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Empirical Quantification of Topobathymetric Lidar System
Resolution Using Modulation Transfer Function
K. W. Sacca
1
and J. P. Thayer
1
1
Active Remote Sensing Laboratory (ARSENL), Ann and H.J. Smead Department of Aerospace Engineering Sciences,
University of Colorado Boulder, Boulder, CO, USA
Abstract Topobathymetric scanning lidar deployed on unmanned aircraft systems is a powerful tool for
high‐resolution mapping of the dynamic interface between topography and bathymetry. However, standardized
methods for empirical resolution validation have not been widely adopted across lidar applications. While
theoretical models of idealized lidar sampling resolution can be used to describe topographical resolution,
misrepresented or unknown behaviors in an instrument, platform, or environment can degrade expected
performance or introduce georeferencing inaccuracies. Furthermore, bathymetric resolution is strongly
dependent on water surface and column conditions. Thus, only empirical methods for evaluating resolution will
provide reliable estimates for both topographic and bathymetric surveys. Presented is an extension of standard
modulation transfer function (MTF) methods used by passive imaging systems applied to high‐resolution
scanning lidar. Compact retroreflectors characterized as point and line sources are employed to empirically
assess effective lidar system resolution through MTF analysis in topographic and bathymetric scenes. These
targets enable MTF analyses using height measurements without reliance on intensity data, promoting
widespread applicability among lidar systems. Empirical MTFs calculated using these targets are compared
against theory‐derived counterparts as empirical measurements elucidate influences by elements that are
unknown or difficult to model. Simulated point cloud data were incorporated into theoretical MTF descriptions
to better represent empirically‐derived topographic MTFs, revealing mirror pointing uncertainties in the across‐
track axis. Similarly, theoretical bathymetric MTFs augmented with simulated, subaqueous data enabled water
surface slope estimation using empirical measurements of submerged retroreflector targets, where rough water
surfaces strongly influenced beam steering and the corresponding point spread MTFs.
1. Introduction
Scanning topobathy lidar instrumentation operating from unmanned aircraft system (UAS) platforms are capable
of topographic and bathymetric mapping at very high resolutions (Mandlburger et al., 2020; Thayer et al., 2022).
Applications that demand high resolutions from topobathymetric lidar, such as object detection and classification,
biomass estimation, change detection, hydrographic surveying, etc., should employ strategies that routinely
validate resolution performance as instrumental or environmental factors can have a significant impact on
effective instrument system resolution (Donnellan et al., 2021; Goulden & Hopkinson, 2010; Hancock
et al., 2019; International Hydrographic Organization, 2022; Kellner et al., 2019; Mandlburger, 2022; National
Oceanic and Atmospheric Administration, 2020). Resolution is a performance metric without a standardized
definition across applications. The resolution of an instrument is defined here as the ability to discern two point
sources from measurements and is distinguished from sampling, which describes the density of measurements.
Point density and point spacing metrics are commonly used to quantify lidar resolution, however, these metrics do
not capture the inherent spatial spread of signal through an optical system that impacts measurement dis-
cernibility. This important distinction serves to convey that resolution is dependent on the sampling and signal
contrast of measurements.
Passive camera imaging in UAS‐, aircraft‐, and spacecraft‐based remote sensing applications regularly leverage
standardized methods for empirically evaluating resolution during operations (Fiete, 1999; Reichenbach
et al., 1991; Williams & Burns, 2001). Since the 1980s and persisting today, ground targets used in resolution
estimation have been designed for modulation transfer function (MTF) analysis methods, which relate signal
contrast to spatial resolution and quantify resolution performance across an instrument's complete range of
resolvable spatial frequencies (Eon et al., 2024; Markham, 1985; Storey, 2001). MTF methods are not limited by a
RESEARCH ARTICLE
10.1029/2024EA004098
Special Collection:
Surface Topography and
Vegetation: Science,
Measurements, and
Technologies
Key Points:
•Standard imaging modulation transfer
function (MTF) methods are applied to
topobathymetric lidar using compact
retroreflector targets to quantify
effective resolution
•Height‐based MTFs are leveraged for
lidar resolution analysis without in-
tensity data and are shown to have
reliable theoretical descriptions
•Empirical lidar MTFs quantified
unknown instrument pointing and
water surface steering errors that
improved theoretical estimates of MTF
Correspondence to:
K. W. Sacca,
Kevin.Sacca@colorado.edu
Citation:
Sacca, K. W., & Thayer, J. P. (2025).
Empirical quantification of
topobathymetric lidar system resolution
using modulation transfer function. Earth
and Space Science,12, e2024EA004098.
https://doi.org/10.1029/2024EA004098
Received 14 NOV 2024
Accepted 9 MAR 2025
Author Contributions:
Conceptualization: K. W. Sacca
Data curation: K. W. Sacca
Formal analysis: K. W. Sacca,
J. P. Thayer
Funding acquisition: K. W. Sacca,
J. P. Thayer
Investigation: K. W. Sacca, J. P. Thayer
Methodology: K. W. Sacca
Project administration: J. P. Thayer
Resources: K. W. Sacca, J. P. Thayer
Software: K. W. Sacca
Supervision: J. P. Thayer
Validation: K. W. Sacca, J. P. Thayer
© 2025. The Author(s).
This is an open access article under the
terms of the Creative Commons
Attribution‐NonCommercial‐NoDerivs
License, which permits use and
distribution in any medium, provided the
original work is properly cited, the use is
non‐commercial and no modifications or
adaptations are made.
SACCA AND THAYER 1 of 16
particular instrument, platform, resolution range, or environment which makes MTF a suitable figure of merit for
standardized resolution analysis.
