Remote sensing, in situ networks and models are now providing unprecedented information for environmental monitoring. To conjunctively use multi-source data nominally representing an identical variable, one must resolve biases existing between these disparate sources, and the characteristics of the biases can be non-trivial due to spatio-temporal variability of the target variable, inter-sensor differences with variable measurement supports. One such example is of soil moisture (SM) monitoring. Triple collocation (TC) based bias correction is a powerful statistical method that is increasingly being used to address this issue, but is only applicable to the linear regime, whereas the non-linear method of statistical moment matching is susceptible to unintended biases originating from measurement error. Since different physical processes that influence SM dynamics may be distinguishable by their characteristic spatio-temporal scales, we propose a multi-timescale linear bias model in the framework of a wavelet-based multi-resolution analysis (MRA). The joint MRA-TC analysis was applied to demonstrate scale-dependent biases between in situ, remotely sensed and modelled SM, the influence of various prospective bias correction schemes on these biases, and lastly to enable multi-scale bias correction and data-adaptive, non-linear de-noising via wavelet thresholding.

Global environmental monitoring requires geophysical measurements from a
variety of sources and sensors to close the information gap. However,
different direct and remote sensing, and model simulation can yield different
estimates due to different measurement supports and errors. Soil moisture
(SM) is one such variable that has garnered increasing interest due to its
influences on atmospheric, hydrologic, geomorphic and ecological processes

In situ networks usually provide point-scale measurements; satellite
retrieval of shallow SM at a mesoscale footprint of 10–50 km must resort to
a homogeneity or dominant-feature assumption, whereas modelled SM depends on
the simplified model parameterization, and the quality, resolution and
availability of forcing data. Subsequently, the spatial (lateral and
vertical) variability of SM can lead to systematically different measurements
regarded as

Triple collocation (TC)

Given the possible (time)scale dependency in biases and errors, we propose an
extension to TC analyses to include wavelet-based multi-resolution analysis
(MRA)

The paper is organised as follows. Section

We consider in situ, satellite-retrieved and modelled SM over Australia. For
an in-depth study, we consider point-scale and pixel-scale SM estimates at
K1 monitoring site (147.56

The satellite SM was retrieved by AMSR-E (Advanced Microwave Scanning
Radiometer for Earth Observing System; AMS) of the AQUA satellite. The
retrieval is based on an inversion of the forward radiative transfer model of
a vegetation-masked soil surface, relating observed brightness temperature to
soil dielectric constant estimates. A dielectric mixing model is then used to
related the dielectric constant to volumetric SM. The combined C/X-band

Spatial variability of land surface and rainfall over Kyeamba Creek. The cross denotes the location of the K1 monitoring station, and the dashed (solid) box is the pixel area of AMS (MER).

The modelled SM is taken from MERRA (Modern Era Retrospective-analysis for
Research and Applications) – Land produced by the Catchment land surface
model GEOS version 5.7.2. The MERRA atmospheric re-analysis is driven by a
vast collection of in situ observations of atmospheric and surface winds,
temperature, and humidity, and remote sensing of precipitation and radiation

The three data are co-located spatially via nearest neighbour and temporally
at around the satellite overpass times of 1.30 a.m./p.m. LT. Their time
series are plotted in blue in the first panels of Fig.

Continental-scale AMS and MER data over Australia are also considered. The
continent has great variability in climatic and land surface characteristics.
Most of the northern regions experience a tropical savannah (Aw)
Köppen–Geiger climate as classified by

The observed Kyeamba SM (denoted by blue curves

MRA of INS, AMS and MER SM at Kyeamba.

The 1-D orthogonal discrete wavelet transform (DWT) enables MRA of a time
series

In Eq. (

Comparisons of correlation

The detail and approximated time series of Kyeamba's SM are illustrated in
subsequent panels of Fig.

The differences between the details of the three SM are apparent on the
finest scales, with AMS and MER showing greater variability and amplitude
compared to INS. However, the similarity of their temporal patterns, in both
details and approximations, grows with increasing scales

MRA enables direct comparisons between any two representations

Before proceeding, we recall that weak

With these considerations, we first examine the correlations between the
three data. For the detail time series (Fig.

