture measuring techniques and factors affecting the moisture dynamics: A comprehensive review. *Sustainability*, 14(18):11538, 2022.

[8] Teros12 user manual.

[9] Mrd & ecom-smos-mrd, mission objectives and scientific requirements of the soil moisture and ocean salinity (smos) mission.

[10] Hassan A.K.M. Bhuiyan, Heather McNairn, Jarrett Powers, Matthew Friesen, Anna Pacheco, Thomas J. Jackson, Michael H. Cosh, Andreas Colliander, Aaron Berg, Tracy Rowlandson, Paul Bullock, and Ramata Magagi. Assessing SMAP soil moisture scaling and retrieval in the carman (canada) study site. *Vadose Zone Journal*, 17(1):1–14, January 2018.

[11] O. Merlin, A.G. Chehbouni, Y.H. Kerr, E.G. Njoku, and D. Entekhabi. A combined modeling and multispectral/multiresolution remote sensing approach for disaggregation of surface soil moisture: application to SMOS configuration. *IEEE Transactions on Geoscience and Remote Sensing*, 43(9):2036–2050, September 2005.

[12] François Gibon, Arnaud Mialon, Philippe Richaume, Yann H. Kerr, Alireza Mahmoodi, Nemesio Jose Rodriguez-Fernandez, Roberto Sabia, and Alexander Boresch. Technical note 1: Smos validation and committed areas. Technical Report SO-TN-CB-GS-0104, CESBIO, 2022.

[13] Beatriz MOLERO RODENAS. *Différence d'échelle spatiale entre les mesures satellitaires et in situ d'humidité du sol : analyse par des approches spatio-temporelle*. PhD thesis, University of Toulouse, SDU2E, 2017.

[14] Y H Kerr, P. Waldteufel, P. Richaume, J-P Wigneron, P Ferrazzoli, and R Gurney. SMOS level 2 processor for soil moisture - Algorithm Theoretical Based Document (ATBD) . Technical Report SO-TN-ESL-SM-GS-0001, Issue 3.e, CBSA, 2011. 121 p.

[15] Wouter Dorigo, Irene Himmelbauer, Daniel Aberer, Lukas Schremmer, Ivana Petrakovic, Luca Zappa, Wolfgang Preimesberger, Angelika Xaver, Frank Annor, Jonas Ardö, Dennis Baldocchi, Günter Blöschl, Heye Bogena, Luca Brocca, Jean-Christophe Calvet, Julio J. Camarero, Giorgio Capello, Minha Choi, Michael C. Cosh, Jerome Demarty, Nick van de
Giesen, Istvan Hajdu, Karsten H. Jensen, Kasturi Devi Kanniah, Ileen de Kat, Gottfried Kirchengast, Pankaj Kumar Rai, Jenni Kyrouac, Kristine Larson, Suxia Liu, Alexander Loew, Mahta Moghaddam, José Martínez Fernández, Cristian Mattar Bader, Renato Morbidelli, Jan P. Musial, Elise Osenga, Michael A. Palecki, Isabella Pfeil, Jarret Powers, Jaakko Ikonen, Alan Robock, Christoph Rüdiger, Udo Rummel, Michael Strobel, Zhongbo Su, Ryan Sullivan, Torbern Tagesson, Mariette Vreugdenhil, Jeffrey Walker, Jean Pierre Wigneron, Mel Woods, Kun Yang, Xiang Zhang, Marek Zreda, Stephan Dietrich, Alexander Gruber, Peter van Oevelen, Wolfgang Wagner, Klaus Scipal, Matthias Drusch, and Roberto Sabia. The international soil moisture network: serving earth system science for over a decade. *Hydrology and Earth System Sciences Discussions*, pages 1–83, 2021.

[16] Harm-Jan F. Benninga, Rogier van der Velde, and Zhongbo Su. Sentinel-1 soil moisture content and its uncertainty over sparsely vegetated fields. *Journal of Hydrology X*, 9:100066, December 2020.

[17] François Gibon, Arnaud Mialon, Philippe Richaume, Nemesio Rodríguez-Fernández, Daniel Aberer, Alexander Boresch, Raffaele Crapolicchio, Wouter Dorigo, Alexander Gruber, Irene Himmelbauer, et al. Estimating the uncertainties of satellite derived soil moisture at global scale. *Science of Remote Sensing*, page 100147, 2024.

[18] Claude Elwood Shannon. A mathematical theory of communication. *The Bell system technical journal*, 27(3):379–423, 1948.

[19] Lou Jost. Entropy and diversity. *Oikos*, 113(2):363–375, May 2006.

[20] ISO. and BIPM OIML. *Guide to the Expression of Uncertainty in Measurement*. Aenor, 1993.

[21] François Gibon, Arnaud Mialon, Philippe Richaume, Nemesio Rodríguez-Fernández, Daniel Aberer, Alexander Boresch, Raffaele Crapolicchio, Wouter Dorigo, Alexander Gruber, Irene Himmelbauer, Wolfgang Preimesberger, Roberto Sabia, Pietro Stradiotti, Monika Tercjak, and Yann H. Kerr. Estimating the uncertainties of satellite derived soil moisture at global scale. *Science of Remote Sensing*, 10:100147, December 2024.

[22] Susha Lekshmi S.U., D.N. Singh, and Maryam Shojaei Baghini. A critical review of soil moisture measurement. *Measurement*, 54:92–105, 2014.