Across the field of lidar, there is no unifying standard method for empirical resolution analysis. It is common to
use fundamental instrument models to estimate resolvability theoretically (Lichti, 2004; Ullrich & Pfennigba-
uer, 2016), or utilize vicarious calibration targets in large‐scale surveys, such as from aircraft or spacecraft
platforms (Collin et al., 2008; Kuester et al., 2010; McCarthy et al., 2022). Hancock et al. (2019) have demon-
strated high fidelity lidar measurement and noise models to validate empirical data, but the model is specific to
waveform lidar and does not consider MTF. This study aims to apply MTF to any lidar instrument. MTF has not
yet been widely practiced or adopted in the field of lidar, but previous work has proved MTF can be applied to
both intensity and height measurements of lidar allowing for multimodal resolution performance estimation
(Albota et al., 2017; Goesele et al., 2003; Miles et al., 2002,2010; Stevens et al., 2011). However, MTF has a
variety of implementations and specific methods for geoscience applications that have not been fully demon-
strated to satisfy the needs of all lidar systems.
Shallow water bathymetry is a unique application area for scanning lidar due to the higher depth‐penetration
capability of pulsed lasers compared to sunlight for passive imaging systems (Forfinski‐Sarkozi & Par-
rish, 2019; Mandlburger, 2022; Parrish et al., 2013; Schwarz et al., 2019; Thayer et al., 2022). Proposed missions
like NASA's Surface Topography and Vegetation (STV) (Donnellan et al., 2024) with targeted observables
demanding high resolutions, such as shallow‐water bathymetry, will require empirical means to validate effective
resolution to ensure scientific objectives are met. Shallow waters are areas of great interest for frequent mapping
but are also very challenging areas to acquire accurate georeferenced data due to dynamic water surface and
column effects (Parrish et al., 2019). Empirical measurements of MTF represent the entire imaging chain,
including environmental factors, making it uniquely suited to characterizing the effective resolution of an in-
strument in the presence of unknown influences (Boreman, 2001; Schott, 2007).
Presented here are empirical methods for MTF‐based resolution performance analysis of a UAS‐based topobathy
lidar instrument. Section 2describes the fundamental relationships between MTF and point spread function
(PSF), previous applications of MTF to lidar, and provides a description of the UAS‐based lidar instrument used
for empirical results. Section 3covers the theoretical descriptions of MTF using fundamental definitions, an
overview of the empirical methods employed here for lidar point cloud analysis, and a description of the
retroreflector targets designed for use in topobathymetric surveys to assess resolution and satisfy needs of groups
like NASA's STV community. Empirical application of standard MTF methods to lidar data are presented in
Section 4, alongside several theoretical models of MTF that support empirical results for topographic and
bathymetric environments. Section 5summarizes and highlights conclusions and impacts of MTF on top-
obathymetric lidar applications.
2. Background
Modulation, M, (also known as contrast) is the ratio of the bias level and the amplitude of sinusoidal or square
wave signals of a particular frequency, ξ, in object and image space and is expressed as (Boreman, 2001):
M(ξ) = Imax Imin
Imax +Imin
(1)
Modulation transfer is a metric that quantifies an instrument's ability to reproduce real spatial frequencies and
associated contrast observed in an imaged scene, defined as:
MTF(ξ) = Mimage
Mobject
(2)
where MTF(ξ)is the MTF at frequency ξand is defined as the ratio of recorded image modulation to the true
object modulation where MTF =1 signifies perfect image reproduction of a feature (Boreman, 2001;
Schott, 2007). However, due to realistic sampling limitations, imaging instruments impart a minimum spatial
spread to signals that bounds the range of reproducible frequencies. The “instrument system MTF” encompasses
all contributing sources of signal spread on a measurement, such as noise, platform, and environmental factors.
Visualization: K. W. Sacca
Writing – original draft: K. W. Sacca
Writing – review & editing: K. W. Sacca,
J. P. Thayer
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Boreman (2001) is one of the most thorough references containing theoretical derivations of MTF, imaging
relationships between MTF and PSF, line spread function (LSF), and edge spread function (ESF), and the
cascading nature of system MTF into individual component MTFs. Interested readers are encouraged to seek that
source for more detailed descriptions of MTF. The important aspect of MTF that has been leveraged extensively
in remote sensing image applications is that PSF and MTF are related via the Fourier transform. Therefore,
empirical PSF measurements can be used to evaluate MTF. LSF and ESF also have similar definitions that relate
to MTF that translate to physical measurements of line and edge sources.
Direct measurements of modulation, or contrast, using Equation 1with samples of square‐wave targets
geometrically arranged into specific frequencies produces a function similar to MTF known as the contrast
transfer function (CTF) (Boreman, 2001; Boreman & Yang, 1995). Contrast transfer function is discretely
sampled using pairs of these targets while MTF is a continuous function defined by a sinusoid response (Boreman
& Yang, 1995; Coltman, 1954). Previous applications of empirical MTF methods to an airborne, scanning,
topographic lidar mainly involved discretized CTF measurements and an approximation of MTF using funda-
mental relationships between square and sinusoid functions (Miles et al., 2010; Stevens et al., 2011). Point, line,
and edge spread measurements of compact targets produce direct and continuous estimates of MTF without the
need for large square wave targets, making it an ideal technique for straightforward integration into geospatial
lidar survey operations (Boreman & Yang, 1995; Reichenbach et al., 1991). It is this aspect of MTF analysis that
is applied in this study. For an ideal image, f(x,y), and an instrument response function (IRF), h(x,y), the in-
strument image function can be written as (Boreman, 2001):
g(x,y) = f(x,y)∗h(x,y) (3)
For point source targets, f(x,y)can be written as a delta function, δ(x,y), such that the instrument image function
convolved with the delta algebraically describes how to empirically measure the optical transfer function (OTF).