Furthermore, we observe that

Difference in SD (in units of m

Next, Fig.

Figure

In order to quantify observed differences between the data, we propose a

Importantly, the model allows for different scaling coefficients between
scales, i.e.

By using a third independently derived representation (

Consider now the bias correction of

Following our theoretical model in Sect.

Bias correction of AMS and MER (as

For illustrations, we correct the biases in AMS and MER SM with respect to
INS SM at Kyeamba using the above five schemes. Using the above notations,
AMS and MER are treated as

Figure

RMSD (in units of m

The MRA of the corrected data

The results of bulk, A/S and MS linear rescaling can be readily interpreted.
For bulk (Fig.

By construction, the MS rescaling uses the estimated

The bulk and A/S CDF methods produced very similar results with each other,
and also with their linear counterparts. There is signal and noise
suppression, but the scale-level biases are retained. The signal components
of

Time series of AMS SM at Kyeamba treated by various bias correction
schemes. The use of WT-based de-noising has also been demonstrated
in

In summary, the MRA of the bulk and A/S schemes highlights the deficiency of
using a correction scheme that does not take into account the scale
variability of bias and the differences in noise statistics between the two
data. The improvements in RMSD and correlation between the corrected

The last example presents an impetus to consider noise removal prior to bias
correction and produce a simpler error structure in the bias-corrected data

One commonly used transformation is soft thresholding

The prescription, which is essentially a two-stage operation, was applied to
AMS for comparisons with the previous results. The first stage of de-noising
leads to smoothing of the time series, improved

This work combines MRA and TC in a new analysis framework with increased capacity to provide a more comprehensive view of the inter-data relations on short and long timescales. TC (or CDF) rescaling can be exploited on individual scales to reduce scale-specific multiplicative biases, and provide “prior” knowledge of noise for calibrating a WT-based de-noising filter. As a demonstration of principle, these methods are applied to SM data from in situ and satellite sensors and a land surface model. Using MRA, we found that the three data exhibit significantly different wavelet spectra and variable degrees of agreement on different timescales. On fine scales, the contribution of noise is most prominent, undermining the correlation between the data sets. By contrast, the biases are most apparent on coarse scales. Furthermore, these biases are non-systematic across timescales in the study region and across spatial locations over Australia, and the signal-to-noise ratios vary with scales and between the various data, pointing to the need to use correction schemes that are capable of handling such complexities.

These observations raised concerns about the possible inadequate treatment of
SM data in the linear regime, even with anomaly/seasonal decomposition.
Scale-by-scale linear rescaling based on a MRA-TC analysis framework offers
a more comprehensive treatment of different biases on different scales, but
error characteristics are found to be modified by variable rescaling, and can
lead to undesirable noise amplification. The method of removing biases and
noise on individual scales offers a remedy, although a few caveats should be
noted. First, TC analysis requires a strong instrument and large sample, and
in cases where these prerequisites are not met, we resort to sub-optimal
estimation and rescaling methods. Second, the issue of non-stationarity in
errors and scaling has not been addressed so far, and this can lead to biased
estimates of the correction parameters for rescaling and de-noising. Despite
this, DWT offers additional degree of freedom in translation parameter

MRA enables the (bulk) variance var(

Similarly, wavelet covariance cov(

Starting with the scale-level affine model of Eqs. (

We thank Wade Crow for valuable discussions and Clara Draper for her
critiques of the early drafts. We acknowledge gratefully the feedback of
Simon Zwieback, two anonymous reviewers, Wolfgang Wagner, and Editor Niko
Verhoest in the refinement of our manuscript. We also thank all who
contributed to the data sets used in this study. Kyeamba in situ data were
produced by colleagues at Monash University and the University of Melbourne
who have been involved in the OzNet programme. AMSR-E data were produced by
Richard de Jeu and colleagues at Vrije University Amsterdam and NASA. The
MERRA-Land data set was provided by NASA Goddard Earth Sciences Data and
Information Services Center (GES DISC). The land cover/use map was produced
by merging land cover