The IRF is the convolution of each component impulse response, or PSF, present in the system. Through the
Fourier transform, we can represent Equation 3in the frequency domain for the point source case with N
component transfer functions as (Boreman, 2001):
G(ξ,η) = δ(ξ,η) × H(ξ,η) = δ(ξ,η) × H1(ξ,η) × H2(ξ,η) × ⋯×HN(ξ,η) (4)
where H(ξ,η)is the overall OTF and G(ξ,η)represents the empirical measurement of the OTF using a point
source target. The OTF can contain any number of component transfer functions, corresponding to any optical
element, sampling architecture, environmental influences, etc. That describe the end‐to‐end imaging system. We
can define MTF as (Boreman, 2001):
MTF ≡ |H(ξ,η)| (5)
where the real component, or magnitude, of the OTF is defined as MTF, and this theoretical description of MTF is
leveraged to obtain empirical measurements of MTF in lidar.
Welsh and Gardner (1989) describe the practical relationships between PSF and OTF that determine resolution
limitations fundamental to MTF theory, but does not cover MTF or the effective PSF obtained through multiple
single‐detector measurements which are relevant to scanning lidar systems. Dolin (2013) presents a theoretical
framework for measuring the MTF of a water surface for bathymetric lidar applications but requires auxiliary
imaging instrumentation that is not relevant across the range of applications utilizing scanning lidar. A high
priority of this work is to provide a universally relevant approach to lidar system MTF that can be obtained
empirically.
Miles et al. (2010) demonstrated empirical, intensity‐based, PSF‐derived MTF with aerial topographic lidar.
However, among the two methods implemented, one produced PSF estimates that mismatched observations and
the other relied on the removal of background noise. This can introduce artificial features or contrast in the PSF
resulting in an MTF that is not fully representative of the system. Lichti (2004) demonstrated the comparison in
resolution performance between scanning topographic lidar instrumentation using MTF but only through theo-
retical models of fundamental sampling behaviors that do not capture noise or extrinsic contributions to effective
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resolution. While theoretical descriptions of MTF can address high‐level instrument trade spaces, robust per-
formance estimates demand empirical methods to validate salient instrument specifications, as will be shown in
Section 4. Goesele et al. (2003) demonstrated an example of empirical, height‐based MTF measurements using
knife edge targets inspired by imaging standards, but did not demonstrate the translation of their method to real‐
world lidar applications or relate empirical MTFs to component MTFs that can be derived theoretically.
The empirical methods presented here utilize point and line source targets to independently evaluate MTF using
standardized image methods adapted to lidar and are supported by theoretical descriptions of MTF. The incor-
poration of lidar height‐based MTF methods into geospatial applications could have far reaching and lasting
impact on how lidar data is collected, validated, and standardized.
The instrument used for empirical demonstrations is a prototype UAS‐based scanning topobathy lidar system
from Orion Space Solutions (2025) (formerly LiteWave Technologies) that produces scan patterns represented by
the illustration in Figure 1. This Geiger‐mode lidar operates with a 532 nm green laser pulsed at 20 kHz and a
scanning mechanism operating at 70 Hz that produces a linear cross‐track scan pattern spanning ±15° through
nadir. All data used to demonstrate these MTF analysis methods were received from Orion and are georeferenced,
GPS‐corrected, and boresight‐calibrated 3‐D point clouds. The point cloud data contains 3‐D XYZ coordinates
and measurement timestamps. The topographic MTF analyses are performed on data collected across several
surveys and different locations. The bathymetric MTF analyses are performed on data collected from a near‐shore
area of the Gulf of Mexico near Panama City, FL under WMO Code 1 and 2 wave conditions (World Meteo-
rological Organization, 2014). Nominally the instrument was operated from above‐ground altitudes of 15–25 m at
velocities of 2–3 m/s and the laser's full beam divergence was approximately 3 mrad. For the purpose of
demonstrating lidar MTF analysis, measurement timestamps were used to extract single, near‐nadir swaths to
reduce uncertainties that are described in Section 4.1.
3. Methods
3.1. Theoretical MTF Descriptions
MTF is evaluated using a source of signal contrast as a function of feature size or spatial frequency. For three‐
dimensional lidar point clouds collected from an aerial platform, a key source of measurement contrast for
discerning groups of points is height, Z. The height of points across a scene is analogous to intensity across a 2‐D
image for which boundaries between groups of measurements can be drawn and features can be distinguished.
Figure 1. Example sampling diagram for a scanning topobathy lidar instrument illustrating along‐track and across‐track
sample distances relative to the flight and scan axes. Here, the scan axis is orthogonal to the flight path and the color scale
indicates bathymetry depths.
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Not all lidar instruments record intensity along with the range measurement, but MTF can be evaluated using
relative height as the source of contrast across a spatial domain. If additional sources of contrast, such as
backscatter intensity, are available, MTF analysis can be applied to them to provide additional information for
evaluating a system's overall MTF. The methods presented here use relative point height as the contrasting
element in the MTF analysis.
Theoretical descriptions of MTF components can be estimated using properties of spatial sampling. The sampling
scheme of lidar can vary among instruments, but sampling properties can be translated between camera imaging
and lidar systems with comparably dense sampling patterns. The MTF methods outlined here rely on complete
measurements of the PSF, which requires dense and overlapping or contiguous sampling. For lidar systems that
produce sparse point clouds with no overlap between projected sample areas, the PSF will not be accurately
represented by measurements with very few point source samples. Void regions, or gaps between points, observed
in point clouds do not capture or illustrate the beam footprints that may have produced overlapping or contiguous
samples. Beam footprints must be considered when evaluating MTF as they have a direct impact on the PSF.
Therefore, it is important to highlight that these MTF methods are intended for point clouds with dense, over-
lapping or contiguous samples typical of “high resolution” lidar instruments.
Figure 1illustrates an example sampling pattern for the scanning Orion lidar system where the scan rate and laser
repetition rate determine the effective across‐track sampling, while the effective along‐track sampling is deter-
mined by the complete scan cycle rate and platform velocity. The fundamental components of MTF defined in
passive imaging that are applicable to scanning lidar schemes represented by Figure 1are the footprint and
sampling MTFs in the along‐ and across‐track axes (Boreman, 2001). In imaging, footprint MTF represents the
resolution capability determined by the ground‐projected size of pixels through the optical system's focal length.
The equivalent footprint MTF in lidar is the ground‐projected laser spot size determined by the angular beam
divergence of the transmitted beam and the range to the ground, also called the instantaneous field‐of‐view
(IFOV). Sampling MTF in imaging is representative of effective sample spacing determined by the ground‐
projected pixel pitch distance and in lidar it is defined by the distance between the centers of two adjacent
ground‐projected pulses. The equation for these MTFs as a function of spatial frequency are both represented by:
MTF(ξ) = |sinc(ξx)| =
sin(πξx)
πξx
(6)
where xis the 1/e2full beam width of the projected spot when evaluating footprint MTF or the ground sample
distance between projected spot centers when evaluating sampling MTF (Boreman, 2001). These sinc functions
represent 1‐D functions for the along‐ and across‐track axes that can be derived individually and combined to
describe the 2‐D sampling pattern.
These fundamental components of MTF are typically identical between the along‐ and across‐track axes in
camera imaging due to square pixels with pixel pitch ≈pixel width (Schott, 2007; Smith, 2008). However, the
sampling mechanism illustrated in Figure 1demonstrates that the along‐ and across‐track components of sam-
pling can have different extents depending on mirror rotation rate, platform velocity, etc. Therefore, theoretical
along‐track sampling and footprint MTFs should be defined separately from the across‐track counterparts using
knowledge of the lidar's effective sampling parameters. Figure 2demonstrates the difference between the axial
footprint and sampling MTF components derived theoretically for the along and across track axes. The overall
system MTF is defined by the combination of fundamental MTF components that impact the effective instrument
spatial resolution and is commonly referred to as the cascading property of MTF (Boreman, 2001; Schott, 2007).
An example of a theoretical system MTF consisting of only along‐ and across‐track sampling and footprint MTFs
is shown in Figure 2, where it can be deduced that the across‐track sampling is the limiting resolution component
for 3‐D imaging using this ideal instrument. An example noise equivalent modulation (NEM) threshold is drawn,
where the minimum detectable modulation corresponds to the intersection between NEM and MTF for quanti-
tative evaluations of limiting resolution for an individual component or a complete system. NEM is described
further in Section 3.4 for quantitative evaluations of noise relative to an MTF. For ideal instruments with no
external influences on sampling, the lidar system MTF can be reliably estimated using these theoretical
descriptions.
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Real lidar systems have a variety of measurement influences due to intrinsic noise sources, platform perturba-
tions, and external media for which their respective MTF components, not represented in Figure 2, impact res-
olution and can be difficult to derive theoretically. However, through robust instrument simulations, the unknown
or unquantified components of MTF that correspond to errors and uncertainties can be obtained and quantified.
The simulated bathymetric point cloud shown in Figure 3was created for the evaluation of the system MTF from
theoretical principles for the lidar instrument described in Section 2, a nominal operations parameter set, and a
realistic bathymetric environment. This model is built from raytracing methods where for each laser shot, several
rays are traced about a pointing vector through the environment to determine the spatial envelope of the beam as a
function of range. Samples are simulated as intersections between projected rays and a surface in the scene and are
registered to the rays' corresponding pointing vector. Retroreflective targets can be approximated as point re-
flectors where the simulation registers a hit on target any time the projected beam overlaps with a target's ge-
ometry. For bathymetric simulations, the water surface steers the rays as a function of slope and as a result, the
subaqueous rays are distorted, producing irregular footprints and sample‐to‐sample spacing. The time‐variant
Tessendorf water surface model described in Tessendorf (2001) is implemented to produce a dynamic water
surface that can be classified to WMO standards quantitatively by wave heights and the Douglas Sea Scale or
Beaufort Scale (World Meteorological Organization, 2014). Without intrinsic or extrinsic sources of uncertainty,
the point cloud generated exactly matches the expected sampling map for which the footprint, sampling, and
effective system MTFs are accurately represented by evaluations of Equation 6. As sources of uncertainty are
introduced, any impacts on effective resolution can be quantified through MTF analysis of this synthetic data
instead of basic theoretical functions.
Physical targets that produce empirical measurements of MTF through PSF and LSF analyses elucidate the total
impact of MTF components without the need for complete theoretical descriptions. For active remote sensing
instrumentation, point reflector targets can be used to empirically measure the OTF, or IRF, analogous to point
sources in passive remote sensing applications. Corner cube retroreflectors (CCRs) have historically been used in
laser applications requiring cooperative point source targets like satellite and aircraft tracking, analyzing atmo-
spheric distortion, and georeferencing validation of lidar measurements (Lucy et al., 1966; Lutomirski & War-
ren, 1975; L. Magruder et al., 2005; L. A. Magruder et al., 2020). Under geometric conditions where a CCR target
is sufficiently smaller than an instrument's IFOV, the three‐dimensional spatial spread among lidar samples of a
CCR represents the system PSF that can be used to derive MTF (Stevens et al., 2011). Similarly, linear targets that
Figure 2. Theoretically‐derived sampling and footprint MTFs with the estimated system modulation transfer function (MTF)
for a scanning lidar instrument on an unmanned aircraft systems platform with elliptical footprints due to off‐nadir scan
angles. A representative noise equivalent modulation (NEM) threshold is drawn illustrating the effective cutoff frequency of
the system.
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satisfy point source conditions, that is, narrow widths and demonstrate retroreflective properties that maximize
detectability, can be used to empirically measure LSF. For scanning lidar systems with sampling like Figure 1,
LSF‐derived MTF is a more informative metric because the measured system MTF can be decomposed to
individually assess along‐ and across‐track resolution performance. The axial components of MTF empirically
measured using LSF can be directly compared to theoretical MTFs derived using Equation 6, while PSF‐derived
MTFs incorporate both along‐ and across‐track sampling effects and is therefore comparable to system MTF.
3.2. Point‐ and Line‐Reflector Target Descriptions
The point‐ and line‐reflector targets shown in Figure 4are used to measure PSFs and axial LSFs respectively for a
UAS‐based scanning topobathy lidar where the contrasting signal used for MTF analysis is height. These targets
Figure 3. 3‐D bathymetric lidar point cloud generated from an instrument sampling model, representative of the Orion Space
Solutions system, and time‐variant Tessendorf water surface model covering an area of approximately 100 m
2. The average
wave height of the water surface simulated here is approx. 8 cm and classified as WMO Sea State 1 by Beaufort scale number or
Douglas Sea Scale. Several 10 mm ×2 m line source targets and 50 mm diameter point source targets with simulated reflective
properties are placed in the scene 30 cm above the bathymetric surface at 3 m depth for use in robust, theoretical assessment of
MTF. Inset image is keyed for reorientation and shows a top‐down view of the outlined region of the point cloud (a 3 ×3 m
area), illustrating the dense sampling pattern produced by the simulated instrument across several linear targets.
Figure 4. Image of retroreflector‐woven rope line source targets and corner cube retroreflector (CCR) point source targets
positioned above a level ground surface (left) and the corresponding 3‐D lidar point cloud containing the targets shaded by
height (right). Image cutouts depict close‐up photographs of the retroreflector targets.
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were selected with topographic and bathymetric lidar applications in mind due to their extremely high backscatter
efficiencies, compact form factor, and capability to estimate MTF above and below a water surface. One
important aspect to consider when selecting a bathymetric CCR is the method of retroreflection. Many
commercially available CCRs rely on total internal reflection (TIR) as opposed to specular reflection off mirrored
facets to produce retroreflection. However, it can be proven that the critical angle required for TIR‐based
retroreflection is not always achieved for a glass‐water interface depending on the dimensions or the refractive
index of the CCR (Hecht, 2017). The 50 mm diameter CCR targets used here and shown in Figure 4are silver‐
plated to produce retroreflection when underwater and the outside substrates have been epoxy‐coated to protect
the mirrored surface from degradation.
The lidar system used for demonstrating MTF produces dense sampling maps with overlapping spots in the along‐
track axis. From 15 m altitude, the estimated nadir footprint diameter is 65 mm, with 24 and 90 mm along‐ and
across‐track sample distances respectively. From 25 m altitude, expected nadir footprint diameters are 110 mm,
with 24 and 150 mm along‐ and across‐track sample distances respectively. These estimates indicate strong
overlap in the along‐track but not in the across‐track. The 50 mm CCRs were sized to ensure target “hits” at nadir
in the across‐track axis for these sample distance and footprint estimates where smaller CCRs could result in
“misses”. Smaller CCRs satisfy geometric point source conditions more easily, but larger CCRs also increase
detectability in bathymetric surveys. Therefore, the largest CCRs that adequately approximate point source
conditions for this demonstration instrument were selected. Additionally, any PSF errors originating from poor
point source approximations can be quantified and compensated using convolution analysis for two finite‐area
sources (Gaskill, 1978).
For LSF analysis, the 8 mm diameter, 2 m long retroreflective rope shown in Figure 4is deployed horizontally
level and suspended above the ground using stakes or screw anchors. The 8 mm diameter easily satisfies geo-
metric point source conditions, but in this case, occasional “misses” do not impact the LSF result. Due to the
extended length of the LSF target, a sufficient number of samples are produced (see Figures 5and 6) such that any
impacts to the LSF due to “misses” are effectively negated. The reflective filament woven throughout the rope
consists of retroreflective microspheres that function underwater. LSF‐based MTF analysis offers added utility
over PSF analysis for discerning along‐track and across‐track system effects. These retroreflective targets
maximize the likelihood of detection within the effective IFOV (see Figure 1) as point and line sources, making
them ideal targets for empirical MTF analysis of lidar instruments. The Geiger‐mode lidar instrument used is
resilient against saturation effects that can make retroreflective materials challenging to use with other types of
lidar instruments. Other instruments can still leverage retroreflectors but relevant detector sensitivities that would
determine maximum target size or minimum separation distance should be considered.
3.3. Empirical PSF‐ and LSF‐Derived MTFs
Figure 5a illustrates a topographic region of interest (ROI) around a line‐reflector target suspended 30 cm above a
flat surface (shown in Figure 4) to isolate the two signal sources used to estimate MTF from the LSF. In this
example, the instrument's across‐track axis is defined as X, the along‐track axis as Y, and the vertical axis as Z.
The accompanying Zhistogram in Figure 5b shows a clear discernibility between points corresponding to the line
target and those associated with the ground. A line is fit to the points corresponding only to the linear target, then
3‐D affine transformations are applied to the ROI point cloud to align the line fit orthogonal to the X‐Zplane to
minimize the line spread, as shown in Figure 6a. It is important to configure the line target to be parallel to the flat
background surface to minimize the X‐Zspread of the background. With the Y‐axis effectively removed, this 2‐D
point cloud is then interpolated along the X‐axis with fine sampling (example shown in Figure 6a uses 1 mm
sampling) to produce a supersampled LSF, analogous to the LSFs used for MTF analysis in imaging using the
slant‐edge MTF method (Burns, 2000). Like the slant‐edge MTF method, a low‐pass Tukey filter is applied to the
normalized LSF to effectively smooth out MTFs from discontinuities caused by high frequency noise (Burns
et al., 2022). In the example shown in Figure 6b,α=0.8 is used to preserve the existing LSF features whereas an
equivalent Hann filter (α=1.0) would narrow the LSF and artificially boost the resulting MTF in a way that is not
representative of the lidar system. MTF is obtained by taking the Fourier transform of the Tukey‐filtered,
normalized LSF with many examples shown in Section 4.
The process to obtain a PSF‐derived MTF using a CCR target is nearly identical to the LSF‐based method and the
primary difference is how the 3‐D point cloud is transformed into the X‐Zplane. Instead of a line fit, a centroid is
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fit to the 3‐D distribution of points corresponding to the CCR target and is set as the origin of the X‐Yplane. Points
within the resulting Cartesian quadrants I and III are rotated about the centroid Z‐axis clockwise, maintaining their
radial centroid distance, into the X‐Zplane, while points within quadrants II and IV are rotated counter‐clockwise
into the X‐Zplane. Due to this rotation of the along‐track spread into the across‐track axis, the PSF‐derived MTF
is closer in theory to the system MTF line illustrated in Figure 2. The resulting 2‐D point cloud resembles
Figure 6a and subsequent processing is identical to the LSF MTF method to obtain a PSF‐derived MTF.
3.4. Quantitative Evaluation of MTF
Evaluating MTF quantitatively is helpful for deriving metrics that relate MTF to physical dimensions or for
comparing resolution performance between different instruments using metrics other than limiting resolution.
The limiting resolution of the instrument is commonly determined using a constant modulation transfer threshold
(often 5%, 10%, or 20%) in imaging and in previous applications of CTF to lidar (Holst, 2008; Lichti, 2004;
Figure 6. X‐Zview of 3‐D lidar point cloud shown in Figure 5a with a line spread function (LSF) fit (a) and the corresponding
normalized, Tukey‐filtered LSF used to compute MTF (b). This Tukey filter was configured using α=0.8 to prevent
artificial narrowing of the LSF. An offset in Z was applied to the point cloud from Figure 5to produce a zero‐mean background.
Figure 5. (a) 3‐D lidar point cloud of line target sampled from a UAS platform. (b) The corresponding histogram as a function
of height, demonstrating clear contrast between target and background. Measurements in green correspond to the target while
black points correspond to the background beneath the target. The line target in this example is oriented parallel to the path of
motion, therefore the spread of points in the point cloud is indicative of the across‐track resolution performance of the lidar
instrument system.
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Stevens et al., 2011). However, constant thresholds like these are subjectively determined and do not sufficiently
describe noise or meaningfully compare instruments. A quantitative method for determining the limiting reso-
lution is deriving an instrument system's NEM threshold, also called minimum detectable modulation (Bore-
man, 2001). NEM captures the cumulative impact from noise and systematic effects that impact effective contrast,
which for height‐based MTF analysis represents the minimum vertical relief required to reliably discern an object
from its surroundings. Additionally, due to the relationship between lidar's range‐time measurement mode and the
vertical coordinate of measurements used as the source of contrast for MTF, the IRF is a component of NEM.
NEM can be illustrated using histograms like Figure 5b or PSF/LSF plots like Figure 6a where the spread in the
Z‐axis around objects that act as Dirac delta functions in range‐time, such as a suspended line target or a level
surface, demonstrate an effective signal‐to‐noise ratio (SNR) related to Mand NEM.
SNR =M
NEM (7)
where for normalized LSFs, M≈1 such that:
NEM =1
SNR =N
S(8)
where Nis the noise, or the random spread in Zvalues, defined here as the full width at half maximum of the
background histogram in Z, and Sis the mean signal, defined here as the mean difference in Zbetween the target
and background. The limiting resolution is equal to one half‐cycle of the cutoff frequency found at the intersection
of MTF and NEM, or:
d=λ
2=1
2f(9)
where dis the limiting target width and fis the cutoff frequency (Schott, 2007).
4. Results
4.1. Topographic MTF
The along‐track resolution of the UAS platform topobathy lidar instrument system described in Section 2was
evaluated through the sampling of a line source target oriented perpendicular to the flight axis (opposite to
Figure 5) to produce the empirical MTF shown in Figure 7representing along‐track resolution performance. The
theoretical MTF was estimated using the product of fundamental footprint and sampling MTFs obtained using
Equation 6for the along‐track axis components only. In this result, the strong agreement between empirical and
theoretical MTFs demonstrates that these fundamental components of MTF provide a close estimate of true
along‐track resolution performance for topographic targets. Evaluating MTF using the empirically‐derived NEM
threshold suggests a limiting spatial frequency of approximately 8.75 cycles per meter, or a limiting resolution of
5.71 cm using Equation 9to invert spatial frequency into an equivalent target diameter.
The across‐track resolution of the system was evaluated using a line source target oriented parallel to the flight
axis, represented by Figure 5, producing the empirical MTF shown in Figure 8. However, in this case the same
theoretical description of MTF used to evaluate along‐track performance does not agree well with the empirical
measurements in the across‐track axis. This is an indication that there are additional components of MTF not
captured by the theoretical estimate using only footprint and sampling MTFs. Because the lidar's scan axis is
aligned in the across‐track axis of flight, scan angle pointing uncertainties have a greater impact in the across‐
track axis than the along‐track axis in terms of effective point spread or line spread. Therefore, the lidar mea-
surement simulation tool described in Section 3.1 was used to reproduce expected sampling patterns for which
pointing uncertainties could also be introduced. The simulation used a retroreflective line source target and
identical flight and sampling parameters to generate a point cloud for which an augmented theoretical MTF could
be evaluated and compared to the empirical MTF. A random error was applied to the pointing vector prior to
raytracing each laser shot to simulate pointing uncertainty and its impact on MTF. Monte Carlo simulations
(N=500) that varied the mean pointing error were generated, from which a pointing error of approximately ±0.2°
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was found to minimize the difference between MTFs obtained from simulated data and the empirical MTF shown
in Figure 8. Additional sources of instrument or platform error may be contributing to the observed ±0.2 degrees
of error, but the axial dependence of this error leads to the conclusion that pointing error is the dominant source of
error. Consultation with the manufacturer Orion Space Solutions confirmed this level of pointing uncertainty for
this instrument.
Figure 7. Empirical and theoretical LSF‐derived MTFs evaluated in the along‐track axis for a target oriented perpendicular to
the flight path (opposite to Figure 5). The intersection of the MTF curve and the NEM threshold corresponds to the minimum
detectable modulation of the system to determine the limiting resolution.
Figure 8. Empirical and theoretical LSF‐derived MTFs evaluated in the across‐track axis for a target oriented parallel to the
flight path. The empirical MTF plotted corresponds to the point cloud and LSFs shown in Figures 5and 6. Simulated point
cloud data was used to enable models of instrument pointing error as an additional component of MTF. The augmented
theoretical model with an instrument pointing error of ±0.2° in the across‐track axis produces an MTF estimate that strongly
aligns with empirical MTF results.
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Evaluating the across‐track MTF using NEM suggests a limiting spatial frequency of approximately 3.80 cycles
per meter, or an equivalent resolution limit of 13.2 cm for the across‐track axis of the instrument. Comparing
across‐track to along‐track performance, these empirical and theoretical MTF results demonstrate the poorest
resolution in the instrument system is the effective across‐track sampling. This is due to the nominal ground
sample distances produced by the scanning configuration and the pointing uncertainties in the scan axis, which is
likely caused by measurement errors in mirror angle position or platform orientation determined by the in-
strument's inertial measurement unit.
Figure 9compares empirical and theoretical MTFs between perpendicular single‐swath and multi‐swath point
clouds containing a line target. The single‐swath, along‐ and across‐track MTFs bound the spectrum of single‐
swath MTFs for line targets with arbitrary orientation angles relative to the flight axis. However, for multi‐
swath point clouds, due to the cascading nature of MTF, it is expected that mixtures of along‐track, across‐
track, and other effects stack on the overall result, producing an effectively lower system resolution (Bore-
man, 2001; Schott, 2007; Smith, 2008). This predictable behavior is observed in Figure 9, for both the empirical
and simulated theoretical MTFs for multi‐pass mixtures of different flight angles with respect to the target axis.
The blending of along‐ and across‐track sampling effects into a single MTF result is also a characteristic of PSF‐
derived MTFs for which each axial component contributes to the overall spread of a point source target. This is
also similar to a system MTF which is the product of each component of MTF present in the end‐to‐end system
description.
Aggregating multiple swaths' point clouds introduces additional components of MTF that impact resolution
performance and can often be neglected in short, single‐swath analyses. Single swaths using this demonstration
instrument produced sufficient target samples for these LSF MTF results due to the extended length of the target.
However, for PSF MTF analyses where single swaths do not produce sufficient target samples, aggregating
multiple swaths is necessary to completely describe the PSF. Sources of uncertainty that impact point geore-
ferencing processes, such as GPS drift errors that vary over time or boresighting errors that vary with flight
heading, are potential sources of additional point or line spread that effectively decrease the overall system MTF
(Stevens et al., 2011; Thayer et al., 2022). Depending on the lidar system, sources of uncertainty such as these
could introduce meaningful or negligible components of MTF as part of the complete instrument system MTF
description. In the example shown in Figure 9, the augmented theoretical description of MTF for a multi‐swath
point cloud aligns strongly with the empirical MTF without modeling sources of error like GPS drift or boresight
Figure 9. Empirical and theoretical MTFs for point clouds collected using single swaths over a line target and a point cloud
collected using multiple swaths comprised of a mixture of flight path orientations with respect to the axis of the line target.
NEM thresholds are derived empirically from the point cloud data used for MTF analysis to evaluate their respective MTFs.
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errors in the theoretical description, implying these error components of MTF are of low impact. However, for
systems with extremely high‐resolution capability these sources of error may have measurable impact on the
resolution limit and could be interrogated through MTF methods.
4.2. Bathymetric MTF
Figure 10 demonstrates that MTF can also be used to quantify unknown environmental impacts on effective
resolution, such as the dynamic water surface and column effects for bathymetric lidar applications. In the
example shown by Figure 10a, PSF‐derived MTF was calculated using multiple aggregated swaths of top-
obathymetric point clouds containing measurements of one CCR target deployed on a beach and another CCR
target deployed to a depth of 3 m offshore. Multiple swaths were needed to produce sufficient samples of the PSF
for MTF evaluation. The theoretical description of MTF aligns strongly with the empirical MTF corresponding to
the topographic CCR, as found in Figures 8and 9.
Figure 10b illustrates that the same theoretical MTF description used for the topographic CCR, when applied to
the bathymetric CCR, results in a misalignment between theoretical and empirical MTFs. This suggests a similar
situation as the across‐track LSF topographic MTF example in Figure 8where the known components of MTF do
not adequately describe the empirical results, and in this case is due to the intervening water medium. The water
surface introduces additional steering errors and pulse stretching caused by wave slopes resulting in optical path
length and direction uncertainties, while the water column introduces beam spread through forward scattering
processes, resulting in larger projected spot sizes (Measures, 1984; Mobley, 1995). These errors introduce three‐
dimensional point spread that impact MTF. The increased spread in the X‐Yplane increases the PSF width,
effectively decreasing the MTF width, while the increased spread in the Z‐axis corresponds to higher levels of
NEM, shown in Figure 10b, resulting in reduced resolution limits as effective contrast in Zis reduced.
In addition to the augmented theoretical MTF description that includes instrument pointing errors, simulated point
cloud data was enhanced further using the Tessendorf water surface model (Tessendorf, 2001). This model
provided surface waves whose variation in slopes produced steering errors in both the along‐ and across‐track
axes in the simulated point cloud. Minimizing the difference between empirical and theoretical MTFs gener-
ated using Monte Carlo simulations (N=100), a steering error of ±6.0° was determined to best‐represent the
overall effects by the water on MTF. As previously mentioned, water surface steering errors are not the only
component of MTF introduced by the water. However, these results demonstrate that further augmentations to the
theoretical description of MTF using simulated lidar point cloud data can encapsulate the overall extent of effects
caused by external sources of unknown severity. Applying known physical processes to those sources, the
resulting theoretical MTF agrees well with the empirical MTF. Therefore, empirical MTF analyses can also be
Figure 10. (a) Empirical and theoretical topographic MTFs for a CCR target set up on the beach where the theoretical MTF
model is the same as Figures 8and 9. (b) Empirical and theoretical bathymetric MTFs for a CCR target submerged to 3 m
depth offshore. Image inserts show the CCR targets in their respective deployments. The theoretical bathy MTF was
augmented with simulated point cloud data to estimate steering error of subaqueous optical paths through rough water
surfaces. An augmented theoretical bathy MTF using ±6.0 degrees of water surface steering errors in both the across‐ and
along‐track axes aligns with empirical MTF results.
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performed to reliably capture and quantify the impacts on resolution from unknown environmental influences as
well as known instrument system properties.
5. Conclusions
Theoretical and empirical MTFs have been shown to strongly agree using MTF methods commonly applied in
passive camera imaging applications that have been adapted for lidar measurement modes. Through demon-
strations using height as the contrasting signal across a three‐dimensional point cloud scene generated by a UAS‐
based topobathy lidar, MTF analysis has quantified instrumental and environmental sampling behaviors and,
more broadly, demonstrates that MTF analysis can serve as a robust means for evaluating resolution performance
for any lidar mapping system. While PSF‐derived MTFs provided an overall evaluation of resolution combining
along‐ and across‐track instrument effects, LSF‐derived MTFs enabled the axial decomposition of MTF to
separately describe the along‐ and across‐track characteristics for the scanning lidar system employed in this
study. Components of MTF previously missing from theoretical descriptions were identified and incorporated
into model simulations resulting in augmented theoretical MTFs that strongly aligned with empirical MTFs
determined from compact calibration targets. Unknown intrinsic pointing errors in the across‐track axis were
quantified using robust point cloud simulation tools to produce synthetic MTFs that match empirically‐derived
MTFs, shown in Figures 8and 9. A similar approach was taken to quantify the extrinsic beam steering un-
certainties by a rough water surface that influenced resolvability in bathymetric scenes, shown in Figure 10.
MTF's ability to quantify intrinsic and extrinsic influences on resolution strongly supports the adoption of MTF as
a standard in hydrographic surveying and other high resolution bathymetry applications that do not have stan-
dardized empirical methods for quantifying resolution.
Previous applications of MTF in lidar mainly involved using very large tribar targets that indirectly captured MTF
through empirically derived measures of CTF (Miles et al., 2010; Stevens et al., 2011). Direct measurements of
MTF through PSF and LSF methods utilizing edge, line, and point source targets have extensive heritage in
passive camera imaging and has been successfully applied here to a scanning topobathy lidar. Additionally, these
compact targets can be easily acquired, transported, and deployed for standardized use in high resolution lidar
surveys to add resolution validation, ground control points, and georeferencing validation capabilities to post‐
processing pipelines (L. A. Magruder et al., 2020). While ESF‐based lidar MTFs have been demonstrated by
Goesele et al. (2003), dynamic bathymetric environments inhibit the feasibility of edge source targets compared
to compact point and line sources. Retroreflective sources maximize the detectability of MTF targets making
them ideal for robust estimation of effective resolution in bathymetric lidar applications where unknown envi-
ronmental factors can cause large departures from theoretical representations of resolution capability. These
targets enabled the empirical estimation of water surface slopes in shallow waters, adding context to measure-
ments with otherwise unknown environmental information. Additionally, deploying a series of bathymetric MTF
targets at known depths enables direct measures of effective spatial resolution as a function of depth, which can
greatly inform bathymetric applications such as underwater target detection and classification (Sacca &
Thayer, 2023).
MTF represents a unifying metric for resolution validation of camera imaging and lidar instruments alike,
enabling direct performance comparisons and providing empirical references of resolution for data fusion ap-
plications. Large‐scale surveys utilizing a suite of instrumentation, such as NASA's STV program, can leverage
MTF as a common metric to calibrate lidar and camera measurements collected at different times and across
various platforms or environmental conditions. Standardizing the utility of MTF to lidar will improve the quality
and potential of point cloud analysis and data science with empirical evidence of effective resolution to support
theoretical resolution estimates. These MTF methods can also be extended to include lidar instrument systems
that record radiometric intensity of backscattered pulses, combining the use of height and intensity as contrasting
signal sources to perform multi‐modal MTF analysis.
Data Availability Statement
Lidar point cloud data supplied by Orion Space Solutions (2025) were used in the production of this manuscript.
The subsets of empirical topographic and bathymetric point cloud data used in the presented MTF analyses and
the processing software used to generate the presented figures are available under the MIT license from GitHub at
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https://github.com/UCBoulder/3DMTF (Sacca, 2024). Matplotlib version 3.7.3 (Caswell et al., 2023; Hunt-
er, 2007) was used to create figures and is available under the Matplotlib license at https://matplotlib.org.
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Acknowledgments
This research was partially supported by
NASA FINESST Grant Number
80NSSC21K1597. UAS‐based lidar data
was collected in collaboration with Orion
Space Solutions, Louisville, CO 80027
USA. This work was also supported in part
by the U.S. Department of Defense,
through the Strategic Environmental
Research and Development Program
(SERDP‐ESTCP, 2024) Contract Number
W912HQ22C0042. Views, opinions, and/
or findings contained in this report are
those of the authors and should not be
construed as an official Department of
Defense position or decision unless so
designated by other official
documentation.
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Earth and Space Science
10.1029/2024EA004098
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